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#### Page No 196:

#### Question 1:

Suppose you have three resistors, each of value 30 Ω. List all the different resistances you can obtain using them.

#### Answer:

(a) When the three resistors are connected in series:

The resultant resistance, *R _{eq} = R + R + R *= 90 $\Omega $

(b) When the three resistors are connected in parallel:

The resultant resistance of the combination,

$\frac{1}{{R}_{eq}}=\frac{1}{R}+\frac{1}{R}+\frac{1}{R}=\frac{3}{R}=\frac{3}{30}=\frac{1}{10}\phantom{\rule{0ex}{0ex}}$

$\Rightarrow {R}_{eq}=10\Omega $

(c) When two of the resistors are connected in parallel and this combination is connected in series with the third resistor:

Let

*R'*be the resultant resistance of the two resistors connected in parallel to each other. Therefore,

$\frac{1}{R\text{'}}=\frac{1}{R}+\frac{1}{R}=\frac{2}{R}=\frac{2}{30}=\frac{1}{15}$

$\Rightarrow R\text{'}=15\Omega $

Now, the net resistance of the combination of the resistors,

*R*=

_{eq}*R' + R =*15 + 30 = 45 $\Omega $

(d) When two of the resistors are connected in series and the combination is connected to the third resistor in parallel:

Let

*R'*be the resultant resistance of the series in combination. Therefore,

*R' = R + R =*30 + 30 = 60 $\Omega $

Now, let the net resultant of the combination be

*R*. So,

_{eq}$\frac{1}{{R}_{eq}}=\frac{1}{R\text{'}}+\frac{1}{R}=\frac{1}{60}+\frac{1}{30}=\frac{3}{60}=\frac{1}{20}$

$\Rightarrow {R}_{eq}=20\Omega $

#### Page No 196:

#### Question 2:

A proton beam is moving from east to west. Is there an electric current? If yes, in what direction?

#### Answer:

Yes, there is an electric current in the direction of the proton beam. Conventionally, the direction of electric current is same as that of the positive charge flow.

#### Page No 196:

#### Question 3:

In an electrolyte, the positive ions move from left to right and the negative ions from right to left. Is there a net current? If yes, in what direction?

#### Answer:

Yes, there is a net current that flows from left to right. By convention, the direction of current is along the flow of positive charge. As the positive ions move from left to right, current also moves from left to right. Also, since the negative ions move from right to left, current moves from left to right. Thus, the movement of net current is from left to right.

#### Page No 196:

#### Question 4:

In a TV tube, electrons are accelerated from the rear to the front. What is the direction of the current?

#### Answer:

Conventionally, the direction of current is along the direction of flow of positive charge and opposite the direction of flow of negative charge. In a TV tube, current will flow from the front to the rear, as the direction of current is opposite the direction in which electrons are moving.

#### Page No 196:

#### Question 5:

Drift speed is defined as *v _{d}* = Δ

*l*/Δ

*t,*where Δ

*l*

*is the distance drifted in a long time Δ*

*t*. Why don't we define drift speed as the limit of Δ

*l*/Δ

*t*as Δ

*t*→ 0?

#### Answer:

An electron drifts under the influence of an external electric field. During the course of this motion, an electron follows a very random path. So, we have to take the average for a very long time. Hence, we cannot define drift speed as the limit of Δ*l*/Δ*t** *as Δ*t *→ 0 because this is a very short interval of time that is not enough to get the desired result.

#### Page No 196:

#### Question 6:

One of your friends says that he has read in the previous chapters that there can be no electric field inside a conductor. Consequently, there can be no current through it. What is the fallacy of this argument?

#### Answer:

You have studied in the previous chapters that when a static charge is given to a metal, the charge resides on its surface and there is no electric field inside it. But this is not the case when a charge is in motion. Current can flow through a conductor only when an electric field is established inside it. An electric field exists inside the conductor, as it is connected to the battery. Thus, current flows through the conductor.

#### Page No 196:

#### Question 7:

When a current is established in a wire, the free electrons drift in the direction opposite to the current. Does the number of free electrons in the wire continuously decrease?

#### Answer:

No, the number of electrons in the wire remains constant. The electrons that drift and move to the positive terminal of the battery, under the influence of the external electric field, are replaced by the battery in the circuit. As a result, the total number of free electrons in the wire is always constant.

#### Page No 196:

#### Question 8:

A fan with copper winding in its motor consumes less power compared to a similar fan with aluminium winding. Explain.

#### Answer:

A copper wire has higher conductance than an aluminium wire. So, a copper wire offer less resistance to the current flow than an aluminium wire. Thus, there is more heat dissipation in an aluminium wire than in a copper wire. This is why a fan with copper winding in its motor consumes less power compared to a similar fan with aluminium winding.

#### Page No 196:

#### Question 9:

The thermal energy developed in a current-carrying resistor is given by *U* = *i*^{2} *Rt* and also by *U* = *Vit*. Should we say that *U *is proportional to *i*^{2} or *i*?

#### Answer:

In the expression *U = Vit,* voltage *V* and current *i* are variables for a given time interval. So, based on this expression we cannot say that *U* is proportional to *i*. In the expression *U* = *i*^{2}*Rt**,* the resistance *R* is fixed for a circuit for a given time interval. So, based on this expression, we can say that *U* is proportional to *i*^{2} and not* i*.

#### Page No 196:

#### Question 10:

Consider a circuit containing an ideal battery connected to a resistor. Do "work done by the battery" and "the thermal energy developed" represent two names of the same physical quantity?

#### Answer:

Yes, the "work done by the battery" and "the thermal energy developed" represent two names of the same physical quantity. The work done by the battery on the resistor is dissipated by the resistor in the form of thermal energy. Hence the " work done by the battery " and " the thermal energy developed " represent two names of the same physical quantity.

However, a non-ideal battery, in this case, would have to do extra work in order to overcome the internal resistance of the battery. Hence, the work done by a non-ideal battery will not be equal to the thermal energy developed.

#### Page No 196:

#### Question 11:

Is work done by a battery always equal to the thermal energy developed in electrical circuit? What happens if a capacitor is connected in the circuit?

#### Answer:

No, the work done by a battery is not always equal to the thermal energy developed in the electrical circuit. In case of a non-ideal battery, the work done by the battery is the sum of the thermal energy developed in the electric circuit and the thermal energy developed in the internal resistance of the battery. In case of a capacitor, the work done by the battery is equal to *C V ^{2}*. An amount of energy equal to $\frac{1}{2}$

*C V*is stored in it when it is fully charged, which is a form of electrical energy and not a form of thermal energy. During the charging of the capacitor, $\frac{1}{2}$

^{2}*C*

*V*

^{2}

*of energy is lost in the form of heat and electromagnetic radiation.*

^{ }#### Page No 196:

#### Question 12:

A non-ideal battery is connected to a resistor. Is work done by the battery equal to the thermal energy developed in the resistor? Will your answer change if the battery is ideal?

#### Answer:

No, the work done by a non-ideal battery is not equal to the thermal energy developed in the resistor, as energy is spent to overcome the internal resistance of the battery and the resistance of the wire that connects the circuit elements/resistor to the battery. However, the resistance of the wire is generally negligible.

Yes, the answer will change if the battery is ideal. An ideal battery has no internal resistance. Hence, the work done by an ideal battery will be equal to the thermal energy developed in the resistor, assuming that the resistance of the wires used for connection is negligible.

#### Page No 196:

#### Question 13:

Sometimes it is said that "heat is developed" in a resistance when there is an electric current in it. Recall that heat is defined as the energy being transferred due to temperature difference. Is the statement in quotes technically correct?

#### Answer:

Yes, the given statement is correct. When charge flows through a conductor, its electric potential energy decreases. This loss in electric potential energy appears as increased heat energy of the resistor. Thus, heat energy is developed in a resistor when there is an electric current in it.

#### Page No 196:

#### Question 14:

We often say, "A current is going through the wire." What goes through the wire, the charge or the current?

#### Answer:

When there is a transfer of charge through a wire, current is said to be flowing through it. It is the electron/charge that drifts through the wire. The assertions "charge is going" and "current is going through the wire" are correct, as they signify the same thing, that is, flow of charge.

#### Page No 196:

#### Question 15:

Would you prefer a voltmeter or a potentiometer to measure the emf of a battery?

#### Answer:

A potentiometer is preferred to measure the emf of a battery, as it gives a more accurate result. This is because a potentiometer uses the null method to measure emf and it hardly draws any current from the primary circuit.

When a voltmeter is used in the circuit, its equivalent resistance is connected parallel to some element of the circuit. This changes the overall current in the circuit and, hence, the potential difference to be measured also changes. The error can be minimised if the equivalent resistance of the voltmeter is increased. However, we also need to keep in mind the heat dissipated due to high resistance while deciding the value of resistance of the voltmeter.

Hence a potentiometer is preferred.

#### Page No 196:

#### Question 16:

Does a conductor become charged when a current is passed through it?

#### Answer:

No, a conductor does not become charged when a current is passed through it. The free electrons present in the valence shell in a circuit drift from a lower potential to a higher potential and, thus, current is produced. A battery does not provide any extra electrons or charge to the circuit. It just provides a potential difference across two points, which helps in creating an electric field. This further helps in moving the electrons along the conductor.

#### Page No 196:

#### Question 17:

Can the potential difference across a battery be greater than its emf?

#### Answer:

The potential difference across a battery cannot be greater than its emf. Basically, emf is the maximum potential difference between the terminals of a battery when the terminals are not connected externally to an electric circuit. When the same battery is connected to an electric circuit, current flows in the closed circuit. When current flows, the potential difference across the terminals of the battery is decreased as some potential drop due to its internal resistance.

Due to the internal resistance in the battery, the potential difference across it is less than its emf. However, for an ideal battery, potential difference and emf are equal.

#### Page No 196:

#### Question 1:

A metallic resistor is connected across a battery. If the number of collisions of the free electrons with the lattice is somehow decreased in the resistor (for example, by cooling it), the current will

(a) increase

(b) decrease

(c) remain constant

(d) become zero

#### Answer:

(a) increase

If the number of collisions of the free electrons with the lattice is decreased, then the drift velocity of the electrons increases.

Current *i* is directly proportional to the drift velocity '*V*_{d}' and is given by the following relation:

*i = neAV _{d}_{ ,}* where '

*n*' is the number density of electrons and '

*A'*is the area of the cross-section of the conductor.

So, we can easily see that current increases with increase in drift velocity.

#### Page No 196:

#### Question 2:

Two resistors *A* and *B *have resistances *R _{A}*

_{ }and

*R*, respectively, and

_{B}*R*<

_{A}*R*. The resistivities of their materials are

_{B}*ρ*and ρ

_{A}_{B}.

(a) ρ

_{A}

_{ }> ρ

_{B}

(b) ρ

_{A}

_{ }= ρ

_{B}

(c) ρ

_{A}

_{ }< ρ

_{B}

(d) The information is not sufficient to find the relation between ρ

_{A}

_{ }and ρ

_{B}.

#### Answer:

(d) The information is not sufficient to find the relation between ρ_{A}_{ }and ρ_{B}.

The resistance *R* of a conductor depends on its resistivity *ρ*, length *l* and cross-sectional area *A*. Thus,

$R=\rho \frac{l}{A}$

From the given comparison of resistances, we cannot derive the correct relation between the resistances. We also need to know the cross-sectional areas and the lengths of both the conductors before concluding about their resistivities. Only then can the relation between the resistivities be found.

The information is not sufficient to find the relation between ρ_{A}_{ }and ρ_{B}.

Hence, the correct option is (d)

#### Page No 196:

#### Question 3:

The product of resistivity and conductivity of a cylindrical conductor depends on

(a) temperature

(b) material

(c) area of cross section

(d) None of these

#### Answer:

(d) None of these

The relation between the resistivity *ρ* and conductivity *σ* of a material is given by

$\rho =\frac{1}{\sigma}\phantom{\rule{0ex}{0ex}}\Rightarrow \rho \times \sigma =1$

The product of conductivity and resistivity is unity. So, this product does not depend on any of the given quantities.

#### Page No 196:

#### Question 4:

As the temperature of a metallic resistor is increased, the product of its resistivity and conductivity

(a) increases

(b) decreases

(c) remains constant

(d) may increase or decrease

#### Answer:

(d) may increase or decrease

The product of resistivity and conductivity is independent of temperature. As the temperature of a metallic resistor is increased, the resistivity increases and conductivity decreases. Hence, both the conductivity and resistivity of the metallic resistor nullify the effect of the change in temperature.

#### Page No 196:

#### Question 5:

In an electric circuit containing a battery, the charge (assumed positive) inside the battery

(a) always goes from the positive terminal to the negative terminal

(b) may go from the positive terminal to the negative terminal

(c) always goes from the negative terminal to the positive terminal

(d) does not move

#### Answer:

(b) may go from the positive terminal to the negative terminal

In the study of electric current, the direction opposite the flow of electrons is regarded as the direction of flow of positive charge. In a battery, positive charge flows from the negative terminal to the positive terminal when it is discharging (connected to external circuit). But when the battery is charged, the positive charge flows from the positive terminal to the negative terminal.

#### Page No 196:

#### Question 6:

A resistor of resistance *R* is connected to an ideal battery. If the value of* R *is decreased, the power dissipated in the resistor will

(a) increase

(b) decrease

(c) remain unchanged

#### Answer:

(a) increase

As the resistance is connected to an ideal battery, it provides a constant potential difference across the two terminals. The internal resistance of the battery is also zero.

The power dissipated in the resistor,

*P* = $\frac{{V}^{2}}{R}$

*V* is constant; hence V $\propto \frac{1}{{R}^{}}$

Thus, if the value of the resistance is decreased, the power dissipated in the resistor will increase.

#### Page No 196:

#### Question 7:

A current passes through a resistor. Let *K*_{1} and *K*_{2}_{ }represent the average kinetic energy of the conduction electrons and the metal ions, respectively.

(a) *K*_{2} < *K*_{2}_{ }

(b) *K*_{1} = *K*_{2}_{ }

(c) *K*_{1} > *K*_{2}_{ }

(d) Any of these three may occur

#### Answer:

(c) *K*_{1} > *K*_{2}_{ }

The metal ions are bound at their positions and vibrate due to collisions with electrons and due to thermal energy. The conduction electrons are free to move. They get energy from the electric field set inside the conductor by connection with a battery and due to thermal motion. The velocity of the electrons is high.

Thus, the kinetic energy of the electrons is greater than the kinetic energy of the metal ions.

#### Page No 196:

#### Question 8:

Two resistors *R* and 2*R* are connected in series in an electric circuit. The thermal energy developed in *R *and 2*R* are in the ratio

(a) 1 : 2

(b) 2 : 1

(c) 1 : 4

(d) 4 : 1

#### Answer:

(a) 1 : 2

Thermal energy developed across a resistor,

*U = i ^{2}Rt* ,

where

*i*is the current flowing through the resistor of resistance

*R*for time

*t.*Since the resistors are connected in series, the current flowing through both the resistors is same and the time for which the current flows is also same.

Thus, the ratio of the thermal energy developed in

*R*and

*2 R*is 1 : 2.

#### Page No 196:

#### Question 9:

Two resistances *R* and 2*R** *are connected in parallel in an electric circuit. The thermal energy developed in *R* and 2*R* are in the ratio

(a) 1 : 2

(b) 2 : 1

(c) 1 : 4

(d) 4 : 1

#### Answer:

(b) 2 : 1

Thermal energy developed in the resistances,

*H *=$\frac{{V}^{2}}{R}t$

Heat developed in the resistance *R*, *H*_{1}=$\frac{{V}^{2}}{R}t$

Heat developed in the resistance 2*R**,* *H*_{2} = $\frac{{V}^{2}}{2R}t$

Thus, Heat developed in the resistance *R *and 2*R* are in ratio 2:1.

#### Page No 196:

#### Question 10:

A uniform wire of resistance 50 Ω is cut into 5 equal parts. These parts are now connected in parallel. The equivalent resistance of the combination is

(a) 2 Ω

(b) 10 Ω

(c) 250 Ω

(d) 6250 Ω

#### Answer:

(a) 2 Ω

Resistance of a wire is directly proportional to its length.

So, when we cut the wire into 5 equal parts, the resistance of each part will be 10 Ω.

On connecting these wires in parallel, the net resistance will be

$\frac{1}{R}=\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}+\frac{1}{10}\phantom{\rule{0ex}{0ex}}\Rightarrow R=2\mathrm{\Omega}$

#### Page No 197:

#### Question 11:

Consider the following two statements:

(A) Kirchhoff's junction law follows from conservation of charge.

(B) Kirchhoff's loop law follows from conservative nature of electric field.

(a) *A *and *B* are correct.

(b) *A* is correct but *B *is wrong.

(c) *B* is correct but* A* is wrong.

(d) *A* and* B* are wrong.

#### Answer:

(a) *A *and *B* are correct.

According to Kirchhoff's junction law, the net charge coming towards a point should be equal to the net charge going away from that point in the same time. It follows from the principle of conservation of charge.

The loop law follows from the fact that electrostatic force is a conservative force and the work done by it in any closed path is zero.

#### Page No 197:

#### Question 12:

Two non-ideal batteries are connected in series. Consider the following statements:

(A) The equivalent emf is larger than either of the two emfs.

(B) The equivalent internal resistance is smaller than either of the two internal resistances.

(a)* A* and *B* are correct.

(b)* A* is correct but *B* is wrong.

(c) *B* is correct but *A* is wrong.

(d) *A* and *B* are wrong.

#### Answer:

(b)* A* is correct but *B* is wrong.

Let the emfs of the batteries be *e*_{1} and *e*_{2}, and their respective resistances be *r*_{1} and *r*_{2}.

Since the batteries are connected in series, the equivalent emf will be the sum of the emf of the two batteries ( *e = **e*_{1} + *e*_{2}) .

Thus*, e > **e*_{1} and *e > **e*_{2}

Thus, the equivalent emf is larger than either of the two emfs. Hence, statement A is correct.

Since the batteries are connected in series, the equivalent internal resistance (*r*) of the combination will be the sum of the internal resistance of the two batteries ( *r = **r*_{1} + *r*_{2}).

*r > **r*_{1} and *r >* *r*_{2}

Thus, the equivalent internal resistance is greater that either of the two resistances. Hence, statement B is wrong.

#### Page No 197:

#### Question 13:

Two non-ideal batteries are connected in parallel. Consider the following statements:

(A) The equivalent emf is smaller than either of the two emfs.

(B) The equivalent internal resistance is smaller than either of the two internal resistances.

(a) *A* and *B* are correct.

(b) *A* is correct but *B* is wrong.

(c) *B* is correct but *A *is wrong.

(d) *A* and *B* are wrong.

#### Answer:

(c) *B* is correct but *A *is wrong.

The equivalent emf ${\epsilon}_{0}$ , when two non-ideal batteries of emfs ${\epsilon}_{1}$ and ${\epsilon}_{2}$ and internal resistances *r _{1}* and

*r*respectively are connected in parallel to each other, is given by:

_{2}${\epsilon}_{0}=\frac{{\epsilon}_{1}{r}_{1}+{\epsilon}_{2}{r}_{2}}{{r}_{1}+{r}_{2}}$

Thus, the equivalent emf is greater than either of the emfs. Thus, statement A is wrong.

For the parallel combination of the batteries of internal resistance

*r*and

_{1}*r*, the equivalent internal resistance

_{2}*r*is given as

$r=\frac{{r}_{1}{r}_{2}}{{r}_{1}+{r}_{2}}$

Thus, the value of the resultant resistance is even smaller than either resistance.

#### Page No 197:

#### Question 14:

The net resistance of an ammeter should be small to ensure that

(a) it does not get overheated

(b) it does not draw excessive current

(c) it can measure large currents

(d) it does not appreciably change the current to be measured

#### Answer:

(d) it does not appreciably change the current to be measured

The ammeter is connected in series with the circuit to measure the net amount of current flowing through the circuit. If the net resistance of the ammeter is high, then the amount of current that the circuit draws will have error. Hence, we cannot accurately measure the amount of current that the circuit draws from the voltage source. Hence, the net resistance or the equivalent resistance of an ammeter should be small enough to ensure that it does not appreciably change the current to be measured.

#### Page No 197:

#### Question 15:

The net resistance of a voltmeter should be large to ensure that

(a) it does not get overheated

(b) it does not draw excessive current

(c) it can measure large potential differences

(d) it does not appreciably change the potential difference to be measured

#### Answer:

(d) it does not appreciably change the potential difference to be measured

To measure potential difference across any circuit element, the voltmeter is connected in parallel to that circuit element. Let *R _{eq}* be the equivalent resistance of the voltmeter and

*V*be the potential difference across the voltmeter.

Then, the current through the voltmeter,

*i*= $\frac{V}{{R}_{eq}}$

Hence, the deflection in the voltmeter is proportional to the current

*i*and, hence, proportional to

*V*. However, when the voltmeter is used in a circuit, its resistance

*R*

_{eq}is connected in parallel to some circuit element. This might change the overall resistance of the circuit and, hence, the current. Consequently, the potential difference to be measured is also changed. To minimise the error due to this, the equivalent resistance

*R*of the voltmeter should be large. When a large resistance is connected in parallel to a small resistance, the equivalent resistance is only slightly less than the smaller one.

_{eq}#### Page No 197:

#### Question 16:

Consider a capacitor-charging circuit. Let *Q*_{1} be the charge given to the capacitor in a time interval of 10 ms and *Q*_{2} be the charge given in the next time interval of 10 ms. Let 10 μC charge be deposited in time interval *t*_{1} and another 10 μC charge be deposited in the next time interval* **t*_{2}.

(a) *Q*_{1} > *Q*_{2}, *t*_{1} >* **t*_{2}

(b) *Q*_{1} > *Q*_{2}, *t*_{1} <* **t*_{2}

(c) *Q*_{1} < *Q*_{2}, *t*_{1} >* **t*_{2}

(d) *Q*_{1} < *Q*_{2}, *t*_{1} <* **t*_{2}

#### Answer:

(b) *Q*_{1} > *Q*_{2}, *t*_{1} <* **t*_{2}

The charge *Q* on the plates of a capacitor at time *t* after connecting it in a charging circuit,

$Q=\epsilon C(1-{e}^{-t/RC})$,

where

ε = emf of the battery connected in the charging circuit

*C* = capacitance of the given capacitor

*R* = resistance of the resistor connected in series with the capacitor

The charge developed on the plates of the capacitor in first 10 mili seconds is given by

${Q}_{1}=\epsilon C(1-{e}^{-10\times {10}^{-3}/RC})$

The charge developed on the plates of the capacitor in first 20 mili seconds is given by

$Q\text{'}=\epsilon C(1-{e}^{-20\times {10}^{-3}/RC})$

The charge developed at the plates of the capacitor in the interval *t* = 10 mili seconds to 20 mili seconds is given by

${Q}_{2}=Q\text{'}-{Q}_{1}\phantom{\rule{0ex}{0ex}}{Q}_{2}=\left[\epsilon C(1-{e}^{-20\times {10}^{-3}/RC})\right]-\left[\epsilon C(1-{e}^{-10\times {10}^{-3}/RC})\right]\phantom{\rule{0ex}{0ex}}\Rightarrow {Q}_{2}=\left[\epsilon C({e}^{-10\times {10}^{-3}/RC}-{e}^{-20\times {10}^{-3}/RC})\right]\phantom{\rule{0ex}{0ex}}\Rightarrow {Q}_{2}=\left[\epsilon C\left({e}^{-10\times {10}^{-3}/RC}\right)(1-{e}^{-10\times {10}^{-3}/RC})\right]\phantom{\rule{0ex}{0ex}}$

Comparing *Q*_{1}_{ }with *Q*_{2}_{ }

${Q}_{1}=\epsilon C(1-{e}^{-10\times {10}^{-3}/RC}),{Q}_{2}=\epsilon C{e}^{-10\times {10}^{-3}/RC}(1-{e}^{-10\times {10}^{-3}/RC})\phantom{\rule{0ex}{0ex}}\frac{{Q}_{1}}{{Q}_{2}}=\frac{1}{{e}^{-10\times {10}^{-3}/RC}}\phantom{\rule{0ex}{0ex}}{e}^{-10\times {10}^{-3}/RC}1\phantom{\rule{0ex}{0ex}}\therefore {Q}_{1}{Q}_{2}$

For second part of the question

10 μC charge be deposited in a time interval *t*_{1} and the next 10 μC charge is deposited in the next time interval* **t*_{2}.

The time taken for 10 μC charge to develop on the plates of the capacitor is *t*_{1}

$10\mathrm{\mu C}=\epsilon C(1-{e}^{-{t}_{1}/RC})$ ...(1)

The time taken for 20 μC charge to develop on the plates of the capacitor is *t'*

$20\mathrm{\mu C}=\epsilon C(1-{e}^{-{t}_{2}/RC})$ ....(2)

Dividing (2) by (1)

$\frac{20\mathrm{\mu C}}{10\mathrm{\mu C}}=\frac{\epsilon C(1-{e}^{-{t}_{2}/RC})}{\epsilon C(1-{e}^{-{t}_{1}/RC})}\phantom{\rule{0ex}{0ex}}2=\frac{(1-{e}^{-{t}_{2}/RC})}{(1-{e}^{-{t}_{1}/RC})}\phantom{\rule{0ex}{0ex}}2(1-{e}^{-{t}_{1}/RC})=(1-{e}^{-{t}_{2}/RC})\phantom{\rule{0ex}{0ex}}{e}^{-{t}_{2}/RC}=2{e}^{-{t}_{1}/RC}-1$

Taking natural log both side,

$\frac{{t}_{2}}{RC}=\mathrm{ln}\left(2\right)+\frac{{t}_{1}}{RC}$

$\Rightarrow $*t*_{1} <* **t*_{2}

#### Page No 197:

#### Question 1:

Electrons are emitted by a hot filament and are accelerated by an electric field, as shown in the figure (32-Q1). The two stops at the left ensure that the electron beam has a uniform cross-section.

(a) The speed of the electrons is more at *B* than at *A*.

(b) The electric current is from left to right.

(c) The magnitude of the current is larger at *B* than at *A*.

(d) The current density is more at *B* than at *A*.

Figure

#### Answer:

(a) The speed of the electrons is more at *B* than at *A*.

Let the potentials at A and B be *V*_{A} and *V*_{B}.

As potential, $E=-\frac{dV}{dr}$ ,

potential increases in the direction opposite to the direction of the electric field.

Thus, *V*_{A} < *V*_{B}

Potential energy of the electrons at points A and B:

*U*_{A}_{ }= $-$*e**V*_{A}

*U*_{B}_{ }= $-$*e**V*_{B}

Thus, *U*_{A}_{ }> *U*_{B}

Let the kinetic energy of an electron at points A and B be *K*_{A }and *K*_{B} respectively.

Applying the principle of conservation of mechanical energy, we get:

*U*_{A}_{ }+ *K*_{A} = *U*_{B}_{ }*+ K*_{B}

As, *U*_{A}_{ }> *U*_{B},

*K*_{A} < *K*_{B}

Therefore, the speed of the electrons is more at *B* than at *A*.

#### Page No 197:

#### Question 2:

A capacitor with no dielectric is connected to a battery at *t *= 0. Consider a point *A* in the connecting wires and a point *B* between the plates.

(a) There is no current through *A*.

(b) There is no current through *B*.

(c) There is a current through *A *as long as the charging is not complete.

(d) There is a current through *B* as long as the charging is not complete.

#### Answer:

(b) There is no current through *B*.

(c) There is a current through *A *as long as the charging is not complete.

As the capacitor is connected to the battery at *t* = 0, current flows through the wire up to the time the capacitor is charged. As the capacitor is completely charged, the potential difference across the capacitor is equal to the terminal potential of the battery. So, the current stops flowing and does not pass through the point between the plates of the capacitor, as there is no medium for the flow of charge.

#### Page No 197:

#### Question 3:

When no current is passed through a conductor,

(a) the free electrons do not move

(b) the average speed of a free electron over a large period of time is zero

(c) the average velocity of a free electron over a large period of time is zero

(d) the average of the velocities of all the free electrons at an instant is zero

#### Answer:

(c) the average velocity of a free electron over a large period of time is zero

(d) the average of the velocities of all the free electrons at an instant is zero

When no current is passed through a conductor, it means that there is no net charge transfer through it. The free electrons are in random motion. The net charge transfer through the cross-section of the conductor is zero. As there is no net charge transfer at any instant, it means that the average of the velocities of all the free electrons at an instant is zero. If we see the motion of each electron as its motion is random and there is no net transfer of charge. Thus, it can be concluded that the average velocity of a free electron over a large period of time is zero.

#### Page No 197:

#### Question 4:

Which of the following quantities does not change when a resistor connected to a battery is heated due to the current?

(a) Drift speed

(b) Resistivity

(c) Resistance

(d) Number of free electrons

#### Answer:

(d) Number of free electrons

As the resistor connected to the battery is heated, the thermal energy of the electrons increases. Thus, the relaxation time of the electrons will decrease and so will the drift velocity. Resistivity is inversely proportional to the relaxation time; thus, resistivity will increase with decrease in relaxation time. Therefore, resistance will also change. The number of electrons in a resistor will remain same.

#### Page No 197:

#### Question 5:

As the temperature of a conductor increases, its resistivity and conductivity change. The ratio of resistivity to conductivity

(a) increases

(b) decreases

(c) remains constant

(d) may increase or decrease, depending on the actual temperature

#### Answer:

(a) increases

Resistivity (*ρ*) of a conductor is the reciprocal of the its conductivity(*σ*). Thus,

$\rho =\frac{1}{\sigma}$

The ratio of the resistivity and the conductivity,

$\frac{\rho}{\sigma}=\frac{\rho}{1/\rho}={\rho}^{2}$

As resistivity increases with temperature, the square of resistivity will also increase. Hence, the ratio of resistivity and conductivity will increase with increase in temperature.

#### Page No 197:

#### Question 6:

A current passes through a wire of non-uniform cross-section. Which of the following quantities are independent of the cross-section?

(a) The charge crossing in a given time interval

(b) Drift speed

(c) Current density

(d) Free-electron density

#### Answer:

(a) The charge crossing in a given time interval

(d) Free-electron density

Drift speed and current density are inversely proportional to the area of cross-section of a wire. Thus, they are dependent on the cross-section. The charge crossing in a given time interval is independent of the area of cross-section of the wire. Free electron density is the total number of free electrons per unit volume of the wire. The density of free electrons depends on the distribution of the free electrons throughout the volume of the wire. It does not depend on the cross-section of the wire.

#### Page No 197:

#### Question 7:

Identify the correct options.

(a) An ammeter should have small resistance.

(b) An ammeter should have large resistance.

(c) A voltmeter should have small resistance.

(d) A voltmeter should have large resistance.

#### Answer:

(a) An ammeter should have small resistance.

(d) A voltmeter should have large resistance.

The ammeter is connected in series in the circuit whose current is to be measured. If the net resistance of the ammeter is high, then the amount of current that the circuit draws will have error . So, we won't be able to accurately measure the amount of current that the circuit draws from the voltage source. Thus, it should have a small resistance.

To measure the potential difference across any circuit element, the voltmeter is connected in parallel to that circuit element. If the resistance of the voltmeter is large, then maximum voltage drop occurs across the voltmeter and it will measure the correct value of the potential.

#### Page No 198:

#### Question 8:

A capacitor of capacitance 500 μF is connected to a battery through a 10 kΩ resistor. The charge stored in the capacitor in the first 5 s is larger than the charge stored in the next.

(a) 5 s

(b) 50 s

(c) 500 s

(d) 500 s

#### Answer:

(a) 5 s

(b) 50 s

(c) 500 s

(d) 500 s

The charge (*Q*) on the capacitor at any instant *t*,

$Q=CV(1-{e}^{-t/RC})$,

where

*C* = capacitance of the given capacitance

*R* = resistance of the resistor connected in series with the capacitor

*RC* = (10 × 10^{3}) × (500 × 10^{$-$6}) = 5 s

The charge on the capacitor in the first 5 seconds,

${Q}_{0}=CV(1-{e}^{-5/5})=CV\times 0.632\phantom{\rule{0ex}{0ex}}$

The charge on the capacitor in the first 10 seconds,

${Q}_{1}=CV(1-{e}^{-10/5})\phantom{\rule{0ex}{0ex}}{Q}_{1}=CV(1-{e}^{-2})=0.864\times CV$

Charge developed in the next 5 seconds,

*Q*' = *Q*_{1} $-$ *Q*_{0}

*Q*' = *CV*(0.864 $-$ 0.632) = 0.232 *CV*

The charge on the capacitor in the first 55 seconds,

${Q}_{2}=CV(1-{e}^{-55/5})\phantom{\rule{0ex}{0ex}}{Q}_{2}=CV(1-{e}^{-11})=0.99\times CV$

Charge developed in the next 50 seconds,

*Q*' = *Q*_{2} $-$ *Q*_{0}

*Q*' = *CV*(0.99 $-$ 0.632) = 0.358 *CV*

Charge developed in the first 505 seconds,

${Q}_{3}=CV(1-{e}^{-500/5})=CV(1-{e}^{-100})\approx CV$

Charge developed in the next 500 seconds,

*Q*' = *CV* (1$-$ 0.632) = 0.368 *CV*

Thus, the charge developed on the capacitor in the first 5 seconds is greater than the charge developed in the next 5,50, 500 seconds.

Disclaimer : Out of the four given options, two options are same.

#### Page No 198:

#### Question 9:

A capacitor *C*_{1}_{ }of capacitance 1 μF and a capacitor *C*_{2}_{ }of capacitance 2 μF are separately charged by a common battery for a long time. The two capacitors are then separately discharged through equal resistors. Both the discharge circuits are connected at* t* = 0.

(a) The current in each of the two discharging circuits is zero at *t *= 0.

(b) The currents in the two discharging circuits at *t* = 0 are equal but not zero.

(c) The currents in the two discharging circuits at *t* = 0 are unequal.

(d) *C*_{1} loses 50% of its initial charge sooner than *C*_{2} loses 50% of its initial charge.

#### Answer:

(b) The currents in the two discharging circuits at *t* = 0 are equal but not zero.

(d) *C*_{1} loses 50% of its initial charge sooner than *C*_{2} loses 50% of its initial charge.

Let the voltage of the battery connected to the capacitors be *V*. Both the capacitors will charge up to the same potential (*V*).

The charge on the capacitors *C*_{1}_{ }is *C*_{1}*V* = (1 μF)×*V*

The charge on the capacitors *C*_{2}_{ }is *C*_{2}*V** =* (2 μF)×*V*

The charge on the discharging circuit at an instant *t*,

$Q=CV{e}^{-t/RC}$

The current through the discharging circuit,

$\frac{\mathrm{d}Q}{\mathrm{d}t}=-\frac{CV}{RC}{\mathrm{e}}^{-t/RC}=\frac{\mathit{V}}{\mathit{R}}{\mathrm{e}}^{-\mathrm{t}/\mathrm{R}\mathrm{C}}$

At *t* = 0, the current through the discharging circuit will be $\frac{V}{R}$ for both the capacitors.

Let the time taken by the capacitor C_{1} to lose 50% of the charge be *t*_{1}.

${Q}_{1}=\frac{{C}_{1}V}{2}\phantom{\rule{0ex}{0ex}}\frac{{C}_{1}V}{2}={C}_{1}V{e}^{-{t}_{1}/RC}\phantom{\rule{0ex}{0ex}}\frac{1}{2}={e}^{-{t}_{1}/RC}$

Taking natural log on both sides, we get:

$-\mathrm{ln}2=-\frac{{t}_{1}}{R{C}_{1}}\phantom{\rule{0ex}{0ex}}{t}_{1}=R{C}_{1}\mathrm{ln}2$

Similarly,

Time taken for capacitor *C*_{2}: ${t}_{2}=R{C}_{2}\mathrm{ln}2$

As, *C*_{1} < *C*_{2}, *t*_{1} < *t*_{2}

Thus, we can say that *C*_{1} loses 50% of its initial charge sooner than *C*_{2} loses 50% of its initial charge.

#### Page No 198:

#### Question 1:

The amount of charge that passes in time *t *through a cross-section of a wire is

*Q*(*t*) = *At*^{2} + *Bt* +* C*.

(a) Write the dimensional formulae for *A*, *B* and *C*.

(b) If the numerical values of *A*, *B* and *C* are 5, 3 and 1, respectively, in S.I units, find the value of the current at *t* = 5 s.

#### Answer:

(a) Amount of charge,

*Q*(*t*) =* **At*^{2} + *Bt* +* **C*

We can only add the terms with the same dimensions. So, all the individual terms will have dimensions equal to the dimensions of the charge.

Comparing the dimensions of each term separately, we get:

$A{t}^{2}=Q\phantom{\rule{0ex}{0ex}}\Rightarrow A=\frac{Q}{{t}^{2}}\phantom{\rule{0ex}{0ex}}Q=It\phantom{\rule{0ex}{0ex}}\Rightarrow A=\frac{I}{t}=\left[{\mathrm{AT}}^{-1}\right],$

where *I* = current through the wire

Now,

$Bt=Q\phantom{\rule{0ex}{0ex}}\Rightarrow B=\frac{Q}{t}=I=\left[\mathrm{A}\right]$

Also,

$C=Q\phantom{\rule{0ex}{0ex}}\Rightarrow C=\left[\mathrm{AT}\right]$

(b) Current *I* = rate of flow of charge

$\Rightarrow I=\frac{\mathrm{d}Q}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}t}\left(A{t}^{2}+Bt+C\right)\phantom{\rule{0ex}{0ex}}\Rightarrow I=2At+B\phantom{\rule{0ex}{0ex}}A=5,B=3\mathrm{and}t=5\mathrm{s}\phantom{\rule{0ex}{0ex}}\Rightarrow I=2\times 5\times 5+3=53\mathrm{A}$

#### Page No 198:

#### Question 2:

An electron gun emits 2.0 ×10^{16} electrons per second. What electric current does this correspond to?

#### Answer:

Rate of emission of electrons = 2 ×10^{16} per second

Total charge passing through the gun = 2 ×10^{16} ×1.6 × 10^{–19 }= 3.2 × 10^{–3} per second

We know:

Current*, I* = rate of flow of charge

∴ *I* = 3.2 × 10^{–3} A

#### Page No 198:

#### Question 3:

The electric current existing in a discharge tube is 2.0 μA. How much charge is transferred across a cross-section of the tube in 5 minutes?

#### Answer:

Let *Q* amount of charge be transferred across the cross-section of the tube.

Given:

Current through the discharge tube, *I* = 2 μA = 2 × 10^{–6} A

Time taken to transfer charge, *t *= 5 mins = 300 s

We know:

*Q* = *It*

∴ *Q *= 2 × 10^{–6} × 300 = 6× 10^{–4} C

#### Page No 198:

#### Question 4:

The current through a wire depends on time as* i = **i*_{0} + α*t*,

where *i*_{0} = 10 A and α = 4 As^{–1}. Find the charge that crosses through a section of the wire in 10 seconds.

#### Answer:

Current through the wire varies with time as

*i** *= *i*_{0}_{ }+ *α t*

Time interval, *t* = 10 s

*i*_{0} = 10 A

*α* = 4 A/sec

We know that for variable current,

$q={\int}_{0}^{t}i\mathrm{d}t\phantom{\rule{0ex}{0ex}}\Rightarrow q={\int}_{0}^{t}\left({i}_{0}+\alpha t\right)\mathrm{d}t\phantom{\rule{0ex}{0ex}}\Rightarrow q={i}_{0}t+\frac{\alpha {t}^{2}}{2}\phantom{\rule{0ex}{0ex}}\therefore q=10\times 10+4\times \frac{{\left(10\right)}^{2}}{2}\phantom{\rule{0ex}{0ex}}=100+200\phantom{\rule{0ex}{0ex}}=300\mathrm{C}$

#### Page No 198:

#### Question 5:

A current of 1.0 A exists in a copper wire of cross-section 1.0 mm^{2}. Assuming one free electron per atom, calculate the drift speed of the free electrons in the wire. The density of copper is 9000 kg m^{–3}.

#### Answer:

Given:

Current*, i *= 1 A

Area of cross-section, *A* = 1 mm^{2} = 1 × 10^{–6} m^{2}

Density of copper, $\rho $ = 9000 kg/m^{3}

Length of the conductor = *l*

Also,

Mass of copper wire = Volume × density

$\Rightarrow m\mathit{=}A\times l\times \rho \phantom{\rule{0ex}{0ex}}\mathit{\Rightarrow}m\mathit{=}A\times l\times 9000\mathrm{kg}$

We know that the number of atoms in molecular mass *M* = N_{A}

∴ Number of atoms in mass *m*, *N* = $\left(\frac{{\mathrm{N}}_{\mathrm{A}}}{M}\right)m$,

where N_{A }is known as Avagadro's number and is equal to 6 × 10^{23} atoms.

*$\Rightarrow N=\left(\frac{{\mathrm{N}}_{\mathrm{A}}}{\mathrm{M}}\right)m\phantom{\rule{0ex}{0ex}}\Rightarrow N=\left(\frac{{\mathrm{N}}_{\mathrm{A}}}{\mathrm{M}}\right)\times A\times l\times 9000\phantom{\rule{0ex}{0ex}}$*

Also, it is given that

No. of free electrons = No. of atoms

Let *n* be the number of free electrons per unit volume

$n=\frac{\mathrm{Number}\mathrm{of}\mathrm{electrons}}{\mathrm{Volume}}\phantom{\rule{0ex}{0ex}}=\frac{{\mathrm{N}}_{\mathrm{A}}\times A\times l\times 9000}{M\times A\times l}\phantom{\rule{0ex}{0ex}}=\frac{{\mathrm{N}}_{\mathrm{A}}\times 9000}{M}\phantom{\rule{0ex}{0ex}}=\frac{6\times {10}^{23}\times 9000}{63.5\times {10}^{-3}}\phantom{\rule{0ex}{0ex}}\therefore i\mathit{=}{V}_{\mathrm{d}}nAe\phantom{\rule{0ex}{0ex}}\Rightarrow {V}_{d}=\frac{1}{{\displaystyle \frac{6\times {10}^{23}\times 9000}{63.5\times {10}^{-3}}\times {10}^{-6}\times 1.6\times {10}^{-19}}}\phantom{\rule{0ex}{0ex}}=\frac{63.5\times {10}^{-3}}{6\times {10}^{23}\times 9000\times {10}^{-6}\times 1.6\times {10}^{-19}}\phantom{\rule{0ex}{0ex}}=\frac{63.5\times {10}^{-3}}{6\times {10}^{26}\times 9\times {10}^{-6}\times 1.6\times {10}^{-19}}\phantom{\rule{0ex}{0ex}}=\frac{63.5\times {10}^{-3}}{6\times 9\times 16}\phantom{\rule{0ex}{0ex}}=0.073\times {10}^{-3}\mathrm{m}/\mathrm{s}\phantom{\rule{0ex}{0ex}}=0.073\mathrm{mm}/\mathrm{s}$

#### Page No 198:

#### Question 6:

A wire of length 1 m and radius 0.1 mm has a resistance of 100 Ω. Find the resistivity of the material.

#### Answer:

Let *ρ* be the resistivity of the wire.

Given:

Length*, l* = 1 m

Radius*, r *= 0.1 × 10^{–3} m

Area, *A* = $\mathrm{\pi}{r}^{2}$

Resistance, *R* = 1000 Ω

We know:

$R=\rho \frac{l}{A}\phantom{\rule{0ex}{0ex}}\Rightarrow \rho =\frac{RA}{l}\phantom{\rule{0ex}{0ex}}=\frac{1000\times \mathrm{\pi}\times 0.1\times 0.1\times {10}^{-6}}{1}\phantom{\rule{0ex}{0ex}}=\mathrm{\pi}\times {10}^{-6}\mathrm{\Omega m}$

#### Page No 198:

#### Question 7:

A uniform wire of resistance 100 Ω is melted and recast as a wire whose length is double that of the original. What would be the resistance of the wire?

#### Answer:

Let

Resistivity of the wire = *ρ*

Original length of the wire = *l*

New length of the wire = *l'*

Original area of the wire = *A*

New area of the wire = *A' *

Original resistance of the wire = *R* = 100 Ω

New resistance of the wire = *R'*

Given:

*l'* = 2*l*

The volume of the wire remains constant. So,

$Al=A\text{'}l\text{'}\phantom{\rule{0ex}{0ex}}\Rightarrow A\text{'}=\frac{A}{2}$

We know:

$R=\frac{\rho l}{A}\phantom{\rule{0ex}{0ex}}\Rightarrow R\text{'}=\frac{\rho l\text{'}}{A\text{'}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{R\text{'}}{R}=\frac{l\text{'}A}{lA\text{'}}\phantom{\rule{0ex}{0ex}}\because l\text{'}=2l,A\text{'}=\frac{A}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{R\text{'}}{R}=\frac{\left(2l\right)A}{l\left({\displaystyle \frac{A}{2}}\right)}=4\phantom{\rule{0ex}{0ex}}\Rightarrow R\text{'}=4R=400\mathrm{\Omega}$

#### Page No 198:

#### Question 8:

Consider a wire of length 4 m and cross-sectional area 1 mm^{2} carrying a current of 2 A. If each cubic metre of the material contains 10^{29 }free electrons, find the average time taken by an electron to cross the length of the wire.

#### Answer:

Given:

Current through the wire, *i* = 2 A

Length of the wire*, l *= 4 m

Area of cross section, *A* = 1 mm^{2} = 1 × 10^{–6} m^{2}

Number of electrons per unit volume, *n* = 10^{29}

We know:

$i=nA{V}_{d}e\phantom{\rule{0ex}{0ex}}\Rightarrow {V}_{d}=\frac{i}{nAe}\phantom{\rule{0ex}{0ex}}=\frac{2}{{10}^{29}\times {10}^{-6}\times 1.6\times {10}^{-19}}\phantom{\rule{0ex}{0ex}}\Rightarrow {V}_{\mathrm{d}}=\frac{1}{8000}\mathrm{m}/\mathrm{s},$

where *V _{d}* is the drift speed.

Let

*t*be the time taken by an electron to cross the length of the wire.

$\Rightarrow t=\frac{\text{L}\mathrm{ength}\mathrm{of}\text{the}\mathrm{wire}}{\text{D}\mathrm{rift}\mathrm{speed}}\phantom{\rule{0ex}{0ex}}=\frac{l}{{V}_{\mathrm{d}}}\phantom{\rule{0ex}{0ex}}\therefore t=\frac{4}{{\displaystyle \frac{1}{8000}}}\phantom{\rule{0ex}{0ex}}=32\times {10}^{3}\mathrm{s}$

#### Page No 198:

#### Question 9:

What length of a copper wire of cross-sectional area 0.01 mm^{2} will be needed to prepare a resistance of 1 kΩ? Resistivity of copper = 1.7 × 10^{–8} Ωm

#### Answer:

Given:

Resistivity of copper, *ρ*_{cu}* _{ }*= 1.7 × 10

^{–8}Ωm

Area of cross-section,

*A*= 0.01 mm

^{2}

^{ }= 0.01 × 10

^{–6}m

^{2}

Required resistance

*, R*= 1 kΩ = 10

^{3}Ω

Let

*l*be the required length of the copper wire.

We know:

$R=\frac{\rho l}{A}\phantom{\rule{0ex}{0ex}}\Rightarrow l=\frac{RA}{{\rho}_{\mathrm{cu}}}\phantom{\rule{0ex}{0ex}}\Rightarrow l=\frac{{10}^{3}\times 0.01\times {10}^{-6}}{1.7\times {10}^{-8}}\phantom{\rule{0ex}{0ex}}\therefore l=0.58\times {10}^{3}\mathrm{m}=0.6\mathrm{km}$

#### Page No 198:

#### Question 10:

Figure (32.E1) shows a conductor of length* l *with a circular cross-section. The radius of the cross-section varies linearly from *a* to *b*. The resistivity of the material is ρ. Assuming that* b – a* << *l,* find the resistance of the conductor.

Figure 32.E1

#### Answer:

Let us consider a small element strip of length *dx* at a distance *x* from one end, as shown below.

Let the resistance of the small element strip be *dR*. Let the radius at that point be *c* .

Then, the resistance of this small strip,

$\mathrm{d}R=\frac{\rho \mathrm{d}x}{\mathrm{\pi}{c}^{2}}...\left(\mathrm{i}\right)\phantom{\rule{0ex}{0ex}}\mathrm{tan\theta}=\frac{c-\mathrm{a}}{x}=\frac{\mathrm{b}-\mathrm{a}}{L}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{c-\mathrm{a}}{x}=\frac{\mathrm{b}-\mathrm{a}}{L}\phantom{\rule{0ex}{0ex}}\Rightarrow L\times \left(c-\mathrm{a}\right)=x\times \left(\mathrm{b}-\mathrm{a}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow Lc-\mathrm{La}=xb-xa\phantom{\rule{0ex}{0ex}}$

Differentiating w.r.t to *x*, we get:

$\mathrm{L}\frac{\mathrm{d}c}{\mathrm{d}x}-0=\mathrm{b}-\mathrm{a}\phantom{\rule{0ex}{0ex}}dx=\frac{Ldc}{\left(b\mathit{-}a\right)}...\left(\mathrm{ii}\right)$

Substituting the value of *dx *in equation (i), we get:

$\mathrm{d}R=\frac{\rho L\mathrm{d}c}{\mathrm{\pi}{c}^{2}\left(b-a\right)}\phantom{\rule{0ex}{0ex}}\mathrm{d}R=\frac{\rho L}{\mathrm{\pi}\left(b-a\right)}\xad\xb7\frac{\mathrm{d}c}{{c}^{2}}\phantom{\rule{0ex}{0ex}}$

Integrating *dR* from *a* to *b*, we get:

${\int}_{0}^{R}dR=\frac{\rho L}{\mathrm{\pi}\left(\mathit{b}\mathit{-}\mathit{a}\right)}{\int}_{a}^{b}\frac{\mathrm{d}c}{{c}^{2}}\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{\rho L}{\mathrm{\pi}\left(\mathit{b}\mathit{-}\mathit{a}\right)}{\left[\frac{-1}{c}\right]}_{a}^{b}\phantom{\rule{0ex}{0ex}}=\frac{\rho L}{\mathrm{\pi}\left(\mathit{b}\mathit{-}\mathit{a}\right)}\left(\frac{-1}{\mathrm{b}}-\frac{-1}{\mathrm{a}}\right)\phantom{\rule{0ex}{0ex}}=\frac{\rho L}{\mathrm{\pi ab}}$

#### Page No 198:

#### Question 11:

A copper wire of radius 0.1 mm and resistance kΩ is connected across a power supply of 20 V. (a) How many electrons are transferred per second between the supply and the wire at one end? (b) Write down the current density in the wire.

#### Answer:

Given:

Radius of the wire, *r = *0.1 mm *= *10^{-4} m

Resistance, *R* = 1 kΩ = 10^{3} Ω

Voltage across the ends of the wire,* V* = 20* *V

(a) Let *q* be the charge transferred per second and *n* be the number of electrons transferred per second.

We know:

$i=\frac{V}{R}\phantom{\rule{0ex}{0ex}}\Rightarrow i=\frac{20\mathrm{V}}{{10}^{3}\mathrm{\Omega}}\phantom{\rule{0ex}{0ex}}\Rightarrow i=20\times {10}^{-3}=2\times {10}^{-2}\mathrm{A}\phantom{\rule{0ex}{0ex}}q\mathit{=}it\phantom{\rule{0ex}{0ex}}\Rightarrow q=2\times {10}^{-2}\times 1\phantom{\rule{0ex}{0ex}}\Rightarrow q=2\times {10}^{-2}\mathrm{C}$

Also,

*q* = *ne*

$\Rightarrow n=\frac{q}{e}=\frac{2\times {10}^{-2}}{1.6\times {10}^{-19}}\phantom{\rule{0ex}{0ex}}\Rightarrow n=1.25\times {10}^{17}$

(b) Current density of a wire,

$j=\frac{i}{A}\phantom{\rule{0ex}{0ex}}\Rightarrow j=\frac{2\times {10}^{-2}}{3.14\times {10}^{-8}}\phantom{\rule{0ex}{0ex}}\Rightarrow j=6.37\times {10}^{5}\mathrm{A}/{\mathrm{m}}^{2}$

#### Page No 198:

#### Question 12:

Calculate the electric field in a copper wire of cross-sectional area 2.0 mm^{2} carrying a current of 1 A.

The resistivity of copper = 1.7 × 10^{–8} Ω m

#### Answer:

Given:

Area of cross-section*, A* = 2 × 10^{–6 }m^{2}

Current* *through the wire*, i* = 1 A

Resistivity of copper, *ρ *= 1.7 × 10^{–8} Ωm

Resistance of a wire,

*$R=\rho \frac{l}{A}$*

Also from Ohm's Law, voltage across a wire,

$V=iR=\frac{i\rho l}{A}$

The electric field of the wire,

$E=\frac{V}{l}\phantom{\rule{0ex}{0ex}}\Rightarrow E=\frac{i\rho l}{Al}=\frac{i\rho}{A}\phantom{\rule{0ex}{0ex}}\Rightarrow E=\frac{1\times 1.7\times {10}^{-8}}{2\times {10}^{-6}}\phantom{\rule{0ex}{0ex}}\Rightarrow E=8.5\mathrm{V}/\mathrm{m}$

#### Page No 198:

#### Question 13:

A wire has a length of 2.0 m and a resistance of 5.0 Ω. Find the electric field existing inside the wire if it carries a current of 10 A.

#### Answer:

Given:

Length of the wire, *l *= 2 m

Resistance of the wire, *R* = 5 Ω

Current through the wire, *i* = 10 A

From Ohm's Law, the potential difference across the wire,

*V* = *iR** *

*∴ V*= 10 × 5 = 50 V

Electric field,

$\mathrm{E}=\frac{\mathrm{V}}{l}\phantom{\rule{0ex}{0ex}}=\frac{50}{2}\phantom{\rule{0ex}{0ex}}=25\mathrm{V}/\mathrm{m}$

#### Page No 198:

#### Question 14:

The resistance of an iron wire and a copper wire at 20°C are 3.9 Ω and 4.1 Ω, respectively. At what temperature will the resistance be equal? Temperature coefficient of resistivity for iron is 5.0 × 10^{–3} K^{–1} and for copper, it is 4.0 × 10^{–3} K^{–1}. Neglect any thermal expansion.

#### Answer:

Given:

Resistance *R*_{Fe} of the iron wire at 20°C = 3.9 Ω

Resistance *R*_{Cu} of the copper wire at 20°C = 4.1 Ω

Temperature coefficient *α*_{Fe} for iron = 5.0 × 10^{–3} K^{–1}

Temperature coefficient *α*_{Cu} for copper = 4.0 × 10^{–3} K^{–1}

Let

The temperature at which the resistance be equal = T

Resistance of iron wire at T °C = *R*_{Fe}'

Resistance of copper wire at T °C = *R*_{Cu}'

We know:

$R={R}_{0}(1+\alpha \u2206T)$

Here, Δ*T** *= *T - 20*

$\Rightarrow {R}_{\mathrm{Fe}}\text{'}={R}_{\mathrm{Fe}}\left[1+{\alpha}_{\mathrm{Fe}}\left(T-20\right)\right]\phantom{\rule{0ex}{0ex}}{R}_{\mathrm{Cu}}\text{'}={R}_{\mathrm{Cu}}\left[1+{\alpha}_{\mathrm{Cu}}\left(T-20\right)\right]\phantom{\rule{0ex}{0ex}}{R}_{\mathrm{Fe}}\mathit{\text{'}}\mathit{=}{R}_{\mathrm{Cu}}\text{'}\phantom{\rule{0ex}{0ex}}\mathit{\Rightarrow}{R}_{\mathrm{Fe}}\left[1+{\alpha}_{\mathrm{Fe}}\left(T-20\right)\right]={R}_{\mathrm{Cu}}\left[1+{\alpha}_{\mathrm{Cu}}\left(T-20\right)\right]\phantom{\rule{0ex}{0ex}}\Rightarrow 3.9\left[1+5\times {10}^{-3}\left(T-20\right)\right]=4.1\left[1+4\times {10}^{-3}\left(T-20\right)\right]\phantom{\rule{0ex}{0ex}}\Rightarrow 3.9+3.9\times 5\times {10}^{-3}\left(T-20\right)=4.1+4.1\times 4\times {10}^{-3}\left(T-20\right)\phantom{\rule{0ex}{0ex}}\Rightarrow 4.1\times 4\times {10}^{-3}\left(T-20\right)-3.9\times 5\times {10}^{-3}\left(T-20\right)=-4.1+3.9\phantom{\rule{0ex}{0ex}}\Rightarrow 16.4\left(T-20\right)-19.5\left(T-20\right)=-0.2\times {10}^{3}\phantom{\rule{0ex}{0ex}}\Rightarrow \left(T-20\right)\left(-3.1\right)=-0.2\times {10}^{3}\phantom{\rule{0ex}{0ex}}\Rightarrow T-20=64.5\phantom{\rule{0ex}{0ex}}\Rightarrow T=84.5\xb0\mathrm{C}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

#### Page No 198:

#### Question 15:

The current in a conductor and the potential difference across its ends are measured by an ammeter and a voltmeter. The meters draw negligible currents. The ammeter is accurate but the voltmeter has a zero error (that is, it does not read zero when no potential difference is applied). Calculate the zero error if the readings for two different conditions are 1.75 A, 14.4 V and 2.75 A, 22.4 V.

#### Answer:

Let the magnitude of zero error in the voltmeter reading be *V.*

We need to subtract the zero error from the readings obtained under the two given conditions to obtain the true value of potential difference.

Under both the conditions ,the resistance of the wire will not change.

$\Rightarrow {R}_{1}={R}_{2}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{V}_{1}}{{I}_{1}}=\frac{{V}_{2}}{{I}_{2}}$

$\frac{{I}_{1}R}{{I}_{2}R}=\frac{{V}_{1}}{{V}_{2}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{1.75}{2.75}=\frac{14.4-V}{22.4-V}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{0.35}{0.55}=\frac{14.4-V}{22.4-V}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{7}{11}=\frac{14.4-V}{22.4-V}\phantom{\rule{0ex}{0ex}}\Rightarrow 7\times \left(22.4-V\right)=11\left(14.4-V\right)\phantom{\rule{0ex}{0ex}}\Rightarrow 156.8-7V=158.4-11V\phantom{\rule{0ex}{0ex}}\Rightarrow \left(7-11\right)V=156.8-158.4\phantom{\rule{0ex}{0ex}}\Rightarrow -4V=-1.6\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{V}=0.4\mathrm{V}$

Magnitude of zero error, V = 0.4 V, which can either be positive or negative. Positive or negative zero error just indicates that the needle of the voltmeter is to the right or left of the zero marked on the device if zero voltage is applied across the voltmeter.

#### Page No 198:

#### Question 16:

Figure (32-E2) shows an arrangement to measure the emf *ε* and internal resistance *r* of a battery. The voltmeter has a very high resistance and the ammeter also has some resistance. The voltmeter reads 1.52 V when the switch *S* is open. When the switch is closed, the voltmeter reading drops to 1.45 V and the ammeter reads 1.0 A. Find the emf and the internal resistance of the battery.

Figure 32-E2

#### Answer:

(a)

When the switch *S* is opened, loop 2 will be open and no current will pass through the ammeter. Loop 1 will be closed; so, a current will flow through it. But since the voltmeter has very high resistance, compared to the internal resistance of the battery, the voltage-drop across the internal resistance can be ignored, compared to the voltage drop across the voltmeter. So, the voltage appearing across the voltmeter will be almost equal to the emf of the battery.

∴ *ε* = 1.52 V

(b) When the switch is closed, current will pass through the circuit in loop 2. In this case, there will be a voltage drop across *r* due to current *i* flowing through it.

Applying the loop rule, we get:

*ε – **ir* = 1.45

⇒ 1.52 *– **ir* = 1.45

⇒ *ir* = 0.07

⇒ 1.r = 0.07

r = 0.07 Ω

#### Page No 198:

#### Question 17:

The potential difference between the terminals of a battery of emf 6.0 V and internal resistance 1 Ω drops to 5.8 V when connected across an external resistor. Find the resistance of the external resistor.

#### Answer:

Given:

Emf of the battery, *E* = 6 V

Internal resistance, *r* = 1 Ω

Potential difference*, V* = 5.8 V

Let *R* be the resistance of the external resistor.

Applying KVL in the above circuit, we get:

*$E\mathit{-}i\mathit{(}R\mathit{+}r\mathit{)}=0\phantom{\rule{0ex}{0ex}}\Rightarrow i\mathit{=}\frac{E}{R\mathit{+}r}\phantom{\rule{0ex}{0ex}}\mathit{=}\frac{6}{R\mathit{+}1}$*

Also,

$V\mathit{=}E\mathit{-}ir\phantom{\rule{0ex}{0ex}}\Rightarrow 5.8=6-\frac{6}{R+1}\times 1\phantom{\rule{0ex}{0ex}}\frac{6}{R+1}=0.2\phantom{\rule{0ex}{0ex}}R+1=\frac{6}{0.2}=30\phantom{\rule{0ex}{0ex}}R=29\mathrm{\Omega}$

#### Page No 199:

#### Question 18:

The potential difference between the terminals of a 6.0 V battery is 7.2 V when it is being charged by a current of 2.0 A. What is the internal resistance of the battery?

#### Answer:

Given:

Potential difference across the terminals of the battery, *V* = 7.2 V

Emf of the battery, *E* = 6 V

Current flowing through the circuit,* i* = 2 A

As the battery is getting charged, the voltage drop across the terminals of the battery will be equal to the emf of the battery plus the voltage drop across the internal resistance of the battery.

So,* V* = *E* + *ir*

⇒ 7.2 = 6 + 2 × *r*

⇒ 1.2 = 2*r*

⇒ *r* = 0.6 Ω

#### Page No 199:

#### Question 19:

The internal resistance of an accumulator battery of emf 6 V is 10 Ω when it is fully discharged. As the battery gets charged up, its internal resistance decreases to 1 Ω.

The battery in its completely discharged state is connected to a charger that maintains a constant potential difference of 9 V. Find the current through the battery (a) just after the connections are made and (b) after a long time when it is completely charged.

#### Answer:

(a) **When the battery is being charged:**

Emf of the battery, *E* = 6 V

Internal resistance of the battery, *r* = 10 Ω

Potential difference, *V* = 9 V

Net e.m.f. across the resistance while charging = 9 – 6 = 3 V

$\therefore \mathrm{Curre}\text{nt}=\frac{3}{10}=0.3\mathrm{A}$

(b) **When the battery is completely charged: **

Internal resistance, *r*' = 1 Ω

$\therefore \mathrm{Current}=\frac{3}{1}=3\mathrm{A}$

#### Page No 199:

#### Question 20:

Find the value of *i*_{1}/*i*_{2} in the figure (32-E3) if (a) *R* = 0.1 Ω (b) *R* = 1 Ω and (c)* R* = 10 Ω. Note from your answers that in order to get more current from a combination of two batteries, they should be joined in parallel if the external resistance is small and in series if the external resistance is large, compared to the internal resistance.

Figure 32-E3

#### Answer:

(a) For *R* = 0.1 Ω:

Applying KVL in the given circuit, we get:

$0.1{i}_{1}+1{i}_{1}-6+1{i}_{1}-6=0\phantom{\rule{0ex}{0ex}}\Rightarrow 0.1{i}_{1}+1{i}_{1}+1{i}_{1}=12\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{1}=\frac{12}{\left(2.1\right)}=5.71\mathrm{A}\phantom{\rule{0ex}{0ex}}$

Now, consider the given circuit.

Applying KVL in the loop ABCDA, we get:

$0.1{i}_{2}+1i-6=0\phantom{\rule{0ex}{0ex}}\Rightarrow 0.1{i}_{2}+i=6\phantom{\rule{0ex}{0ex}}\Rightarrow i=6-0.1{i}_{2}$

Applying KVL in ADEFA, we get:

$i-6+6-\left({i}_{2}-i\right)1=0\phantom{\rule{0ex}{0ex}}\Rightarrow i-{i}_{2}+i=0\phantom{\rule{0ex}{0ex}}\Rightarrow 2\mathrm{i}-{i}_{2}=0\phantom{\rule{0ex}{0ex}}\Rightarrow 2\left[6-0.1{i}_{2}\right]-{i}_{2}=0\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{2}=10\mathrm{A}$

$\therefore \frac{{i}_{1}}{{i}_{2}}=0.571$

(b) For *R* = 1 Ω:

Applying KVL in the circuit given in figure 1, we get:

$1{i}_{1}+1.{i}_{1}-6+{i}_{1}-6=0\phantom{\rule{0ex}{0ex}}\Rightarrow 3{i}_{1}=12\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{1}=4\phantom{\rule{0ex}{0ex}}$

Now, for figure 2:

Applying KVL in ABCDA, we get:

${i}_{2}+i-6=0\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{2}+i=6$

Applying KVL in ADEFA, we get:

$i-6+6-\left({i}_{2}-i\right)1=0\phantom{\rule{0ex}{0ex}}\Rightarrow i-{i}_{2}+i=0\phantom{\rule{0ex}{0ex}}\Rightarrow 2i-{i}_{2}=0\phantom{\rule{0ex}{0ex}}\Rightarrow 2\left[6-{i}_{2}\right]-{i}_{2}=0\phantom{\rule{0ex}{0ex}}\Rightarrow 12-3{i}_{2}=0\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{2}=4\mathrm{A}\phantom{\rule{0ex}{0ex}}\therefore \frac{{i}_{1}}{{i}_{2}}=1$

(c) For R = 10 Ω:

Applying KVL in the circuit given in figure 1, we get:

$10{i}_{1}+1{i}_{1}-6+1{i}_{1}-6=0$

$\Rightarrow 12{i}_{1}=12\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{1}=1\phantom{\rule{0ex}{0ex}}$

Now, for figure 2:

Applying KVL in ABCDA, we get:

$10{i}_{2}+i-6=0\phantom{\rule{0ex}{0ex}}\Rightarrow i=6-10{i}_{2}$

Applying KVL in ADEFA, we get:

$i-6+6-\left({i}_{2}-i\right)1=0\phantom{\rule{0ex}{0ex}}\Rightarrow i-{i}_{2}+i=0\phantom{\rule{0ex}{0ex}}\Rightarrow 2i-{i}_{2}=0\phantom{\rule{0ex}{0ex}}\Rightarrow 2\left[6-10{i}_{2}\right]-{i}_{2}=0\phantom{\rule{0ex}{0ex}}\Rightarrow 12-21{i}_{2}=0\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{2}=0.57\mathrm{A}\phantom{\rule{0ex}{0ex}}\therefore \frac{{i}_{1}}{{i}_{2}}=1.75$

#### Page No 199:

#### Question 21:

Consider *N* = *n*_{1}*n*_{2}_{ }identical cells, each of emf *ε *and internal resistance* r. *Suppose *n*_{1}_{ }cells are joined in series to form a line and *n*_{2} such lines are connected in parallel.

The combination drives a current in an external resistance *R*. (a) Find the current in the external resistance. (b) Assuming that *n*_{1} and *n*_{2} can be continuously varied, find the relation between *n*_{1}, *n*_{2}, *R* and *r *for which the current in* R* is maximum.

#### Answer:

(a)

Given:

Emf of one cell = *E*

∴ Total e.m.f. of *n*_{1} cells in one row = *n*_{1}*E*

Total emf of one row will be equal to the net emf across all the *n _{2}* rows because of parallel connection.

Total resistance in one row =

*n*

_{1}

*r*

Total resistance of

*n*rows in parallel $=\frac{{n}_{1}r}{{n}_{2}}$

_{2}_{ }Net resistance of the circuit =

*R*+ $\frac{{n}_{1}r}{{n}_{2}}$

$\therefore \mathrm{Current},I=\frac{{n}_{1}E}{R+{\displaystyle \frac{{n}_{1}r}{{n}_{2}}}}=\frac{{n}_{1}{n}_{2}\mathrm{E}}{{n}_{2}\mathrm{R}+{n}_{1}r}$

(b) From (a),

$I=\frac{{n}_{1}{n}_{2}\mathrm{E}}{{n}_{2}\mathrm{R}+{n}_{1}r}$

For

*I*to be

*maximum,*(

*n*

_{1}

*r*

*+*

*n*

_{2}R) should be minimum

$\Rightarrow {\left(\sqrt{{n}_{1}r}-\sqrt{{n}_{2}\mathrm{R}}\right)}^{2}+2\sqrt{{n}_{1}\mathrm{R}{n}_{2}r}=\mathrm{min}$

It is minimum when

$\sqrt{{n}_{1}r}=\sqrt{{n}_{2}\mathrm{R}}\phantom{\rule{0ex}{0ex}}{n}_{1}r={n}_{2}\mathrm{R}$

*∴ I*is maximum when

*n*

_{1}

*r*

*=*

*n*

_{2}R .

#### Page No 199:

#### Question 22:

A battery of emf 100 V and a resistor of resistance 10 kΩ are joined in series. This system is used as a source to supply current to an external resistance *R*. If *R* is not greater than 100 Ω, the current through it is constant up to two significant digits.

Find its value. This is the basic principle of a *constant-current source*.

#### Answer:

Given:

Emf of the battery, *E* = 100 volt

Resistance in series with battery, *R' *= 10 kΩ = 10000 Ω

External resistance, *R* = (1-100) Ω

When no external resistor is present (*R* = 0), current through the circuit,

$i=\frac{\mathrm{E}}{\mathrm{R}\mathit{\text{'}}}=\frac{100}{1000}=0.01\mathrm{Amp}.\phantom{\rule{0ex}{0ex}}$

When *R* = 1 Ω,

$i=\frac{100}{10000+1}=\frac{100}{10001}\phantom{\rule{0ex}{0ex}}=0.009999\mathrm{A}$

When *R* = 100 Ω,

$i=\frac{100}{10000+100}\phantom{\rule{0ex}{0ex}}=\frac{100}{10100}=0.009900\mathrm{A}$

We can see that up to *R* = 100 Ω, the current does not change up to two significant digits.

#### Page No 199:

#### Question 23:

If the reading of the ammeter, *A*_{1} in the figure (32-E4) is 2.4 A, what will be the readings of ammeters *A*_{2}_{ }and *A*_{3}? Neglect the resistances of the ammeters.

figure 32-E4

#### Answer:

Given*: i _{1}* = 2.4 A

Since the 20 Ω and 30 Ω resistors are connected in parallel, the voltage across them will be the same.

From the figure,

$20{i}_{1}=30\left(i-{i}_{1}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow 50{i}_{1}=30i\phantom{\rule{0ex}{0ex}}\Rightarrow i=\frac{50{i}_{1}}{30}=3.99=4\mathrm{A}\phantom{\rule{0ex}{0ex}}$

∴ The reading of ammeter A3 =

*i*= 4 A

The reading of ammeter A2 =

*i*-

*i*= 1.6 A

_{1}#### Page No 199:

#### Question 24:

The resistance of the rheostat shown in the figure (32-E5) is 30 Ω. Neglecting the meter resistance, find the

figure32-E5

minimum and maximum currents through the ammeter as the rheostat is varied.

#### Answer:

Current will be minimum when the rheostat will have maximum resistance, i.e 30 Ω.

In this case, total resistance of the circuit = $\frac{20\times 10}{30}+30=36.67\mathrm{\Omega}$

The minimum current through the circuit,

${i}_{\mathrm{min}}=\frac{5.5}{36.67}=0.15\mathrm{A}$

Current will be maximum when the rheostat will have minimum resistance, i.e. 0 Ω.

In this case, total resistance of the circuit = $\frac{20\times 10}{30}=6.67\mathrm{\Omega}$

The maximum current through the circuit,

${i}_{\mathrm{max}}=\frac{5.5}{6.67}=0.825\mathrm{A}$

#### Page No 199:

#### Question 25:

Three bulbs, each with a resistance of 180 Ω, are connected in parallel to an ideal battery of emf 60 V.

Find the current delivered by the battery when (a) all the bulbs are switched on, (b) two of the bulbs are switched on and (c) only one bulb is switched on.

#### Answer:

(a) When all the bulbs are switched on, we have a combination of three bulbs in parallel.

The effective resistance of the circuit,

${R}_{\mathrm{eff}}=\frac{180}{3}=60\mathrm{\Omega}\phantom{\rule{0ex}{0ex}}$

The current delivered by the battery,

$i=\frac{60}{60}=1\mathrm{A}$

(b) When two of the bulbs are switched on, they are parallel to each other.

The effective resistance of the circuit,

${R}_{\mathrm{eff}}=\frac{180}{2}=90\mathrm{\Omega}$

The current delivered by the battery,

$i=\frac{60}{90}=0.67\mathrm{A}$

(c) When only one bulb is switched on, the effective resistance of the circuit,

${R}_{\mathrm{eff}}=180\mathrm{\Omega}\phantom{\rule{0ex}{0ex}}$

The current delivered by the battery,

$i=\frac{60}{180}=0.33\mathrm{A}$

#### Page No 199:

#### Question 26:

Suppose you have three resistors of 20 Ω, 50 Ω and 100 Ω. What minimum and maximum resistance can you obtain from these resistors?

#### Answer:

We can obtain maximum resistance when all the resistors are connected in series.

So, maximum resistance,

${R}_{\mathrm{max}}=\left(20+50+100\right)\mathrm{\Omega}=170\mathrm{\Omega}$

The minimum resistance will be obtained when all the resistors are connected in parallel

So, minimum resistance,

${R}_{\mathrm{min}}=\frac{1}{{\displaystyle \frac{1}{20}}+{\displaystyle \frac{1}{50}}+{\displaystyle \frac{1}{100}}}=\frac{100}{8}=12.5\mathrm{\Omega}$

#### Page No 199:

#### Question 27:

A bulb is made using two filaments. A switch selects whether the filaments are used individually or in parallel. When used with a 15 V battery, the bulb can be operated at 5 W, 10 W or 15 W. What should be the resistances of the filaments?

#### Answer:

We know:

$R=\frac{{V}^{2}}{P}$,

where R is the resistance, *V* is the voltage drop and *P* is the power on which the bulb is operated.

We can calculate the resistance of the bulb for the same *V* and different *P*.

The resistances of the bulb for three different powers are:

${R}_{1}=\frac{{\left(15\right)}^{2}}{5}=45\mathrm{\Omega}\phantom{\rule{0ex}{0ex}}{R}_{2}=\frac{{\left(15\right)}^{2}}{10}=22.5\mathrm{\Omega}\phantom{\rule{0ex}{0ex}}{R}_{3}=\frac{{\left(15\right)}^{2}}{15}=15\mathrm{\Omega}$

When the two resistance are used in parallel, the equivalent resistance will be less than their individual resistance.

∴ The two resistance are 45 Ω and 22.5 Ω.

#### Page No 199:

#### Question 28:

Figure (32-E6) shows a part of a circuit. If a current of 12 mA exists in the 5 kΩ resistor, find the currents in the other three resistors. What is the potential difference between the points *A* and *B*?

Figure32-E6

#### Answer:

Let the current flowing through the resistors be marked as shown in the figure.

Given*: i* = 12 mA

The resistors of 20 kΩ and 10 kΩ are connected in parallel. So, the potential difference across their ends will be same, i.e.

$20{i}_{1}=10\left(i-{i}_{1}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow 30{i}_{1}=10i\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{1}=\frac{i}{3}=\frac{12}{3}=4\mathrm{mA}$

∴ The current flowing through the 20 kΩ resistor = *i _{1}* = 4 mA

The current flowing through the 10 kΩ resistor =

*i*-

*i*= 8 mA

_{1}From the figure,

The current flowing through the 100 kΩ resistor =

*i*= 12 mA

The equivalent resistance between the points A and B,

${R}_{\mathrm{net}}=\left(5+\frac{20\times 10}{20+10}+100\right)\mathrm{k\Omega}=111.67\mathrm{k\Omega}$

The potential difference

*V*between the points A and B,

$V=i{R}_{\mathrm{net}}\phantom{\rule{0ex}{0ex}}\Rightarrow V=12\times {10}^{-3}\times 111.67\times {10}^{3}=1340\mathrm{V}$

#### Page No 199:

#### Question 29:

An ideal battery sends a current of 5 A in a resistor. When another resistor of 10 Ω is connected in parallel, the current through the battery is increased to 6 A. Find the resistance of the first resistor.

#### Answer:

Let the resistance of the first resistor be *R*.

If *V* is the potential difference across *R,* then the current through it,

$i=5=\frac{V}{R}$

Now, the other resistor of 10 Ω is connected in parallel with *R*.

It is given that the new value of current through the circuit,* i'* = 6 A

The effective resistance of the circuit, *R'* = $\frac{10R}{10+R}$

Since the potential difference is constant,

$iR=i\text{'}R\text{'}\phantom{\rule{0ex}{0ex}}\Rightarrow 5\times R=6\times \frac{10R}{10+R}\phantom{\rule{0ex}{0ex}}\Rightarrow \left(10+R\right)5=60\phantom{\rule{0ex}{0ex}}\Rightarrow 5R=10\phantom{\rule{0ex}{0ex}}\Rightarrow R=2\mathrm{\Omega}$

#### Page No 199:

#### Question 30:

Find the equivalent resistance of the network shown in the figure (32-E7) between the points *a* and *b*.

figure (32-E7)

#### Answer:

The simplified circuit can be drawn as shown below.

Equivalent resistance $=\frac{1}{{\displaystyle \frac{1}{r}}+{\displaystyle \frac{1}{r}}+{\displaystyle \frac{1}{r}}}=\frac{r}{3}$

#### Page No 199:

#### Question 31:

A wire of resistance 15.0 Ω is bent to form a regular hexagon *ABCDEFA*. Find the equivalent resistance of the loop between the points (a)* A *and *B* (b)* A* and *C* and (c) *A* and *D*.

#### Answer:

(a)

From the figure, it can be seen that between points A and B, the resistance of the first side of the hexagon will be in parallel with the total resistance of the other five sides.

The resistance of the first side can be calculated as shown below.

Resistance of the first leg=$\frac{\text{L}\mathrm{ength}\mathrm{of}\mathrm{the}\mathrm{that}\mathrm{portion}}{\text{T}\mathrm{otal}\mathrm{length}}\times R=\frac{{\displaystyle \frac{1}{6}}}{{\displaystyle \frac{5}{6}}+{\displaystyle \frac{1}{6}}}\times 15=15\times \frac{1}{6}$

Total resistance of the 5 legs=$\frac{{\displaystyle \frac{5}{6}}}{{\displaystyle \frac{1}{6}}+{\displaystyle \frac{5}{6}}}\times 15=\frac{5}{6}\times 15$

∴ The effective resistance between the points A and B,

${R}_{\mathrm{eff}}=\frac{{\displaystyle \frac{15\times 5}{6}}\times {\displaystyle \frac{15}{6}}}{{\displaystyle \frac{15\times 5}{6}}+{\displaystyle \frac{15}{6}}}=\frac{{\displaystyle \frac{15\times 5\times 15}{6\times 6}}}{{\displaystyle \frac{75+15}{6}}}\phantom{\rule{0ex}{0ex}}=\frac{15\times 5\times 15}{6\times 90}=\frac{25}{12}\phantom{\rule{0ex}{0ex}}=2.08\mathrm{\Omega}$

(b) From the figure, it can be seen that between points A and C, the resistance of the first two sides of the hexagon will be in parallel with the total resistance of the other four sides.

Resistance of the two legs =$\frac{\text{L}\mathrm{ength}\mathrm{of}\mathrm{the}\mathrm{that}\mathrm{portion}}{\text{T}\mathrm{otal}\mathrm{length}}\times R=\frac{{\displaystyle \frac{2}{6}}}{{\displaystyle \frac{4}{6}}+{\displaystyle \frac{2}{6}}}\times 15=15\times \frac{2}{6}$

Total resistance of the four legs=$\frac{{\displaystyle \frac{4}{6}}}{{\displaystyle \frac{2}{6}}+{\displaystyle \frac{4}{6}}}\times 15=\frac{4}{6}\times 15$

∴ The effective resistance between the points A and C,

${\mathrm{R}}_{eff}=\frac{{\displaystyle \frac{15\times 4}{6}}\times {\displaystyle \frac{15\times 2}{6}}}{{\displaystyle \frac{15\times 4}{6}}+{\displaystyle \frac{15\times 2}{6}}}=\frac{{\displaystyle \frac{15\times 4\times 2\times 15}{6\times 6}}}{{\displaystyle \frac{60+30}{6}}}\phantom{\rule{0ex}{0ex}}=\frac{15\times 2\times 4\times 15}{6\times 90}\phantom{\rule{0ex}{0ex}}=\frac{10}{3}=3.33\mathrm{\Omega}$

(c) From the figure, it can be seen that between points A and D, the resistance of the first three sides of the hexagon will be in parallel with the total resistance of the other three sides.

Resistance of the three legs = $\frac{\text{L}\mathrm{ength}\mathrm{of}\mathrm{the}\mathrm{that}\mathrm{portion}}{\text{T}\mathrm{otal}\mathrm{length}}\times R=\frac{{\displaystyle \frac{3}{6}}}{{\displaystyle \frac{3}{6}}+{\displaystyle \frac{3}{6}}}\times 15=15\times \frac{3}{6}$

∴ The effective resistance between the points A and D,

${\mathrm{R}}_{\mathrm{eff}}=\frac{{\displaystyle \frac{15\times 3}{6}}\times {\displaystyle \frac{15\times 3}{6}}}{{\displaystyle \frac{15\times 3}{6}}+{\displaystyle \frac{15\times 3}{6}}}=\frac{{\displaystyle \frac{15\times 3\times 3\times 15}{6\times 6}}}{{\displaystyle \frac{90}{6}}}\phantom{\rule{0ex}{0ex}}=\frac{15\times 3\times 3\times 15}{6\times 90}=\frac{15}{4}=3.75\mathrm{\Omega}$

#### Page No 199:

#### Question 32:

Consider the circuit shown in the figure (32-E8). Find the current through the 10 Ω resistor when the switch *S *is (a) open (b) closed.

figure32-E8

#### Answer:

(a) When *S* is open, the resistors will be connected in series, as the current will flow through the 20 Ω resistor. The effective resistance,

${\mathrm{R}}_{\mathrm{eff}}=\left(10+20\right)\mathrm{\Omega}=30\mathrm{\Omega}\phantom{\rule{0ex}{0ex}}$

The current flowing through the circuit,

$i=\left(\frac{3}{30}\right)\mathrm{A}=0.1\mathrm{A}$

(b) When *S* is closed, current will flow through the switch that offers the least resistive path. So, the 20 Ω resistance will be ineffective in this case. The effective resistance in this case,

${R}_{\mathrm{eff}}=10\mathrm{\Omega}\phantom{\rule{0ex}{0ex}}$

The current flowing through the circuit,

$i=\left(\frac{3}{10}\right)\mathrm{A}=0.3\mathrm{A}$

#### Page No 200:

#### Question 33:

Find the currents through the three resistors shown in the figure (32-E9).

Figure32-E9

#### Answer:

From the figure, we can see that current will flow only in loop 1 because current follows the least resistive path. All the current will pass through the wire connected in parallel to the 4 Ω resistor in loop 2.

∴ There will be no current through the 4 Ω resistor in loop 2.

For loop1, applying KVL, we get:

$6i+4i-4+2=0\phantom{\rule{0ex}{0ex}}\Rightarrow 10i=2\phantom{\rule{0ex}{0ex}}\Rightarrow i=0.2\mathrm{A}\phantom{\rule{0ex}{0ex}}$

∴ The current through the 4 Ω and 6 Ω resistors, *i* = 0.2 A

#### Page No 200:

#### Question 34:

Figure (32-E10) shows a part of an electric circuit. The potentials at the points *a, b* and *c* are 30 V, 12 V and 2 V respectively. Find the currents through the three resistors.

Figure 32-E10

#### Answer:

Let the potential at the point *o* be *X* volts.

From the figure,

${i}_{1}=\frac{{V}_{\mathrm{a}}-{V}_{\mathrm{o}}}{10}\phantom{\rule{0ex}{0ex}}{V}_{\mathrm{a}}=30\mathrm{V}\mathrm{and}{V}_{\mathrm{o}}=X\phantom{\rule{0ex}{0ex}}So,{i}_{1}=\frac{30-X}{10}\phantom{\rule{0ex}{0ex}}\mathrm{Similarly},\phantom{\rule{0ex}{0ex}}{i}_{2}=\frac{{V}_{\mathrm{o}}-{V}_{\mathrm{b}}}{20}\phantom{\rule{0ex}{0ex}}=\frac{X-12}{20}\phantom{\rule{0ex}{0ex}}\mathrm{And}\phantom{\rule{0ex}{0ex}}{i}_{3}=\frac{{V}_{\mathrm{o}}-{V}_{\mathrm{c}}}{30}\phantom{\rule{0ex}{0ex}}=\frac{X-2}{30}$

Also, from kirchoff's junction law we have:

*i*_{1} = *i*_{2} + *i*_{3}

$\Rightarrow \frac{30-X}{10}=\frac{X-12}{20}+\frac{X-2}{30}\phantom{\rule{0ex}{0ex}}\Rightarrow 30-X=\frac{X-12}{2}+\frac{X-2}{3}\phantom{\rule{0ex}{0ex}}\Rightarrow 30-X=\frac{3X-36+2X-4}{6}\phantom{\rule{0ex}{0ex}}\Rightarrow 180-6X=5X-40\phantom{\rule{0ex}{0ex}}\Rightarrow 11X=220\phantom{\rule{0ex}{0ex}}\Rightarrow X=\frac{220}{11}=20\mathrm{V}$

Thus, the currents through the three resistors are:

${i}_{1}=\frac{30-20}{10}=1\mathrm{A}\phantom{\rule{0ex}{0ex}}{i}_{2}=\frac{20-12}{20}=\frac{8}{20}=0.4\mathrm{A}\phantom{\rule{0ex}{0ex}}{i}_{3}=\frac{20-2}{30}=\frac{18}{30}=0.6\mathrm{A}$

#### Page No 200:

#### Question 35:

Each of the resistors shown in the figure (32-E11) has a resistance of 10 Ω and each of the batteries has an emf of 10 V. Find the currents flowing through the resistors* a* and *b* in the two circuits.

figure 32-E11

#### Answer:

(*a*)

From the figure,

Potential difference between the two ends of resistor *a* = 10 V

Resistance of resistor *a* =10 Ω

∴ Current through *a* = $\frac{10}{10}=1\mathrm{A}$

Since the two batteries are connected opposite each other,

potential difference between the terminals of *b* = 10 – 10 = 0 V

∴ Current through *b* = 0 A

(b)

From the figure,

Potential difference between the two ends of resistor *a* = 10 V.

Resistance of resistor *a* =10 Ω

∴ Current through *a* = $\frac{10}{10}=1\mathrm{A}$

Since the two batteries are again connected opposite to each other,

Potential difference between terminal of *b* = 10 – 10 = 0 V

∴ Current through *b* = 0 A

#### Page No 200:

#### Question 36:

Find the potential difference *V*_{a} – *V*_{b} in the circuits shown in the figure (32-E12).

figure 32-12

#### Answer:

Applying KVL in loop 1, we get:

${i}_{1}{R}_{2}-{E}_{2}+\left({i}_{1}+{i}_{2}\right){R}_{3}=0\phantom{\rule{0ex}{0ex}}({R}_{2}+{R}_{3}){i}_{1}+{R}_{3}{i}_{2}={E}_{2}...\left(1\right)$

Applying KVL in loop 2, we get:

${i}_{2}{R}_{1}-{E}_{1}+\left({i}_{1}+{i}_{2}\right){R}_{3}=0\phantom{\rule{0ex}{0ex}}\left({R}_{1}+{R}_{3}\right){i}_{2}+{R}_{3}{i}_{1}={E}_{1}...\left(2\right)$

Multiplying equation (1) by (*R _{1}*+

*R*) and (2) by

_{3}*R*and then subtracting (2) from (1), we get:

_{3}${i}_{1}=\frac{{E}_{2}\left({R}_{1}+{R}_{3}\right)-{E}_{1}{R}_{3}}{\left({R}_{1}{R}_{2}+{R}_{2}{R}_{3}+{R}_{3}{R}_{1}\right)}$

Similarly, multiplying equation (1) by

*R*and (2) by (

_{3}*R*+

_{1}*R*), and then subtracting (2) from (1), we get:

_{3}${i}_{2}=\frac{{E}_{1}\left({R}_{2}+{R}_{3}\right)-{E}_{2}{R}_{3}}{\left({R}_{1}{R}_{2}+{R}_{2}{R}_{3}+{R}_{3}{R}_{1}\right)}$

From the figure,

${V}_{\mathrm{a}}-{V}_{\mathrm{b}}=\left({i}_{1}+{i}_{2}\right){R}_{3}\phantom{\rule{0ex}{0ex}}\Rightarrow {V}_{\mathrm{a}}-{V}_{\mathrm{b}}=\left[\frac{{E}_{1}{R}_{2}+{E}_{2}{R}_{1}}{\left({R}_{1}{R}_{2}+{R}_{2}{R}_{3}+{R}_{3}{R}_{1}\right)}\right]{R}_{3}\phantom{\rule{0ex}{0ex}}\Rightarrow {V}_{\mathrm{a}}-{V}_{\mathrm{b}}=\frac{{\displaystyle \frac{{E}_{1}}{{R}_{1}}}+{\displaystyle \frac{{E}_{2}}{{R}_{2}}}}{{\displaystyle \frac{1}{{R}_{1}}}+{\displaystyle \frac{1}{{R}_{2}}}+{\displaystyle \frac{1}{{R}_{3}}}}$

(

*b*)

*The circuit in figure b can be redrawn as shown below:*

We can see that it is similar to the circuit in figure a and, hence, the answer obtained will be same.

#### Page No 200:

#### Question 37:

In the circuit shown in the figure (32-E13), *ε*_{1} = 3 V, *ε*_{2} = 2 V, *ε*_{a} = 1 V and *r*_{1} =* **r*_{2}_{ }= *r*_{3} = 1Ω. Find the potential difference between the points *A* and* B* and the current through each branch.

figure 32-E13

#### Answer:

Applying KVL in loop 1, we get:

${i}_{1}+\left({i}_{1}-{i}_{2}\right)-1+2=0\phantom{\rule{0ex}{0ex}}2{i}_{1}-{i}_{2}=-1...\left(1\right)\phantom{\rule{0ex}{0ex}}$

Similarly, for loop 2:

${i}_{2}-\left({i}_{1}-{i}_{2}\right)-2+3=0\phantom{\rule{0ex}{0ex}}2{i}_{2}-{i}_{1}=-1...\left(2\right)$

Solving (1) and (2), we get:

${i}_{1}={i}_{2}=1\mathrm{A}\phantom{\rule{0ex}{0ex}}$

Potential difference between A and B:

${V}_{\mathrm{A}}-{V}_{\mathrm{B}}={E}_{2}-\left({i}_{1}-{i}_{2}\right)\times 1=2-\left(1-1\right)=2\mathrm{V}$

Current through the top branch = *i _{2}* = 1 A

Current through the middle branch =

*i*-

_{1}*i*= 0 A

_{2}Current through the bottom branch =

*i*= 1 A

_{1}#### Page No 200:

#### Question 38:

Find the current through the 10 Ω resistor shown in the figure (32-E14).

Figure 32-E14

#### Answer:

Applying KVL in loop 1, we get:

3*i** *+ 6*i*_{1} = 4.5 ...(1)

Applying KVL in loop 2, we get:

$\left(i-{i}_{1}\right)10+3-6{i}_{1}=0\phantom{\rule{0ex}{0ex}}10i-16{i}_{1}=-3...\left(2\right)\phantom{\rule{0ex}{0ex}}$

Multiplying equation (1) by 10 and (2) by 3 and then, subtracting (2) from (1), we get:

$-108{i}_{1}=-54\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{1}=\frac{54}{108}=\frac{1}{2}=0.5\phantom{\rule{0ex}{0ex}}$

Substituting the value of* i _{1}* in (1), we get:

$3i+6\times \frac{1}{2}-4.5=0\phantom{\rule{0ex}{0ex}}3i-1.5=0\phantom{\rule{0ex}{0ex}}\Rightarrow i=\frac{1.5}{3}=0.5$

So, current flowing through the 10 Ω resistor =

*i*-

*i*= 0.5 - 0.5 = 0 A

_{1}#### Page No 200:

#### Question 39:

Find the circuit in the three resistors shown in the figure (32-E15).

figure 32-E15

#### Answer:

Applying KVL in loop 1, we get:

$2+\left({i}_{1}-{i}_{2}\right)\times 1-2=0\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{1}={i}_{2}$

Applying KVL in loop 2, we get:

$2+\left({i}_{2}-{i}_{3}\right)\times 1-2-\left({i}_{1}-{i}_{2}\right)\times 1=0\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{2}-{i}_{3}-{i}_{1}+{i}_{2}=0\phantom{\rule{0ex}{0ex}}{i}_{1}={i}_{2}\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{2}-{i}_{3}-{i}_{2}+{i}_{2}=0\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{2}={i}_{3}$

Applying KVL in loop 3, we get:

$2+{i}_{3}-2-\left({i}_{2}-{i}_{3}\right)=0\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{3}=0\phantom{\rule{0ex}{0ex}}{i}_{1}={i}_{2}={i}_{3}\phantom{\rule{0ex}{0ex}}\therefore {i}_{1}={i}_{2}={i}_{3}=0$

#### Page No 200:

#### Question 40:

What should be the value of* R* in the figure (32-E16) for which the current in it is zero?

figure 32-E16

#### Answer:

For current to be zero in the middle branch, the circuit should be balanced. According to the Wheatstone bridge principle, balancing a circuit does not depend on the value of resistance R in the middle branch. The circuit in this case is balanced. Hence, any value of R is acceptable.

#### Page No 200:

#### Question 41:

Find the equivalent resistance of the circuits shown in the figure (32-E17) between the points *a *and *b. *Each resistor has resistance *r*.

figure 32-E17

#### Answer:

(a) The simplified circuit can be drawn as shown below.

Here cdeo forms a balanced Wheatstone bridge; therefore, branch od will become become ineffective.

The simplified circuit will then be as shown below.

The equivalent resistance between points c and e,

${R}_{cd}=\frac{2r\times 2r}{4r}=r$

The equivalent resistance between a and b,

${R}_{ab}=\frac{r\times r}{2r}=\frac{r}{2}$

(b)

Let R_{eff} be the effective resistance of the circuit.Then, from the symmetry of the circuit, we can assume that the current moving along CO enters OB and the current moving along EO enters OD.

Current on CO = current on OB

Current on EO = current on OD

So, the circuit can be simplified as shown below.

From the simplified circuit diagram, effective resistance of the upper half of the circuit will be

$\left[\left[r+\left(\frac{2{r}^{2}}{3r}\right)+r\right]=2r+\left(\frac{2r}{3}\right)=\frac{8}{3}r\right]\phantom{\rule{0ex}{0ex}}\Rightarrow {\mathrm{R}}_{\mathrm{eff}}=\frac{{\displaystyle \frac{8r}{6}}\times 2r}{{\displaystyle \frac{8r}{6}}+2r}\phantom{\rule{0ex}{0ex}}=8{r}^{2}\times \frac{2}{20r}=\frac{8r}{10}=\frac{4r}{5}$

#### Page No 200:

#### Question 42:

Find the current measured by the ammeter in the circuit shown in the figure (32-E18).

figure 32-E18

#### Answer:

From the symmetry of the circuit,we can see that the potential at point C will be equal to the potential at point F. This implies that no current will flow in this branch. Hence, this arm of the circuit can be removed.

Similarly, the potential at point B will be equal to the potential at point G. So, this arm of the circuit can also be removed.

The simplified circuit diagram will be as shown below.

(b)

From the figure, the effective resistance of the circuit,

${R}_{\mathrm{eff}}=\frac{30\times 30}{30+30}=15\mathrm{\Omega}$

The current through the circuit,

$i=\frac{6}{15}=0.4\mathrm{A}$

So, the ammeter reading will be 0.4 A.

#### Page No 201:

#### Question 43:

Consider the circuit shown in the figure (32-E19a). Find (a) the current in the circuit (b) the potential drop across the 5 Ω resistor (c) the potential drop across the 10 Ω resistor (d) Answer the parts (a), (b) and (c) with reference to the figure (32-E19b).

figure 32-E19

#### Answer:

(a)

Applying KVL in the above loop, we get:

$10i+6+5i+12=0\phantom{\rule{0ex}{0ex}}\Rightarrow 10i+5i=-18\phantom{\rule{0ex}{0ex}}\Rightarrow 15i=-18\phantom{\rule{0ex}{0ex}}\Rightarrow i=-\frac{18}{15}=-\frac{6}{5}=-1.2\mathrm{A}$

The negative sign indicates that current is flowing in the direction opposite to our assumed direction.

(b) Potential drop across the 5 Ω resistor= 5*i** *= 5×(-1.2 ) = -6 V

(c) Potential drop across the 10 Ω resistor = 10*i* = (-1.2) × 10 = 12 V

(d)

Applying KVL in the above loop, we get:

$10i+5i+6+12=0\phantom{\rule{0ex}{0ex}}\Rightarrow 15i=-18\phantom{\rule{0ex}{0ex}}\Rightarrow i=-1.2\mathrm{A}$

* *

Potential drop across the 5Ω register = -6 V

Potential drop across the 10Ω register = -12 V

#### Page No 201:

#### Question 44:

Twelve wires, each of equal resistance *r*, are joined to form a cube, as shown in the figure (32-E20). Find the equivalent resistance between the diagonally-opposite points *a *and *f*.

figure 32-E20

#### Answer:

Let *V* be the potential difference between the points a and f. Let current *i *enter a and leave from f. The distribution of current in various branches is shown in figure.

To calculate the potential difference between a and f, consider the path abcf and apply Kirchofff's Law:

$\frac{i}{3}r+\frac{i}{6}r+\frac{i}{3}r=V$

$\Rightarrow \left(\frac{2ir}{3}+\frac{ir}{6}\right)=V\phantom{\rule{0ex}{0ex}}\Rightarrow \left(\frac{5ir}{6}\right)=V\phantom{\rule{0ex}{0ex}}$

The effective resistance between a and f,

${R}_{\mathrm{eff}}\mathit{}=\frac{V}{i}=\frac{5}{6}r$

#### Page No 201:

#### Question 45:

Find the equivalent resistances of the networks shown in the figure (32-E21) between the points* a* and *b*.

figure 32-E21

#### Answer:

(a) The circuit can be simplified stepwise, as shown below.

The effective resistance between the points a and b,

${\mathrm{R}}_{\mathrm{eff}}=\frac{{\displaystyle \frac{5r}{3}}\times r}{{\displaystyle \frac{5r}{3}}+r}=\frac{5r}{8}\phantom{\rule{0ex}{0ex}}$

(b) The circuit can be simplified, as shown below.

The effective resistance between the points a and b,

${\mathrm{R}}_{\mathrm{eff}}=\left(\frac{r}{3}\right)+r=\frac{4r}{3}\phantom{\rule{0ex}{0ex}}$

(c)

From the figure, it can be seen that axbya is a balanced Wheatstone bridge. The resistors in branch xy will, thus, become ineffective. The circuit can be simplified as under

The effective resistance between the points a and b,

${\mathrm{R}}_{\mathrm{eff}}=\left(\frac{2r\times 2r}{2r+2r}\right)=r$

(d) The circuit can be simplified as shown below.

The effective resistance between the points a and b,

${\mathrm{R}}_{\mathrm{eff}}=\frac{r}{4}$

(e) The circuit can be redrawn as shown below.

Now, we can see that the circuit is a balanced Wheatstone bridge. So, the branch xy will become ineffective. Thus, the simplified circuit will become as shown below.

The effective resistance between the points a and b,

${\mathrm{R}}_{\mathrm{eff}}=\left(\frac{2r\times 2r}{2r+2r}\right)=r$

#### Page No 201:

#### Question 46:

An infinite ladder is constructed with 1 Ω and 2 Ω resistors, as shown in the figure (32-E22). (a) Find the effective resistance between the points *A* and *B*. (b) Find the current that passes through the 2 Ω resistor nearest to the battery.

figure 32-E22

#### Answer:

(a) Let the effective resistance of the combination be *R*. The circuit can be redrawn as shown below.

From the figure,

$\frac{2R}{R+2}+1=R\phantom{\rule{0ex}{0ex}}\Rightarrow 3R+2={R}^{2}+2R\phantom{\rule{0ex}{0ex}}\Rightarrow {R}^{2}-R-2=0\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{+1+\sqrt{1+4\times 1\times 2}}{2\times 1}\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{+1+\sqrt{9}}{2.1}=2\mathrm{\Omega}$

(b) Total current sent by the battery $=\frac{6}{R}=\frac{6}{2}=3\mathrm{A}$

Applying Kirchoff's Law in loop 1, we get:

$3\times 1+2i=6\phantom{\rule{0ex}{0ex}}\Rightarrow 2i=3\phantom{\rule{0ex}{0ex}}\Rightarrow i=\frac{3}{2}=1.5\mathrm{A}$

#### Page No 201:

#### Question 47:

The emf ε and the internal resistance r of the battery, shown in the figure (32-E23), are 4.3 V and 1.0 Ω respectively. The external resistance R is 50 Ω. The resistances of the ammeter and voltmeter are 2.0 Ω and 200 Ω respectively. (a) Find the readings of the two 200 Ω respectively. (a) Find the readings of the two meters. (b) The switch is thrown to the other side. What will be the readings of the two meters now?

figure 32-E23

#### Answer:

(a)

The 50 Ω and 200 Ω resistances are in parallel. Their equivalent resistance,

${R}_{\mathrm{eqv}}=\frac{50\times 200}{50+200}=40\mathrm{\Omega}$

This equivalent resistance and the 2 Ω and 1 Ω resistances are connected in series. The effective resistance of the circuit,

${R}_{\mathrm{eff}}=\left(40+2+1\right)=43\mathrm{\Omega}$

In this case, the ammeter will read the total current of the circuit. The current through the ammeter,

$i=\frac{4.3}{43}=0.1\mathrm{A}$

This current will be distributed in the inverse ratio of resistance between the resistances 50 Ω and 200 Ω. The current through the voltmeter,

$i\text{'}=\frac{50}{\left(50+200\right)}i\phantom{\rule{0ex}{0ex}}\Rightarrow i\text{'}=0.02\mathrm{A}$

Reading of the voltmeter = 0.02 × 200 = 4 V

(b)

b

The two branches AB and CD are in parallel. Their equivalent resistance,

${R}_{\mathrm{eqv}}=\frac{52\times 200}{52+200}=41.27\mathrm{A}$

This equivalent resistance is in series with the 1 Ω resistance. The effective resistance of the circuit,

${R}_{\mathrm{eff}}=\left(41.27+1\right)\mathrm{\Omega}=42.27\mathrm{\Omega}$

The total current through the circuit,

$i=\frac{4.3}{42.27}=0.1\mathrm{A}$

In this case, the ammeter will read the current flowing through the 50 Ω resistance, which is *i*_{1}, as shown. The currents in the two parallel branches will distribute in the inverse ratio of the resistances.

$\therefore {i}_{1}=\left(\frac{200}{200+52}\right)i\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{i}=0.08\mathrm{A}$

The current through the voltmeter = *i *- *i _{1}* = 0.02 A

The reading of the voltmeter = 0.02 × 200 = 4 V

#### Page No 201:

#### Question 48:

A voltmeter of resistance 400 Ω is used to measure the potential difference across the 100 Ω resistor in the circuit shown in the figure (32-E24). (a) What will be the reading of the voltmeter? (b) What was the potential difference across 100 Ω before the voltmeter was connected?

figure 32-E24

#### Answer:

(a) The effective resistance of the circuit,

${\mathrm{R}}_{\mathrm{eff}}=\frac{100\times 400}{500}+200=280\mathrm{\Omega}\phantom{\rule{0ex}{0ex}}$

The current through the circuit,

$i=\frac{84}{280}=0.3\mathrm{A}$

Since 100 Ω resistor and 400 Ω resistor are connected in parallel, the potential difference will be same across their ends. Let the current through 100 Ω resistor be *i _{1}* ; then, the current through 400 Ω resistor will be

*i*-

*i*.

_{1}$100{i}_{1}=400\left(i-{i}_{1}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow 500{i}_{1}=400i\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{i}=\frac{4}{5}i=0.24\mathrm{A}$

The reading of the voltmeter = 100 × 0.24 = 24 V

(b) Before the voltmeter is connected, the two resistors 100 Ω resistor and 200 Ω resistor are in series.

The effective resistance of the circuit,

${R}_{\mathrm{eff}}=\left(200+100\right)\phantom{\rule{0ex}{0ex}}=300\mathrm{\Omega}\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

The current through the circuit,

$i=\frac{84}{300}=0.28\mathrm{A}$

∴ Voltage across the 100 Ω resistor = (0.28 × 100) = 28 V

#### Page No 201:

#### Question 49:

The voltmeter shown in the figure (32-E25) reads 18 V across the 50 Ω resistor. Find the resistance of the voltmeter.

figure 32-E25

#### Answer:

Let *R *be the resistance of the voltmeter. For the given circuit, 50 Ω and *R *are in parallel. The equivalent resistance of these two resistors,

${R}_{\mathrm{eqv}}=\frac{50R}{50+R}$

This equivalent resistance is now in series with the 24 Ω resistor. The effective resistance of the circuit,

${R}_{\mathrm{eff}}=\left(\frac{50R}{50+R}+24\right)\mathrm{\Omega}$

Current *t*hrough the circuit,

$i=\frac{30}{{R}_{\mathrm{eff}}}$

Voltage across the 24 Ω resistor, *V _{1}* = $i\times 24=\frac{30}{{R}_{\mathrm{eff}}}\times 24\mathrm{V}$

Voltage across the 50 Ω resistor,

*V*

_{2}_{ }= $30-{V}_{1}=30-\frac{30}{{R}_{\mathrm{eff}}}\times 24\mathrm{V}(\because \mathit{}{V}_{1}+{V}_{2}=30)$

It is given that

*V*= 18 V

_{2}$\Rightarrow 18=30-\frac{30}{{R}_{eff}}\times 24\phantom{\rule{0ex}{0ex}}\Rightarrow 18=30-\frac{30}{\left({\displaystyle \frac{50R}{50+R}}+24\right)}\times 24\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{30}{\left({\displaystyle \frac{74R+1200}{50+R}}\right)}\times 24=30-18\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{30}{\left({\displaystyle \frac{74R+1200}{50+R}}\right)}=\frac{12}{24}=\frac{1}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow 60\left(50+R\right)=\left(74R+1200\right)\phantom{\rule{0ex}{0ex}}\Rightarrow R\approx 130\mathrm{\Omega}\phantom{\rule{0ex}{0ex}}$

#### Page No 201:

#### Question 50:

A voltmeter consists of a 25 Ω coil connected in series with a 575 Ω resistor. The coil takes 10 mA for full-scale deflection. What maximum potential difference can be measured by this voltmeter?

#### Answer:

It is given that for maximum current, *i* = 10 mA, the potential drop across the voltmeter will be maximum.

The effective resistance of the circuit,

${R}_{\mathrm{eff}}=\left(575+25\right)\mathrm{\Omega}=600\mathrm{\Omega}$

The maximum value of potential difference measured,

*V* = R_{eff} × *i*

= 600 × 10 × 10^{−3} = 6 V

#### Page No 201:

#### Question 51:

An ammeter is to be constructed that can read currents up to 2.0 A. If the coil has resistance of 25 Ω and takes 1 mA for full-scale deflection, what should be the resistance of the shunt used?

#### Answer:

Let *R* be the resistance of the shunt used.

Coil resistance*, g *= 25 Ω

Current through the coil, *i*_{g} = 1 mA

Total current *i* through the circuit in which the ammeter is connected = 2 A

Since *R* and *g* are connected in parallel,

$g\times {i}_{\mathrm{g}}=R\left(i-{i}_{\mathrm{g}}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{g\times {i}_{\mathrm{g}}}{i-{i}_{\mathrm{g}}}\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{25\times 1\times {10}^{-3}}{2-{10}^{-3}}\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{R}=1.25\times {10}^{-2}\mathrm{\Omega}$

#### Page No 201:

#### Question 52:

A voltmeter coil has resistance 50.0 Ω and a resistor of 1.15 kΩ is connected in series. It can read potential differences up to 12 volts. If this same coil is used to construct an ammeter that can measure currents up to 2.0 A, what should be the resistance of the shunt used?

#### Answer:

The effective resistance for the voltmeter,

*R*_{eff}_{ }= (1150 + 50) Ω = 1200 Ω

The maximum current *i _{g}_{ }*through the coil for maximum deflection of 12 V,

${i}_{\mathrm{g}}=\left(\frac{12}{1200}\right)\mathrm{A}=0.01\mathrm{A}$

Let

*R*be the resistance of the shunt used for ammeter.

Resistance of the coil, g = 50 Ω

Maximum deflection current,

*i,*through the ammeter = 2 A

Since

*R*and

*g*are connected in parallel,

$g\times {i}_{\mathrm{g}}=R\times \left(i-{i}_{\mathrm{g}}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{g\times {i}_{\mathrm{g}}}{\left(i-{i}_{\mathrm{g}}\right)}\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{50\times 0.01}{2-0.01}=0.251\mathrm{\Omega}$

#### Page No 201:

#### Question 53:

The potentiometer wire AB shown in the figure (32-E26) is 40 cm long. Where should the free end of the galvanometer be connected on AB, so that the galvanometer may show zero deflection?

Figure

#### Answer:

Let D be the null-point of the potentiometer. Since the bridge is balanced at this point,

$\frac{{R}_{\mathrm{AD}}}{{R}_{\mathrm{DB}}}=\frac{8}{12}$

According to the principle of a potentiometer,

$\frac{{R}_{\mathrm{AD}}}{{R}_{\mathrm{DB}}}=\frac{{l}_{\mathrm{AD}}}{{l}_{\mathrm{DB}}}=\frac{8}{12}=\frac{2}{3}\phantom{\rule{0ex}{0ex}}\Rightarrow {l}_{\mathrm{AD}}=\frac{2}{3}{l}_{\mathrm{DB}}\phantom{\rule{0ex}{0ex}}{l}_{\mathrm{AD}}+{l}_{\mathrm{DB}}=40\mathrm{cm}\phantom{\rule{0ex}{0ex}}\Rightarrow {l}_{\mathrm{DB}}\frac{2}{3}+{l}_{\mathrm{DB}}=40\mathrm{cm}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{5}{3}{l}_{\mathrm{DB}}=40\mathrm{cm}\phantom{\rule{0ex}{0ex}}\Rightarrow {l}_{\mathrm{DB}}=40\times \frac{3}{5}=24\mathrm{cm}$

Hence, the null-point is obtained 24 cm from B.

#### Page No 202:

#### Question 54:

The potentiometer wire AB shown in the figure (32−E27) is 50 cm long. When AD = 30 cm, no deflection occurs in the galvanometer. Find *R*.

Figure

#### Answer:

It is given that at point D, there is no deflection in the galvanometer, i.e. the bridge is balanced

$\Rightarrow \frac{{R}_{\mathrm{AD}}}{{R}_{\mathrm{DB}}}=\frac{6}{R}$

According to the principle of a potentiometer,

$\frac{{R}_{\mathrm{AD}}}{{R}_{\mathrm{DB}}}=\frac{{l}_{\mathrm{AD}}}{{l}_{\mathrm{DB}}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{6}{R}=\frac{30}{20}\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{20\times 6}{30}=4\mathrm{\Omega}$

#### Page No 202:

#### Question 55:

A 6-volt battery of negligible internal resistance is connected across a uniform wire AB of length 100 cm. The positive terminal of another battery of emf 4 V and internal resistance 1 Ω is joined to the point A, as shown in the figure (32-E28). Take the potential at B to be zero. (a) What are the potentials at the points A and C? (b) At which point D of the wire AB, the potential is equal to the potential at C? (c) If the points C and D are connected by a wire, what will be the current through it? (d) If the 4 V battery is replaced by a 7.5 V battery, what would be the answers of parts (a) and (b)?

Figure

#### Answer:

(a) Potential difference across AB = Potential at A - Potential at B

Potential at B = 0 V

â‡› Potential at point A = Potential difference across AB = 6 V

Potential difference across AC = Potential at A - Potential at B

â‡› 4 = 6 - Potential at C

â‡› Potential at C = 2 V

(b) Given:

Potential across AD = Potential across AC = 4 V

â‡› Potential across DB = 2 V

$\frac{{V}_{\mathrm{AD}}}{{V}_{\mathrm{BD}}}=\frac{{l}_{\mathrm{AD}}}{{l}_{\mathrm{DB}}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{4}{2}=\frac{{l}_{\mathrm{AD}}}{100-{l}_{\mathrm{AD}}}\phantom{\rule{0ex}{0ex}}\Rightarrow 4\left(100-{l}_{\mathrm{AD}}\right)=2{l}_{\mathrm{AD}}\phantom{\rule{0ex}{0ex}}\Rightarrow 6{l}_{\mathrm{AD}}=400\phantom{\rule{0ex}{0ex}}\Rightarrow {l}_{\mathrm{AD}}=\frac{400}{6}=66.7\mathrm{cm}$

(c) When the points C and D are connected by a wire, current flowing through the wire will be zero because the points are at the same potential.

(d) Potential difference across AC = Potential at A - Potential at C

â‡› 7.5 = 6 - Potential at C

â‡› Potential at C = −1.5 V

Since the potential at C is negative now, this point will go beyond point B, which is at 0 V. Hence, no such point D will exist between the points A and B.

#### Page No 202:

#### Question 56:

Consider the potentiometer circuit as arranged in the figure (32−E29). The potentiometer wire is 600 cm long. (a) At what distance from the point A should the jockey touch the wire to get zero deflection in the galvanometer? (b) If the jockey touches the wire at a distance of 560 cm from A, what will be the current in the galvanometer?

Figure

#### Answer:

Let X be the null point on the wire at a distance *x* cm from point A, as shown.

Given:

Total resistance of the wire AB = 15*r*

Resistance per unit cm = $\frac{15r}{600}$

Resistance of *x* cm of the wire = $\frac{15rx}{600}$

Resistance of (600 - *x *) cm of the wire = $\frac{15r\left(600-x\right)}{600}$

(a) Applying KVL in loop 1, we get:

$\left({i}_{1}+{i}_{2}\right)\frac{15}{600}rx+\frac{15}{600}r\left(600-x\right){i}_{1}+{i}_{1}r=\epsilon ...\left(\mathrm{i}\right)$

Applying KVL in loop 2, we get:

${i}_{2}r+\frac{15}{600}rx\left({i}_{1}+{i}_{2}\right)=\frac{\epsilon}{2}...\left(\mathrm{ii}\right)$

For zero deflection in the galvanometer, *i*_{2} = 0. From equation (ii),

$\frac{15}{600}rx\left({i}_{1}\right)=\frac{\epsilon}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{1}=\frac{20\epsilon}{rx}$

Substituting the values of *i _{1}* and

*i*in equation (i), we get:

_{2}*x*= 320 cm

(b) Putting

*x*= 560 cm and solving equations (i) and (ii), we get:

${i}_{2}=\frac{3\epsilon}{22r}$

#### Page No 202:

#### Question 57:

Find the charge on the capacitor shown in the figure (32-E30).

Figure

#### Answer:

In steady state, the capacitor is fully charged and then, it offers infinite resistance to the direct current flow. So, no current can flow through the capacitor in steady state.

The effective resistance of the circuit,

R_{eff}_{ }= 10 + 20 = 30 Ω

The current *i *through the circuit,

$i=\frac{2}{30}=\frac{1}{15}\mathrm{A}$

Voltage drop across the 10 Ω resistor,

V = *i *×* r*

$=\frac{1}{15}\times 10\phantom{\rule{0ex}{0ex}}=\frac{10}{15}=\frac{2}{3}\mathrm{V}$

Since the potential drops across the capacitor and the 10 Ω resistor are the same,

the charge stored on the capacitor,

*Q* = *CV*

$=6\times {10}^{-6}\times \frac{2}{3}\phantom{\rule{0ex}{0ex}}=4\times {10}^{-6}\mathrm{C}=4\mathrm{mC}$

#### Page No 202:

#### Question 58:

(a) Find the current in the 20 Ω resistor shown in the figure (32-E31). (b) If a capacitor of capacitance 4 μF is joined between the points A and B, what would be the electrostatic energy stored in it in steady state?

Figure

#### Answer:

(a) Applying Kirchoff's voltage law in loop 1, we get:

In the circuit ABEDA,

10*i*_{1} + 20 (*i*_{1} + *i _{2}*) − 5 = 0

⇒ 30

*i*

_{1}+ 20

*i*= 5 ...(i)

_{2}Applying Kirchoff's voltage law in loop 2, we get:

20 (

*i*

_{1}+

*i*) − 5 + 10

_{2}*i*

_{2}= 0

⇒ 20

*i*

_{1}*+*30

*i*

_{2}= 5 ...(ii)

Multiplying equation (i) by 20 and (ii) by 30 and subtracting (ii) from (i), we get:

*i*= 0.1 A

_{2}and

*i*= 0.1 A

_{1}∴ Current through the 20 Ω resistor =

*i*+

_{1}_{ }*i*= 0.1 + 0.1 = 0.2 A

_{2}(b) Potential drop across across AB is,

${V}_{\mathrm{AB}}=0.2\times 20=4\mathrm{V}$

Electrostatic energy stored in the capacitor is given by,

$U=\frac{1}{2}C{{V}_{\mathrm{AB}}}^{2}\phantom{\rule{0ex}{0ex}}U=\frac{1}{2}\times 4\times {10}^{-6}\times (0.2\times 20{)}^{2}\phantom{\rule{0ex}{0ex}}U=32\times {10}^{-6}\mathrm{J}$

#### Page No 202:

#### Question 59:

Find the charges on the four capacitors of capacitances 1 μF, 2 μF, 3 μF and 4 μF shown in the figure (32-E32).

Figure

#### Answer:

When the capacitors are fully charged, they attain steady state and no current flows through them. Then, equivalent resistance of the circuit,

${R}_{\mathrm{eff}}=\frac{3\times 6}{3+6}=2\mathrm{\Omega}\phantom{\rule{0ex}{0ex}}$

Current through the circuit,

$i=\frac{6}{2}=3\mathrm{A}$

Current *i *is divided in the inverse ratio of the resistance in each branch. One branch has resistance of 3 Ω and the other branch has resistance of 6 Ω.

Current *i'* through the 3 Ω branch,

$i\text{'}=\frac{6}{9}i=\frac{2}{3}\times 3\mathrm{A}=2\mathrm{A}\phantom{\rule{0ex}{0ex}}$

Current* i'' *through the 6 Ω branch,

$i\text{'}\text{'}=i-i\text{'}=1\mathrm{A}$

Voltage across the 1 Ω resistor = 2 A × 1 Ω = 2 V

Charge on the 1 μF capacitor = 2 × 1 μF = 2 μC

Voltage across the 2 Ω resistor = 2 Ω × 2 A = 4 V

Charge on the 2 μF capacitor = 4V × 2 μF = 8 μC

Voltage across each 3 Ω resistor = 3 Ω × 1 A = 3 V

Charge on the 4 μF capacitor = 3 × 4 μC = 12 μC

Charge on the 3 μF capacitor = 3 × 3 μC = 9 μC

#### Page No 202:

#### Question 60:

Find the potential difference between the points A and B and between the points B and C of the figure (32−E33) in steady state.

Figure

#### Answer:

Equivalent capacitance of the circuit can be calculatesd as,

C_{eq}:

$1\mathrm{\mu F}\mathrm{is}\mathrm{in}\mathrm{parallel}\mathrm{with},C=\frac{1}{{\displaystyle \frac{1}{(3+3)}}+{\displaystyle \frac{1}{(1+1)}}}=\frac{3}{2}\mathrm{\mu F}\phantom{\rule{0ex}{0ex}}\mathrm{So},{C}_{\mathrm{eq}}=\frac{3}{2}+1=\frac{5}{2}\mathrm{\mu F}$

*V* = 100 V

Total charge in the circuits is,

$Q={C}_{\mathrm{eq}}V=\frac{5}{2}\times 100=250\mathrm{\mu}c$

As the volatge across 1μ*f** *is 100 V, therfore charge stored on 1 μ*f* capacitors = 100 μC

Charge flowing from A to B = (250 $-$ 100) = 150 μC

*C*_{eq} between* *A and B is 6 μ*f*.

Potential drop across AB,

$V=\frac{Q}{C}\phantom{\rule{0ex}{0ex}}\Rightarrow {V}_{\mathrm{AB}}=\frac{150}{6}=25\mathrm{V}$

Potential drop across BC = (100 $-$ 25) = 75 V

#### Page No 202:

#### Question 61:

A capacitance C, a resistance R and an emf ε are connected in series at *t* = 0. What is the maximum value of (a) the potential difference across the resistor (b) the current in the circuit (c) the potential difference across the capacitor (d) the energy stored in the capacitor (e) the power delivered by the battery and (f) the power converted into heat?

#### Answer:

(a) When the charge on the capacitor is zero, it acts as short circuit.

Thus, maximum value of potential difference across the resistor = *ε *(at *t* = 0)â€‹

(b) Maximum value of current in the circuit$=\frac{\mathrm{\epsilon}}{r}$ (at *t* = 0)

(c) Maximum value of potential difference across the capacitor = $\epsilon $

(at *t* = ∞, when the capacitor is fully charged and acts as a open circuit)

(d) Maximum energy stored in the capacitor$=\frac{1}{2}\mathrm{C}{\epsilon}^{2}$ (at *t* = ∞)

(e) Maximum power delivered by the battery$=\frac{{\mathrm{\epsilon}}^{2}}{r}$

(f) Maximum power converted to heat$=\frac{{\mathrm{\epsilon}}^{2}}{r}$

#### Page No 202:

#### Question 62:

A parallel-plate capacitor with plate area 20 cm^{2} and plate separation 1.0 mm is connected to a battery. The resistance of the circuit is 10 kΩ. Find the time constant of the circuit.

#### Answer:

Given:

Area of the plates*, A* = 20 cm^{2} = 20 × 10^{−4} m^{2}

Separation between the plates*, d* = 1 mm = 1 × 10^{−3} m,

Resistance of the circuit, *R* = 10 kΩ

The capacitance of a parallel-plate capacitor,

$C=\frac{{\mathit{\in}}_{\mathit{0}}\mathit{}A}{d}\phantom{\rule{0ex}{0ex}}=\frac{8.85\times {10}^{-12}\times 20\times {10}^{-4}}{1\times {10}^{-3}}\phantom{\rule{0ex}{0ex}}=\frac{8.85\times {10}^{-12}\times 2\times {10}^{-4}}{{10}^{-3}}\phantom{\rule{0ex}{0ex}}=17.7\times {10}^{-12}\mathrm{F}$

Time constant = *CR*

= 17.7 × 10^{−12} × 10 × 10^{3}

= 17.7 × 10^{−8} s

= 0.177 × 10^{−6} s

= 0.18 μs

#### Page No 202:

#### Question 63:

A capacitor of capacitance 10 μF is connected to a battery of emf 2 V. It is found that it takes 50 ms for the charge of the capacitor to become 12.6 μC. Find the resistance of the circuit.

#### Answer:

Given:

Capacitance of the capacitor, *C* = 10 μF = 10^{−5} F

Emf of the battery*, E*= 2 V

Time taken to charge the capacitor completely, *t* = 50 ms = 5 × 10^{−2} s

The charge growth across a capacitor,

*q* = $Q{e}^{-\frac{t}{RC}}$

*Q* = *CE* = 10^{−5} × 2 C

and* q* = 12.6 × 10^{−6} C

$\Rightarrow 12.6\times {10}^{-6}=2\times {10}^{-5}{\mathrm{e}}^{-\frac{5\times {10}^{-2}}{R\times {10}^{-5}}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{12.6\times {10}^{-6}}{2\times {10}^{-5}}={\mathrm{e}}^{-\frac{5\times {10}^{3}}{R}}\phantom{\rule{0ex}{0ex}}\Rightarrow 0.63={\mathrm{e}}^{-\frac{5\times {10}^{3}}{R}}\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{log}0.63=-\frac{5\times {10}^{3}}{R}\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{-5\times {10}^{3}}{-0.2}=2.5\mathrm{k\Omega}$

#### Page No 202:

#### Question 64:

A 20 μF capacitor is joined to a battery of emf 6.0 V through a resistance of 100 Ω. Find the charge on the capacitor 2.0 ms after the connections are made.

#### Answer:

The growth of charge across a capacitor,

$Q={Q}_{0}.{\mathrm{e}}^{-\frac{\mathrm{t}}{\mathrm{RC}}}\phantom{\rule{0ex}{0ex}}{Q}_{0}=\mathit{}CV=20\times 6\times {10}^{-6}\mathrm{C}\phantom{\rule{0ex}{0ex}}\Rightarrow Q=120\times {10}^{-6}.{\mathrm{e}}^{-\frac{2\times {10}^{-3}}{{10}^{2}.20\times {10}^{-6}}}\phantom{\rule{0ex}{0ex}}\Rightarrow Q=120\times {10}^{-6}.{\mathrm{e}}^{-1}=76\mathrm{\mu C}$

#### Page No 202:

#### Question 65:

The plates of a capacitor of capacitance 10 μF, charged to 60 μC, are joined together by a wire of resistance 10 Ω at *t* = 0. Find the charge on the capacitor in the circuit at (a) *t* = 0 (b) *t* = 30 μs (c) *t* = 120 μs and (d) *t* = 1.0 ms.

#### Answer:

Given:

Capacitance of the capacitor, *C* = 10 μF

Initial charge on capacitor, *Q* = 60 μC

Resistance of the circuit, *R* = 10 Ω

(a)Decay of charge on the capacitor,

*$q=Q{\mathrm{e}}^{-\frac{t}{RC}}$*

At* t *=* 0,
q* =

*Q*= 60 μC

(b) At

*t*= 30 μs,

*$q=Q.{\mathrm{e}}^{-\frac{t}{RC}}\phantom{\rule{0ex}{0ex}}\Rightarrow q=Q.{\mathrm{e}}^{-\frac{30\times {10}^{-6}}{10\times 10\times {10}^{-6}}}\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{q}=60.{\mathrm{e}}^{-0.3}\phantom{\rule{0ex}{0ex}}\Rightarrow \mathrm{q}=44\mathrm{\mu C}$*

(c) At

*t*= 120 μs,

*$q=Q.{\mathrm{e}}^{-\frac{t}{RC}}\phantom{\rule{0ex}{0ex}}\Rightarrow q=Q.{\mathrm{e}}^{-\frac{120\times {10}^{-6}}{10\times 10\times {10}^{-6}}}\phantom{\rule{0ex}{0ex}}\Rightarrow q=60.{\mathrm{e}}^{-1.2}\phantom{\rule{0ex}{0ex}}\Rightarrow q=18\mathrm{\mu C}$*

(d) At

*t*= 1 ms,

*$q=Q.{\mathrm{e}}^{-\frac{t}{RC}}\phantom{\rule{0ex}{0ex}}\Rightarrow q=Q.{\mathrm{e}}^{-\frac{1\times {10}^{-3}}{10\times 10\times {10}^{-6}}}\phantom{\rule{0ex}{0ex}}\Rightarrow q=60.{\mathrm{e}}^{-10}\phantom{\rule{0ex}{0ex}}\Rightarrow q=0.003\mathrm{\mu C}$*

#### Page No 202:

#### Question 66:

A capacitor of capacitance 8.0 μF is connected to a battery of emf 6.0 V through a resistance of 24 Ω. Find the current in the circuit (a) just after the connections are made and (b) one time constant after the connections are made.

#### Answer:

Given:

Capacitance, *C* = 8 μF

Emf of the battery, *V*= 6 V

Resistance, *R* = 24

(a) Just after the connections are made, there will be no charge on the capacitor and, hence, it will act as a short circuit. Current through the circuit,

$i=\frac{V}{R}=\frac{6}{24}=0.25\mathrm{A}$

(b) The charge growth on the capacitor,

$q=Q\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)$

One time constant = *RC* = 8 × 24 = 192 × 10^{-6} s

For* t* = *RC*, we have:

$q=Q.\left(1-{\mathrm{e}}^{\frac{-RC}{RC}}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow q=CV\left(1-{\mathrm{e}}^{-1}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow q=8\times {10}^{-6}\times 6\times 0.632\phantom{\rule{0ex}{0ex}}=3.036\times {10}^{-5}\mathrm{C}$

$V=\frac{Q}{C}=\frac{3.036\times {10}^{-5}}{8\times {10}^{-6}}=3.792\mathrm{V}$

Applying KVL in the circuit, we get:

*E *= *V *+ *iR*

⇒ 6 = 3.792 + 24*i*

⇒ *i* = 0.09 A

#### Page No 203:

#### Question 67:

A parallel-plate capacitor of plate area 40 cm^{2} and separation between the plates 0.10 mm, is connected to a battery of emf 2.0 V through a 16 Ω resistor. Find the electric field in the capacitor 10 ns after the connections are made.

#### Answer:

Given:

Area of plates*, A* = 40 cm^{2} = 40 × 10^{−4} m^{2}

Separation between the plates, *d* = 0.1 mm = 1 × 10^{−4} m

Resistance*, R* = 16 Ω

Emf of the battery,${V}_{0}$= 2 V

The capacitance *C* of a parallel plate capacitor,

$C=\frac{{\mathit{\in}}_{0}A}{d}\phantom{\rule{0ex}{0ex}}=\frac{8.85\times {10}^{-12}\times 40\times {10}^{-4}}{1\times {10}^{-4}}\phantom{\rule{0ex}{0ex}}=35.4\times {10}^{-11}\mathrm{F}$

So, the electric field,

$E=\frac{V}{d}=\frac{Q}{Cd}=\frac{{Q}_{0}}{A{\mathit{\in}}_{0}}\left(1-{\mathrm{e}}^{\mathit{-}\frac{t}{RC}}\right)\phantom{\rule{0ex}{0ex}}=\frac{C{V}_{0}}{A{\mathit{\in}}_{0}}\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)\phantom{\rule{0ex}{0ex}}=\frac{35.4\times {10}^{-11}\times 2}{8.85\times {10}^{-12}\times 40\times {10}^{-4}}\left(1-{\mathrm{e}}^{-1.76}\right)\phantom{\rule{0ex}{0ex}}=1.655\times {10}^{4}\phantom{\rule{0ex}{0ex}}=1.7\times {10}^{4}\mathrm{V}/\mathrm{m}$

#### Page No 203:

#### Question 68:

A parallel-plate capacitor has plate area 20 cm^{2}, plate separation 1.0 mm and a dielectric slab of dielectric constant 5.0 filling up the space between the plates. This capacitor is joined to a battery of emf 6.0 V through a 100 kΩ resistor. Find the energy of the capacitor 8.9 μs after the connections are made.

#### Answer:

Given:

Area of the plates, *A* = 20 cm^{2}

Separation between the plates*, d *= 1 mm

Dielectric constant, *k* = 5

Emf of the battery, *E *= 6 V

Resistance of the circuit, *R* = 100 × 10^{3} Ω

The capacitance of a parallel-plate capacitor,

$C=\frac{K{\mathit{\in}}_{0}A}{d}\phantom{\rule{0ex}{0ex}}=\frac{5\times 8.85\times {10}^{-12}\times 20\times {10}^{-4}}{1\times {10}^{-3}}\phantom{\rule{0ex}{0ex}}=\frac{10\times 8.85\times {10}^{-12}\times 20\times {10}^{-4}}{1\times {10}^{-3}}\phantom{\rule{0ex}{0ex}}=88.5\times {10}^{-12}\mathrm{C}$

After the connections are made, growth of charge through the capacitor,

*Q* = *EC *(1 − ${\mathrm{e}}^{-\frac{t}{RC}}$)

= 6 × 88.5 × 10^{−12} (1 − ${e}^{-\frac{8.9}{8.85}}$)

= 335.6 × 10^{−12} C

Thus, energy stored in the capacitor,

$U=\frac{1}{2}\frac{{Q}^{2}}{C}\phantom{\rule{0ex}{0ex}}=\frac{1}{2}\times \frac{335.6\times 335.6\times {10}^{-24}}{88.5\times {10}^{-12}}\phantom{\rule{0ex}{0ex}}=\frac{335.6\times 335.6}{88.5\times 2}\times {10}^{-12}\phantom{\rule{0ex}{0ex}}=6.3\times {10}^{-10}\mathrm{J}$

#### Page No 203:

#### Question 69:

A 100 μF capacitor is joined to a 24 V battery through a 1.0 MΩ resistor. Plot qualitative graphs (a) between current and time for the first 10 minutes and (b) between charge and time for the same period.

#### Answer:

Time constant of the circuit,

*τ* = *R*C

= 1 × 10^{6} × 100 × 10^{−6}

= 100 s

The growth of charge through a capacitor,

*q* = *VC* (1 − ${\mathrm{e}}^{-\frac{\mathit{t}}{\mathit{R}\mathit{C}}}$)

The current through the circuit,

$i=\frac{dq}{dt}\phantom{\rule{0ex}{0ex}}=\frac{VC}{RC}\xb7{e}^{-\frac{t}{RC}}\phantom{\rule{0ex}{0ex}}=\frac{V}{R}.{e}^{-\frac{t}{RC}}\phantom{\rule{0ex}{0ex}}=24\times {10}^{-6}\xb7{e}^{-\frac{t}{100}}$

For* t* = 10 min = 600 s

*q* = 24 × 10^{−4} × (1 − *e*^{−6})

= 23.99 × 10^{−4}

$i=\frac{24}{{10}^{6}}.\frac{1}{e}=5.9\times {10}^{-8}\mathrm{A}$

(a) The plot between current and time for the first 10 minutes is shown below.

(b) The plot between charge and time for the first 10 minutes is shown below.

#### Page No 203:

#### Question 70:

How many time constants will elapse before the current in a charging *RC* circuit drops to half of its initial value? Answer the same question for a discharging *RC* circuit.

#### Answer:

The growth of charge across a capacitor,

$q=Q\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)\phantom{\rule{0ex}{0ex}}q=\frac{Q}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{Q}{2}=Q\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{1}{2}=\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow {e}^{-\frac{t}{RC}}=\frac{1}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{t}{R\mathrm{C}}=\mathrm{ln}2\phantom{\rule{0ex}{0ex}}\text{L}\mathrm{et}t=nRC\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{nRC}{RC}=0.69\phantom{\rule{0ex}{0ex}}\Rightarrow n=0.69$

The decay of charge across a capacitor,

$q=Q{\mathrm{e}}^{-\frac{t}{RC}}\phantom{\rule{0ex}{0ex}}q=\frac{Q}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{Q}{2}=Q{\mathrm{e}}^{-\frac{t}{RC}}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{1}{2}={\mathrm{e}}^{-\frac{t}{RC}}\phantom{\rule{0ex}{0ex}}\text{L}\mathrm{et}\mathit{}t\mathit{=}nRC\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{nRC}{RC}=\mathrm{ln}2\phantom{\rule{0ex}{0ex}}\Rightarrow n=0.69$

#### Page No 203:

#### Question 71:

How many time constants will elapse before the charge on a capacitors falls to 0.1% of its maximum value in a discharging RC circuit?

#### Answer:

The decay of charge across a capacitor,

*q* = $Q{\mathrm{e}}^{\mathit{-}\frac{\mathit{t}}{\mathit{R}\mathit{C}}}$

Here,* q* = 0.1 % and Q = 1 × 10^{−3} Q

⇒ 1 × 10^{−3} Q = $Q{\mathrm{e}}^{\mathit{-}\frac{\mathit{t}}{\mathit{R}\mathit{C}}}$

⇒ ${\mathrm{e}}^{-\frac{t}{RC}}$= 10^{−3}

$\Rightarrow \frac{t}{r\mathrm{C}}=-\left(-6.9\right)=6.9$

Let *t *= *nRC*

$\Rightarrow \frac{nRC}{RC}=6.9\phantom{\rule{0ex}{0ex}}\Rightarrow n=6.9$

#### Page No 203:

#### Question 72:

How many time constants will elapse before the energy stored in the capacitor reaches half of its equilibrium value in a charging RC circuit?

#### Answer:

The equilibrium value of energy in a capacitor,

$U=\frac{1}{2}\frac{{Q}^{2}}{C}$, where *Q *is the steady state charge.

Let *q* be the charge for which energy reaches half its equilibrium. Then,

$\frac{1}{2}\frac{{q}^{2}}{C}=\frac{1}{2}U\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{1}{2}\frac{{q}^{2}}{C}=\frac{1}{2}\left(\frac{1}{2}\frac{{Q}^{2}}{C}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow q=\sqrt{\frac{{Q}^{2}}{2}}$

The growth of charge in a capacitor,

$q=Q\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)\phantom{\rule{0ex}{0ex}}\because q=\sqrt{\frac{{Q}^{2}}{2},}\phantom{\rule{0ex}{0ex}}\sqrt{\frac{{Q}^{2}}{2}}=Q\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{Q}{\sqrt{2}}=Q\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow {\mathrm{e}}^{-\frac{t}{RC}}=\left(1-\frac{1}{\sqrt{2}}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow -\frac{t}{RC}=\mathrm{ln}\left(1-\frac{1}{\sqrt{2}}\right)\phantom{\rule{0ex}{0ex}}\mathrm{Let}t\mathit{=}nRC\phantom{\rule{0ex}{0ex}}\text{So},-\frac{nRC}{RC}=\mathrm{ln}\left(1-\frac{1}{\sqrt{2}}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow n=1.23$

#### Page No 203:

#### Question 73:

How many time constants will elapse before the power delivered by a battery drops to half of its maximum value in an RC circuit?

#### Answer:

Power = CV^{2} = *q* × V

$\mathrm{Now},\frac{q\mathrm{V}}{2}=q\mathrm{V}\times {e}^{-t/rc}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{1}{2}={e}^{-t/rc}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{t}{rc}=-\mathrm{ln}\left(0.5\right)\phantom{\rule{0ex}{0ex}}=-\left(-0.69\right)=0.69$

#### Page No 203:

#### Question 74:

A capacitor of capacitance C is connected to a battery of emf ε at *t* = 0 through a resistance R. Find the maximum rate at which energy is stored in the capacitor. When does the rate have this maximum value?

#### Answer:

The rate of growth of charge for the capacitor,

*q* = ε*C* (1 − ${\mathrm{e}}^{\frac{-\mathrm{t}}{\mathrm{RC}}}$)

Let *E* be the energy stored inside the capacitor. Then,

$E=\frac{{q}^{2}}{2C}=\frac{{\epsilon}^{2}{C}^{2}}{2C}{\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow E=\frac{{\epsilon}^{2}C}{2}{\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)}^{2}$

Let *r* be the rate of energy stored inside the capacitor. Then,

$r=\frac{dE}{dt}=\frac{2{\epsilon}^{2}C}{2}\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)\left(-{\mathrm{e}}^{-\frac{t}{RC}}\right)\left(-\frac{1}{RC}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow r=\frac{{\epsilon}^{2}}{R}\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)\left({\mathrm{e}}^{-\frac{t}{RC}}\right)$

$\frac{dr}{dt}=\frac{{\epsilon}^{2}}{R}\left[\left(-{\mathrm{e}}^{-\frac{t}{RC}}\right)\left(-\frac{1}{RC}\right)\left({\mathrm{e}}^{-\frac{t}{RC}}\right)+\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)\left({\mathrm{e}}^{-\frac{t}{RC}}\right)\left(-\frac{1}{RC}\right)\right]$

For *r *to be maximum, $\frac{dr}{dt}=0$

$\Rightarrow \frac{{\epsilon}^{2}}{R}\left[\left(-{\mathrm{e}}^{-\frac{t}{RC}}\right)\left(-\frac{1}{RC}\right)\left({\mathrm{e}}^{-\frac{t}{RC}}\right)+\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)\left({\mathrm{e}}^{-\frac{t}{RC}}\right)\left(-\frac{1}{RC}\right)\right]=0\phantom{\rule{0ex}{0ex}}\Rightarrow \left[\frac{{\mathrm{e}}^{-{\displaystyle \frac{2t}{RC}}}}{RC}+\frac{{\mathrm{e}}^{-{\displaystyle \frac{2t}{RC}}}}{RC}-\frac{{\mathrm{e}}^{{\displaystyle \frac{-t}{RC}}}}{RC}\right]=0\phantom{\rule{0ex}{0ex}}\Rightarrow 2{\mathrm{e}}^{-\frac{2t}{RC}}={\mathrm{e}}^{-\frac{t}{RC}}\phantom{\rule{0ex}{0ex}}\Rightarrow {\mathrm{e}}^{-\frac{t}{RC}}=\frac{1}{2}\phantom{\rule{0ex}{0ex}}\Rightarrow -\frac{t}{RC}=-\mathrm{ln}2\phantom{\rule{0ex}{0ex}}\Rightarrow t=RC\mathrm{ln}2$

#### Page No 203:

#### Question 75:

A capacitor of capacitance 12.0 μF is connected to a battery of emf 6.00 V and internal resistance 1.00 Ω through resistanceless leads. 12.0 μs after the connections are made, what will be (a) the current in the circuit (b) the power delivered by the battery (c) the power dissipated in heat and (d) the rate at which the energy stored in the capacitor is increasing?

#### Answer:

Given,

Capacitance of capacitor, *C*= 12.0 μF = 12 × 10^{−6} F

Emf of battery,* **V*_{0} = 6.00 V

Internal resistance of battery,* R* = 1 Ω

Time interval,* t * = 12 μs

(a)

Charging current in the circuit is given as,

*i *= *i*_{0}*e*^{−t}^{/RC}

Current at,* t* = 12.0 μs

$i=\frac{{V}_{0}}{R}{e}^{-t/RC}\phantom{\rule{0ex}{0ex}}i=\frac{6}{1}\times {e}^{-1}\phantom{\rule{0ex}{0ex}}i=2.207=2.21\mathrm{A}$

(b)

During charging, charge on the capacitor at any time ''*t*'' is given as

$Q=C{V}_{0}(1-{e}^{-\frac{t}{RC}})$

Work done by battery in in time delivering this charge is,

*W = **QV*_{0}

Power deliver by the battery in time ''*t*'' is,

$P=\frac{C{{V}_{0}}^{2}(1-{e}^{-{\displaystyle \frac{t}{RC}}})}{t}$

Putting,* t* = 12 μs

$P=\frac{12\times {10}^{-6}{{V}_{0}}^{2}(1-{e}^{-{\displaystyle 1}})}{12\times {10}^{-6}}\phantom{\rule{0ex}{0ex}}\Rightarrow P=13.25W$

(c)

Energy stroed in the capacitor at any instant of time is given as,

$U=\frac{1}{2}\frac{{Q}^{2}}{C}\phantom{\rule{0ex}{0ex}}\Rightarrow U=\frac{1}{2}\frac{{C}^{2}{{V}_{0}}^{2}(1-{e}^{-{\displaystyle \frac{t}{RC}}}{)}^{2}}{C}\phantom{\rule{0ex}{0ex}}\Rightarrow U=\frac{1}{2}C{{V}_{0}}^{2}(1-{e}^{-\frac{t}{RC}}{)}^{2}$

Rate at which the energy stored in the capacitor is,

$\frac{dU}{dt}=\frac{1}{2}C{{V}_{0}}^{2}\times 2(1-{e}^{-\frac{t}{RC}})\times ({e}^{-\frac{t}{RC}})\times \frac{1}{RC}$

$\Rightarrow \frac{dU}{dt}=\frac{{{V}_{0}}^{2}}{R}({e}^{-\frac{t}{RC}}-{e}^{-\frac{2t}{RC}})\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{dU}{dt}=\frac{6\times 6}{1}({e}^{-1}-{e}^{-2})\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{dU}{dt}=8.37\mathrm{W}\phantom{\rule{0ex}{0ex}}$

So, the power dissipated in heat = $P-\frac{dU}{dt}$

= 13.25 $-$ 8.37 = 4.87 W

(d)

Rate at which the energy stored in the capacitor is increasing = $\Rightarrow \frac{dU}{dt}=8.37\mathrm{W}$

#### Page No 203:

#### Question 76:

A capacitance C charged to a potential difference V is discharged by connecting its plates through a resistance R. Find the heat dissipated in one time constant after the connections are made. Do this by calculating ∫ *i*^{2}R *dt* and also by finding the decrease in the energy stored in the capacitor.

#### Answer:

Let *Q _{0}* be the initial charge on the capacitor. Then,

*Q*=

_{0}*CV*

The charge on the capacitor at time

*t*after the connections are made,

$Q={Q}_{0}{e}^{-\frac{t}{RC}}$

$i=\frac{dQ}{dt}=-\left(\frac{{Q}_{0}}{RC}\right){\mathrm{e}}^{-\frac{t}{RC}}$

Heat dissipated during time

*t*to

_{1}*t*,

_{2}$U={\int}_{{t}_{1}}^{{t}_{2}}{i}^{2}Rdt\phantom{\rule{0ex}{0ex}}=\frac{{{Q}_{0}}^{2}}{2C}\left({\mathrm{e}}^{-\frac{2{t}_{1}}{RC}}-{\mathrm{e}}^{-\frac{2{t}_{2}}{RC}}\right)$

Time constant =

*RC*

Putting

*t*= 0 and

_{1}*t*=

_{2}*RC*, we get:

$U=\frac{{{Q}_{0}}^{2}}{2C}\left({\mathrm{e}}^{-0}-{\mathrm{e}}^{-2}\right)\phantom{\rule{0ex}{0ex}}\because {Q}_{0}=CV,\phantom{\rule{0ex}{0ex}}U=\frac{CV}{2}\left(1-\frac{1}{{\mathrm{e}}^{2}}\right)$

**Alternative method:**

Heat dissipated at any time = Energy stored at time 0 − Energy stored at time

*t*

$\Rightarrow U=\frac{1}{2}C{V}^{2}-\frac{1}{2}C{V}^{2}{\mathrm{e}}^{-\frac{2t}{RC}}\phantom{\rule{0ex}{0ex}}\because t=RC,\phantom{\rule{0ex}{0ex}}U=\frac{1}{2}C{V}^{2}-\frac{1}{2}C{V}^{2}{\mathrm{e}}^{-2}\phantom{\rule{0ex}{0ex}}\Rightarrow U=\frac{1}{2}C{V}^{2}\left(1-\frac{1}{{\mathrm{e}}^{2}}\right)$

#### Page No 203:

#### Question 77:

By evaluating ∫ *i*^{2}R*dt*, show that when a capacitor is charged by connecting it to a battery through a resistor, the energy dissipated as heat equals the energy stored in the capacitor.

#### Answer:

The growth of charge on the capacitor at time* t *,

$Q={Q}_{0}\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)$

$i=\frac{dQ}{dt}=\left(\frac{{Q}_{0}}{RC}\right){\mathrm{e}}^{-\frac{t}{RC}}$

Heat dissipated during time *t _{1}* to

*t*,

_{2}$U={\int}_{{t}_{1}}^{{t}_{2}}{i}^{2}Rdt\phantom{\rule{0ex}{0ex}}=\frac{{{Q}_{0}}^{2}}{2C}\left({\mathrm{e}}^{-\frac{2{t}_{1}}{RC}}-{\mathrm{e}}^{-\frac{2{t}_{2}}{RC}}\right)\phantom{\rule{0ex}{0ex}}\because {Q}_{0}=C{V}_{0,}\phantom{\rule{0ex}{0ex}}U=\frac{1}{2}C{{V}_{0}}^{2}\left({\mathrm{e}}^{-\frac{2{t}_{1}}{RC}}-{\mathrm{e}}^{-\frac{2{t}_{2}}{RC}}\right)$

The potential difference across a capacitor at any time

*t*,

$V={V}_{0}\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)$

The energy stored in the capacitor at any time

*t,*

$E=\frac{1}{2}C{V}^{2}=\frac{1}{2}C{{V}_{0}}^{2}{\left(1-{\mathrm{e}}^{-\frac{2t}{RC}}\right)}^{2}$

∴ The energy stored in the capacitor from

*t*

_{1}to

*t*

_{2},

$E=\frac{1}{2}C{{V}_{0}}^{2}\left({\mathrm{e}}^{-\frac{2{t}_{1}}{RC}}\right)-\frac{1}{2}C{{V}_{0}}^{2}\left(1-{\mathrm{e}}^{-\frac{2{t}_{2}}{RC}}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow E=\frac{1}{2}C{{V}_{0}}^{2}\left({\mathrm{e}}^{-\frac{2{t}_{1}}{RC}}-{\mathrm{e}}^{-\frac{2{t}_{2}}{RC}}\right)$

#### Page No 203:

#### Question 78:

A parallel-plate capacitor is filled with a dielectric material of resistivity ρ and dielectric constant K. The capacitor is charged and disconnected from the charging source. The capacitor is slowly discharged through the dielectric. Show that the time constant of the discharge is independent of all geometrical parameters like the plate area or separation between the plates. Find this time constant.

#### Answer:

The capacitance of a parallel plate capacitor,

$C=\frac{K{\epsilon}_{0}A}{d}$

The resistance of dielectric material,

$R=\frac{\mathrm{\rho}d}{A}$

Time constant,

$\tau =RC=\mathrm{\rho}K{\epsilon}_{0}$,

which is independent of the plate area or separation between the plates.

#### Page No 203:

#### Question 79:

Find the charge on each of the capacitors 0.20 ms after the switch S is closed in the figure (32-E34).

Figure

#### Answer:

The equivalent capacitance of the circuit,

*${C}_{\mathrm{eqv}}={C}_{1}+{C}_{2}=2+2=4\mathrm{\mu F}$*

The growth of charge through the capacitor,

*q* = *q*_{0}(1 − *e*^{−t}^{/RC})

${q}_{0}=CV=4\times {10}^{-6}\times 6=24\times {10}^{-6}\mathrm{C}\phantom{\rule{0ex}{0ex}}\frac{\mathit{t}}{\mathit{R}\mathit{C}}=\frac{0.20\times {10}^{-3}}{25\times 4\times {10}^{-6}}=2$

⇒ *q* = 24 × 10^{−6} (1 − *e*^{−2})

= 18.4 × 10^{−6} C

This is the total charge on both capacitors. As the capacitors are in parallel, the total charge will be shared between them. Also, both the capacitors are of same capacitance; so, they will share equal amount of charge.

∴ Charge on each capacitor$=\frac{18.4}{2}\mathrm{\mu C}=9.2\mathrm{\mu C}$

#### Page No 203:

#### Question 80:

The switch S shown in figure (32−E35) is kept closed for a long time and is then opened at *t* = 0. Find the current in the middle 10 Ω resistor at *t* = 1 ms.

Figure

#### Answer:

Initially, the switch S was closed; so, the capacitor was getting charged. Also, the two resistors are connected in parallel. The equivalent resistance of the circuit,

$R=\frac{1}{{\displaystyle \frac{1}{10}}+{\displaystyle \frac{1}{10}}}=5\mathrm{\Omega}$

Initially, the switch was closed and the capacitor was getting charged. So, the two resistances were in parallel connection. Hence, their effective resistance will be 5 Ω.

Potential difference across the 10 Ω resistance, $V=\frac{12}{5}\times 10=24\mathrm{V}$

When the switch is opened, the decay of charge through the capacitor,

$Q={Q}_{0}{\mathrm{e}}^{-\frac{t}{RC}}=V\times C{\mathrm{e}}^{-\frac{t}{RC}}\phantom{\rule{0ex}{0ex}}\frac{t}{RC}=\frac{1\times {10}^{-3}}{10\times 25\times {10}^{-6}}=\frac{1000}{250}=4\phantom{\rule{0ex}{0ex}}\Rightarrow Q=24\times 25\times {10}^{-6}\times {\mathrm{e}}^{-4}\phantom{\rule{0ex}{0ex}}=24\times 25\times {10}^{-6}\times 0.0183\phantom{\rule{0ex}{0ex}}=10.9\times {10}^{-6}\mathrm{C}$

Current in the 10 Ω resistor,

$i=\frac{Q}{t}=\frac{10.9\times {10}^{-6}\mathrm{C}}{1\times {10}^{-3}\mathrm{sec}}=11\mathrm{mA}$

#### Page No 203:

#### Question 81:

A capacitor of capacitance 100 μF is connected across a battery of emf 6 V through a resistance of 20 kΩ for 4 s. The battery is then replaced by a thick wire. What will be the charge on the capacitor 4 s after the battery is disconnected?

#### Answer:

Given:

Capacitance of capacitor* ,C* = 100 μF

Emf of battery *,E *= 6V

Resistance* ,R *= 20 kΩ

Time for charging , *t*_{1} = 4 s

Time for discharging ,*t*_{2} = 4 s

During charging of the capacitor, the growth of charge across it,

$\mathrm{Q}=CE\left(1-{\mathrm{e}}^{-\frac{{t}_{1}}{RC}}\right)\phantom{\rule{0ex}{0ex}}\frac{{t}_{1}}{RC}=\frac{4}{20\times {10}^{3}\times 100\times {10}^{-6}}\phantom{\rule{0ex}{0ex}}=2\phantom{\rule{0ex}{0ex}}\Rightarrow Q=6\times {10}^{-4}\left(1-{\mathrm{e}}^{-2}\right)\phantom{\rule{0ex}{0ex}}=5.187\times {10}^{-4}\mathrm{C}$

This is the amount of charge developed on the capacitor after 4s.

During discharging of the capacitor, the decay of charge across it,

$Q\text{'}=Q\left({\mathrm{e}}^{-\frac{t}{RC}}\right)\phantom{\rule{0ex}{0ex}}=5.184\times {10}^{-4}\times {\mathrm{e}}^{-2}\phantom{\rule{0ex}{0ex}}=0.7\times {10}^{-4}\mathrm{C}=70\mathrm{\mu C}$

#### Page No 203:

#### Question 82:

Consider the situation shown in figure (32−E36). The switch is closed at *t *= 0 when the capacitors are uncharged. Find the charge on the capacitor *C _{1} *as a function of time

*t*.

Figure

#### Answer:

The two capacitors are connected in series. Their equivalent capacitance,

${C}_{eqv}=\frac{{C}_{\mathit{1}}{C}_{\mathit{2}}}{{C}_{\mathit{1}}+{C}_{\mathit{2}}}$

The growth of charge in the capacitors,

$q={C}_{\mathrm{eqv}}\epsilon \left(1-{\mathrm{e}}^{-\frac{1}{r{C}_{\mathrm{eqv}}}}\right)$

#### Page No 203:

#### Question 83:

A capacitor of capacitance C is given a charge Q. At *t* = 0, it is connected to an uncharged capacitor of equal capacitance through a resistance R. Find the charge on the second capacitor as a function of time.

#### Answer:

Given:

Initial charge on first capacitor = *Q*

Let *q* be the charge on the second capacitor after time *t*.

According to the principle of conservation of charge, charge on the first capacitor after time *t* = *Q *- *q*.

Let* V _{1}* be the potential difference across the first capacitor and

*V*be the potential difference across the second capacitor after time

_{2}*t*. Then,

${V}_{\mathit{1}}=\frac{Q-q}{C}\phantom{\rule{0ex}{0ex}}{V}_{\mathit{2}}=\frac{q}{C}\phantom{\rule{0ex}{0ex}}\Rightarrow {\mathrm{V}}_{1}-{\mathrm{V}}_{2}=\frac{Q-q}{C}-\frac{q}{C}\phantom{\rule{0ex}{0ex}}=\frac{Q-2q}{C}$

The current through the circuit after time

*t*,

$i=\frac{{V}_{1}-{V}_{2}}{R}=\frac{dq}{dt}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{Q-2q}{CR}=\frac{dq}{dt}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{dq}{Q-2q}=\frac{1}{RC}dt\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{dq}{Q-2q}=\frac{1}{RC}dt\phantom{\rule{0ex}{0ex}}$

Integrating both sides within the limits time =0 to t and charge on the second capacitor varying from q=0 to q, we get:

$\frac{1}{2}\left[\mathrm{ln}\left(Q-2q\right)-\mathrm{ln}Q\right]=\frac{-1t}{RC}\phantom{\rule{0ex}{0ex}}\mathrm{ln}\frac{Q-2q}{Q}=\frac{-2t}{RC}\phantom{\rule{0ex}{0ex}}Q-2q=Q{\mathrm{e}}^{-\frac{2t}{RC}}\phantom{\rule{0ex}{0ex}}2q=Q\left(1-{\mathrm{e}}^{-\frac{2t}{RC}}\right)\phantom{\rule{0ex}{0ex}}q=\frac{Q}{2}\left\{1-{\mathrm{e}}^{-\frac{2t}{RC}}\right\}$

#### Page No 203:

#### Question 84:

A capacitor of capacitance C is given a charge Q. At *t* = 0, it is connected to an ideal battery of emf ε through a resistance R. Find the charge on the capacitor at time *t*.

#### Answer:

Given:

Initial charge given to the capacitor = *Q*

When the capacitor is connected to a battery, it will charge through the battery. The initial charge will also decay.

The growth of charge across the capacitor due to the battery of emf *E,*

$q=CE\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)$

The decay of charge through the capacitor,

$q\text{'}=Q{\mathrm{e}}^{-\frac{t}{RC}}$

The net charge on the capacitor at any time *t*,

${q}_{\mathrm{net}}=q+q\text{'}\phantom{\rule{0ex}{0ex}}=CE\left(1-{\mathrm{e}}^{-\frac{t}{RC}}\right)+Q{\mathrm{e}}^{-\frac{t}{RC}}$

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