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#### Page No 328:

#### Question 1:

What is the reactance of a capacitor connected to a constant DC source?

#### Answer:

The reactance of a capacitor is given by,

${X}_{c}=\frac{1}{\omega C}$

For a DC source,* ω* = 0

$\Rightarrow {X}_{c}=\frac{1}{0\times C}=\infty $

So, for a constant DC source, reactance of a capacitor is infinite.

#### Page No 328:

#### Question 2:

The voltage and current in a series AC circuit are given by V = V_{0}cos ω*t* and *i* = *i*_{0} sin ω*t*. What is the power dissipated in the circuit?

#### Answer:

Voltage, *V *= *V*_{0}cos ω*t*

Current, *i* = *i*_{0} sin ω*t* or *i* = *i*_{0} cos (ω*t $-$ $\frac{\pi}{2}$*)

Power dissipated in an AC circuit is given by,

$P={I}_{rms}{V}_{rms}\mathrm{cos}\varphi $,

where* I _{rms}* = rms value of current

*V*= rms value of voltage

_{rms}*ϕ*= phase difference between current and voltage

Here,

*ϕ*= $\pi $/2

$\Rightarrow \mathrm{cos}\varphi =\mathrm{cos}\frac{\mathrm{\pi}}{2}=0\phantom{\rule{0ex}{0ex}}\therefore P={I}_{rms}{V}_{rms}\times 0=0$

#### Page No 328:

#### Question 3:

Two alternating currents are given by

${i}_{1}={i}_{0}\mathrm{sin}\mathrm{\omega}t\mathrm{and}{i}_{2}={i}_{0}\mathrm{sin}\left(\mathrm{\omega}t+\frac{\mathrm{\pi}}{3}\right)$

Will the rms values of the currents be equal or different?

#### Answer:

The rms value of current is given by,

${i}_{rms}=\frac{{i}_{0}}{\sqrt{2}}$

Since peak value of current *i _{0}* is same for both currents, their rms values will be same.

#### Page No 328:

#### Question 4:

Can the peak voltage across the inductor be greater than the peak voltage of the source in an LCR circuit?

#### Answer:

Let a LCR circuit is connected across an AC supply with the emf *E* = *E*_{0} sin *ωt*.

Let the inductance in the circuit be *L*

Let the net impedence of the circuit be $Z=\sqrt{{R}^{2}+({X}_{\mathrm{L}}-{X}_{\mathrm{C}}{)}^{2}}$

Where,

*R* = resistance in the circuit

*X*_{L}_{ }= reactance due to inductor

*X*_{C}_{ }= reactance due to capacitor

The magnitude of the voltage across the inductor is given by

$V=L\frac{\mathrm{d}i}{\mathrm{d}t}$

The current in the circuit can be written as $I={I}_{0}\mathrm{sin}(\omega t+\varphi )$

Where, *ϕ* is the phase difference between the current and the supply voltage

Thus, the voltage across the inductor can be written as

$V=L{I}_{0}\mathrm{cos}(\omega t+\varphi )$

Thus peak value of the voltage across the inductor is given by

$V=L{I}_{0}\phantom{\rule{0ex}{0ex}}\Rightarrow V=\frac{{E}_{0}}{Z}\times L$

Therefore, the peak voltage across the inductor is given by $V=\frac{{E}_{0}}{Z}\times L$

At resonance *Z = R*,

$V=\frac{{E}_{0}}{R}\times L$,

If $\frac{L}{R}>1$

*V* > *E*_{0}

Therefore if magnitude of $\frac{L}{R}>1$ at resonance the value of the voltage across the inductor will bw greater than the peak value of the supply voltage.

#### Page No 328:

#### Question 5:

In a circuit, containing a capacitor and an AC source, the current is zero at the instant the source voltage is maximum. Is it consistent with Ohm's Law?

#### Answer:

Ohm's Law is valid for resistive circuits only. It is not valid for capacitive or inductive circuits, or a combination of both.

#### Page No 328:

#### Question 6:

An AC source is connected to a capacitor. Will the rms current increase, decrease or remain constant if a dielectric slab is inserted into the capacitor?

#### Answer:

The reactance of a capacitor is given by,

${X}_{c}=\frac{1}{\omega C}$

Also,

$C=\frac{K{\epsilon}_{0}A}{d}$,

where *C* = capacitance

*K* = dielectric constant

*A* = area of plates

*d* = distance between the plates.

$K>1$

$\therefore $ The capacitance *C* of the capacitor will increase on inserting the dielectric slab and, consequently, the reactance *X _{c}* will decrease.

Rms current, ${i}_{rms}=\frac{{\epsilon}_{0}}{\sqrt{2}{X}_{C}}$

Therefore, rms current will decrease.

#### Page No 328:

#### Question 7:

When the frequency of the AC source in an LCR circuit equals the resonant frequency, the reactance of the circuit is zero. Does it mean that there is no current through the inductor or the capacitor?

#### Answer:

The condition for resonance is:

$\frac{1}{\omega C}=\omega L$

The peak current through the circuit is given by,

${i}_{0}=\frac{{V}_{0}}{{\sqrt{{R}^{2}+\left({\displaystyle \frac{1}{\omega C}}-\omega L\right)}}^{2}}\phantom{\rule{0ex}{0ex}}$

From the condition of resonance, we get:

${i}_{0}=\frac{{V}_{0}}{R}$

The current will flow through the all circuit elements. But since the reactance of the capacitor and inductor are equal, the potential difference across them will be equal and opposite and will cancel each other.

#### Page No 328:

#### Question 8:

When an AC source is connected to a capacitor, there is a steady-state current in the circuit. Does it mean that the charges jump from one plate to the other to complete the circuit?

#### Answer:

No. When an AC source is connected to a capacitor, there is a steady in the circuit to transfer change to the plates of the capacitor. This produces a potential difference between the plates. The capacitance is alternatively charged and discharged as the current reverses after each half cycle.

#### Page No 329:

#### Question 9:

A current *i*_{1} = *i*_{0} sin ω*t* passes through a resistor of resistance R. How much thermal energy is produced in one time period? A current* **i*_{2} = −*i*_{0} sin ω*t* passes through the resistor. How much thermal energy is produced in one time period? If *i*_{1} and *i*_{2} both pass through the resistor simultaneously, how much thermal energy is produced? Is the principle of superposition obeyed in this case?

#### Answer:

The thermal energy produced for an AC circuit in one time period is given by,

$H={{I}_{rms}}^{2}\times R\times \frac{2\mathrm{\pi}}{\omega}$

For current, *i*_{1} = *i*_{0} sin ω*t*,

${I}_{rms}=\frac{{i}_{0}}{\sqrt{2}}$

$\Rightarrow H=\frac{{{i}_{0}}^{2}R}{2}\times \frac{2\mathrm{\pi}}{\omega}=\frac{\mathrm{\pi}{{i}_{0}}^{2}R}{\omega}$

For current, *i*_{2} = −*i*_{0} sin ω*t*,

${I}_{rms}=\frac{{i}_{0}}{\sqrt{2}}$

Hence, the same thermal energy will be produced due to this current.

Since, the direction of *i*_{1} and *i*_{2} are opposite and their magnitude is same, the net current through the resistor will become zero when both are passed together*. *Yes, the principle of superposition is obeyed in this case.

#### Page No 329:

#### Question 10:

Is energy produced when a transformer steps up the voltage?

#### Answer:

When a transformer steps up the voltage, the voltage increases but current decreases. Neglecting any loss of energy, the power remains constant and, hence, energy is not produced. It remains constant.

#### Page No 329:

#### Question 11:

A transformer is designed to convert an AC voltage of 220 V to an AC voltage of 12 V. If the input terminals are connected to a DC voltage of 220 V, the transformer usually burns. Explain.

#### Answer:

A transformer is ideally an inductive coil. For an inductor connected across a DC voltage,

$V-L\frac{di}{dt}=0\phantom{\rule{0ex}{0ex}}\Rightarrow V=L\frac{di}{dt}\phantom{\rule{0ex}{0ex}}\Rightarrow \int di=\frac{V}{L}\int dt\phantom{\rule{0ex}{0ex}}\Rightarrow i=\frac{Vt}{L}$

For a DC source, the current across the inductor will increase with time and can reach a very large value, which can burn the transformer.

#### Page No 329:

#### Question 12:

Can you have an AC series circuit in which there is a phase difference of (a) 180° (b) 120° between the emf and the current?

#### Answer:

Let us consider an AC series LCR circuit of angular frequency *ω*. The impedance of the circuit is given by,

$Z=\sqrt{{R}^{2}+{\left(\omega L-\frac{1}{\omega C}\right)}^{2}}$

The phase difference between *V *and *I* is given by,

$\mathrm{tan}\varphi =\left(\frac{\omega L-{\displaystyle \frac{1}{\omega C}}}{Z}\right)$

From the above formula, we can clearly see that

$\varphi \in \left(-\frac{\mathrm{\pi}}{2},\frac{\mathrm{\pi}}{2}\right)$

So, we cannot have a phase difference of 180° or 120°.

#### Page No 329:

#### Question 13:

A resistance is connected to an AC source. If a capacitor is included in the series circuit, will the average power absorbed by the resistance increase or decrease? If an inductor of small inductance is also included in the series circuit, will the average power absorbed increase or decrease further?

#### Answer:

When a capacitor is included in a series circuit, the impedance of the circuit,

$Z=\sqrt{{R}^{2}+{{X}_{C}}^{2}}\phantom{\rule{0ex}{0ex}}$

The power absorbed by a resistor is given by,

$P={{I}_{rms}}^{2}R$

Since impedance increases due to introduction of a capacitor, the rms value of current *I _{rms}* will decrease and, hence, the power absorbed by the resistor will decrease.

When a small inductance is introduced in the circuit, the impedance of the circuit,

$Z=\sqrt{{R}^{2}+{\left({X}_{C}-{X}_{L}\right)}^{2}}$

Since the impedance now decreases a little, the rms value of current will increase and, hence, the power absorbed by the resistor will increase.

#### Page No 329:

#### Question 14:

Can a hot-wire ammeter be used to measure a direct current of constant value? Do we have to change the graduations?

#### Answer:

A hot-wire ammeter measures the rms value of current for an alternating current. So, it can be used to measure the direct current of constant value because that constant value will be equal to the rms value of current. As, the rms value of the current is same as the direct current thus we need not change the graduations.

#### Page No 329:

#### Question 1:

A capacitor acts as an infinite resistance for

(a) DC

(b) AC

(c) DC as well as AC

(d) neither AC nor DC

#### Answer:

(a) DC

For a DC source, *ω* = 0. So, the reactance of the capacitance is given by,

${X}_{C}=\frac{1}{\omega C}=\frac{1}{0}=\infty $

#### Page No 329:

#### Question 2:

An AC source producing emf

ε = ε_{0} [cos (100 π s^{−1})*t* + cos (500 π s^{−1})*t*]

is connected in series with a capacitor and a resistor. The steady-state current in the circuit is found to be *i *= *i*_{1} cos [(100 π s^{−1})*t* + φ_{1}) + *i*_{2} cos [(500π s^{−1})*t* + ϕ_{2}]. So,

(a) *i*_{1} > *i*_{2}

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

(c) *i*_{1} < *i*_{2}

(d) The information is insufficient to find the relation between *i*_{1} and *i*_{2}.

#### Answer:

(c) *i*_{1} < *i*_{2}

The charge on the capacitor during steady state is given by,

$Q=C\epsilon ={\epsilon}_{0}C\left[\mathrm{cos}\left(100{\mathrm{\pi s}}^{-1}\right)t+\mathrm{cos}\left(500{\mathrm{\pi s}}^{-1}\right)t\right]$

The steady state current is, thus, given by,

$i=\frac{dQ}{dt}={\epsilon}_{0}C\times 100\mathrm{\pi}\left[\mathrm{sin}\left(100{\mathrm{\pi s}}^{-1}\right)t\right]+{\epsilon}_{\mathit{0}}C\times 500\mathrm{\pi}\left[\mathrm{sin}\left(500{\mathrm{\pi s}}^{-1}\right)t\right]\phantom{\rule{0ex}{0ex}}\Rightarrow i=100C{\mathrm{\pi \epsilon}}_{0}\mathrm{cos}\left[\left(100{\mathrm{\pi s}}^{-1}\right)\mathrm{t}+{\varphi}_{1}\right]+500C{\mathrm{\pi \epsilon}}_{0}\mathrm{cos}\left[\left(500{\mathrm{\pi s}}^{-1}\right)+{\varphi}_{2}\right]\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{1}=100C{\mathrm{\pi \epsilon}}_{0}\mathit{}{i}_{\mathit{2}}=500C{\mathrm{\pi \epsilon}}_{0}\phantom{\rule{0ex}{0ex}}\therefore {i}_{2}{i}_{1}$

#### Page No 329:

#### Question 3:

The peak voltage of a 220 V AC source is

(a) 220 V

(b) about 160 V

(c) about 310 V

(d) 440 V

#### Answer:

(c) about 310 V

Given:

*V _{rms}* = 220 V

The peak value of voltage is given by,

${V}_{p}=\sqrt{2}\times {V}_{rms}=1.414\times 220=311\mathrm{V}$

#### Page No 329:

#### Question 4:

An AC source is rated 220 V, 50 Hz. The average voltage is calculated in a time interval of 0.01 s. It

(a) must be zero

(b) may be zero

(c) is never zero

(d) $\mathrm{is}\left(220/\sqrt{2}\right)\mathrm{V}$

#### Answer:

(b) may be zero

Let the AC voltage be given by,

$V={V}_{0}\mathrm{sin}\omega t$

Here, ω = 2$\pi $f = 314 rad/s

The average voltage over the given time,

${V}_{avg}=\frac{{\int}_{0}^{0.01}Vdt}{{\int}_{0}^{0.01}dt}=-{V}_{0}{\left[\frac{\mathrm{cos}\omega t}{\omega}\right]}_{0}^{0.01}\phantom{\rule{0ex}{0ex}}=\frac{{V}_{0}}{\omega \times 0.01}\left(1-\mathrm{cos}\omega \left(0.01\right)\right)\phantom{\rule{0ex}{0ex}}=\frac{{V}_{0}}{314\times 0.01}\left(1-\mathrm{cos}\left(314\times 0.01\right)\right)\phantom{\rule{0ex}{0ex}}=\frac{{V}_{0}}{3.14}\left(1-\mathrm{cos\pi}\right)\phantom{\rule{0ex}{0ex}}=\frac{2{V}_{0}}{\mathrm{\pi}}=140.127\mathrm{V}$

Also, when $V={V}_{0}\mathrm{cos}\omega t$,

${V}_{avg}=\frac{{\int}_{0}^{0.01}Vdt}{{\int}_{0}^{0.01}dt}={V}_{0}{\left[\frac{\mathrm{sin}\omega t}{\omega}\right]}_{0}^{0.01}\phantom{\rule{0ex}{0ex}}=\frac{{V}_{0}}{\omega \times 0.01}\left(\mathrm{sin}\omega \left(0.01\right)-0\right)\phantom{\rule{0ex}{0ex}}=\frac{{V}_{0}}{314\times 0.01}\left(\mathrm{sin}\left(314\times 0.01\right)\right)\phantom{\rule{0ex}{0ex}}=\frac{{V}_{0}}{3.14}\left(\mathrm{sin}\pi \right)\phantom{\rule{0ex}{0ex}}=0$

From the above results, we can say that the average voltage can be zero. But it is not necessary that it should be zero or never zero.

#### Page No 329:

#### Question 5:

The magnetic field energy in an inductor changes from maximum to minimum value in 5.0 ms when connected to an AC source. The frequency of the source is

(a) 20 Hz

(b) 50 Hz

(c) 200 Hz

(d) 500 Hz

#### Answer:

(b) 50 Hz

The magnetic field energy in an inductor is given by,

$E=\frac{1}{2}L{i}^{2}$

The magnetic energy will be maximum when the current will reach its peak value, *i _{0}*, and it will be minimum when the current will become zero.

From the above graph of alternating current, we can see that current reduces from maximum to zero in

*T*/4 time, where

*T*is the time period.

So, in this case,

*T*/4 = 5 ms

$\Rightarrow T=20\mathrm{ms}\phantom{\rule{0ex}{0ex}}\therefore \mathrm{Frequency},\nu =\frac{1}{T}=\frac{1}{20\times {10}^{-3}}=50\mathrm{Hz}$

#### Page No 329:

#### Question 6:

Which of the following plots may represent the reactance of a series LC combination?

Figure

#### Answer:

(d)

The reactance of a series LC circuit is given by,

$X={X}_{L}-{X}_{C}=\omega L-\frac{1}{\omega C}\phantom{\rule{0ex}{0ex}}\Rightarrow X=2\mathrm{\pi}fL-\frac{1}{2\mathrm{\pi}fC}$

The correct relation between the reactance and *f* is represented by the graph in option (d).

$X=0$, when ${X}_{L}={X}_{C}$.

Thus plot d represent the reactance of a series LC combination.

#### Page No 329:

#### Question 7:

A series AC circuit has a resistance of 4 Ω and a reactance of 3 Ω. The impedance of the circuit is

(a) 5 Ω

(b) 7 Ω

(c) 12/7 Ω

(d) 7/12 Ω

#### Answer:

(a) 5 Ω

The impedance of the circuit is given by

$Z=\sqrt{{R}^{2}+{X}^{2}}\phantom{\rule{0ex}{0ex}}R=4\mathrm{\Omega}X=3\mathrm{\Omega}\phantom{\rule{0ex}{0ex}}\Rightarrow Z=\sqrt{{4}^{2}+{3}^{2}}=5\mathrm{\Omega}$

#### Page No 329:

#### Question 8:

Transformers are used

(a) in DC circuits only

(b) in AC circuits only

(c) in both DC and AC circuits

(d) neither in DC nor in AC circuits

#### Answer:

(b) in AC circuits only

When a DC supply is provided to a transformer, there will be no change in flux with time across the coils of the transformer. So, there will be no induced emf in the secondary coil due to changing current in the primary coil. Hence, the transformer cannot operate in DC because of the violation of its working principle.

#### Page No 329:

#### Question 9:

An alternating current is given by *i* =* **i*_{1} cos ω*t* + *i*_{2} sin ω*t*. The rms current is given by

(a) $\frac{{i}_{1}+{i}_{2}}{\sqrt{2}}$

(b) $\frac{\left|{i}_{1}+{i}_{2}\right|}{\sqrt{2}}$

(c) $\sqrt{\frac{{i}_{1}^{2}+{i}_{2}^{2}}{2}}$

(d) $\sqrt{\frac{{i}_{1}^{2}+{i}_{2}^{2}}{\sqrt{2}}}$

#### Answer:

(c) $\sqrt{\frac{{i}_{1}^{2}+{i}_{2}^{2}}{2}}$

Given:

*i* =* **i*_{1} cos ω*t* + *i*_{2} sin ω*t*

The rms value of current is given by,

${i}_{rms}=\sqrt{\frac{{\int}_{0}^{T}{i}^{2}dt}{{\int}_{0}^{T}dt}}\phantom{\rule{0ex}{0ex}}i={i}_{1}\mathrm{cos}\omega t+{i}_{2}\mathrm{sin}\omega t\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}{i}_{\mathrm{rms}}=\sqrt{\frac{{\int}_{0}^{T}{\left({i}_{1}\mathrm{cos}\omega t+{i}_{2}\mathrm{sin}\omega t\right)}^{2}dt}{{\int}_{0}^{T}dt}}\phantom{\rule{0ex}{0ex}}{i}_{\mathrm{rms}}=\sqrt{\frac{{\int}_{0}^{T}\left({i}_{1}^{2}{\mathrm{cos}}^{2}\omega t+{i}_{2}^{2}{\mathrm{sin}}^{2}\omega t+2{i}_{1}{i}_{2}\mathrm{sin}\mathrm{\omega t}\mathrm{cos}\omega t\right)dt}{{\int}_{0}^{T}dt}}\phantom{\rule{0ex}{0ex}}{i}_{\mathrm{rms}}=\sqrt{\frac{{\int}_{0}^{T}\left({i}_{1}^{2}{\displaystyle \frac{(\mathrm{cos}2\omega t+1)}{2}}+{i}_{2}^{2}{\displaystyle \frac{(1-\mathrm{cos}2\omega t)}{2}}+{i}_{1}{i}_{2}\mathrm{sin}2\omega t\right)dt}{{\int}_{0}^{T}dt}}\phantom{\rule{0ex}{0ex}}[\because {\mathrm{cos}}^{2}\omega t=\frac{(\mathrm{cos}2\omega t+1)}{2},{\mathrm{sin}}^{2}\omega t=\frac{(1-\mathrm{cos}2\omega t)}{2}]\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

We know that, *T* = 2π

Integrating the above expression

${i}_{\mathrm{rms}}=\sqrt{\frac{{\displaystyle \frac{1}{2}}{i}_{1}^{2}\left({\int}_{0}^{2\pi}1dt+{\int}_{0}^{2\pi}\mathrm{cos}2\omega tdt\right)+{i}_{2}^{2}\left({\int}_{0}^{2\pi}1dt-{\int}_{0}^{2\pi}\mathrm{cos}2\omega tdt\right)+{i}_{1}{i}_{2}{\int}_{0}^{2\pi}\mathrm{sin}2\omega tdt}{{\int}_{0}^{2\pi}dt}}$

The following integrals become zero

${\int}_{0}^{2\pi}\mathrm{cos}2\omega tdt=0={\int}_{0}^{2\pi}\mathrm{sin}2\omega tdt\phantom{\rule{0ex}{0ex}}$

Therefore, it becomes

${i}_{\mathrm{rms}}=\sqrt{\frac{{\displaystyle \frac{{i}_{1}^{2}}{2}}\left({\int}_{0}^{2\mathrm{\pi}}1dt\right)+{\displaystyle \frac{{i}_{2}^{2}}{2}}\left({\int}_{0}^{2\mathrm{\pi}}1dt\right)}{{\int}_{0}^{2\pi}dt}}\phantom{\rule{0ex}{0ex}}{i}_{\mathrm{rms}}=\sqrt{\frac{{\displaystyle \frac{{i}_{1}^{2}}{2}}\times 2\mathrm{\pi}+{\displaystyle \frac{{i}_{2}^{2}}{2}}\times 2\mathrm{\pi}}{2\mathrm{\pi}}}\phantom{\rule{0ex}{0ex}}\Rightarrow {i}_{\mathrm{rms}}=\sqrt{\frac{{i}_{1}^{2}+{i}_{2}^{2}}{2}}$

#### Page No 329:

#### Question 10:

An alternating current of peak value 14 A is used to heat a metal wire. To produce the same heating effect, a constant current *i* can be used, where *i* is

(a) 14 A

(b) about 20 A

(c) 7 A

(d) about 10 A

#### Answer:

(d) about 10 A

The rms value of an alternating current is equivalent to the constant current. So, the heating effect produced is actually measured in terms of the rms value, in case of alternating current. The constant current is, thus, equal to the rms value of alternating current, which is given by,

${I}_{rms}=\frac{{I}_{peak}}{\sqrt{2}}=\frac{14}{\sqrt{2}}=9.9\approx 10\mathrm{A}$

#### Page No 329:

#### Question 11:

A constant current of 2.8 A exists in a resistor. The rms current is

(a) 2.8 A

(b) about 2 A

(c) 1.4 A

(d) undefined for a direct current

#### Answer:

(a) 2.8 A

The constant current is equal to the rms value of current. So,

*I _{rms}*

_{ }= 2.8 A

#### Page No 329:

#### Question 1:

An inductor, a resistance and a capacitor are joined in series with an AC source. As the frequency of the source is slightly increased from a very low value, the reactance

(a) of the inductor increases

(b) of the resistor increases

(c) of the capacitor increases

(d) of the circuit increases

#### Answer:

(a) of the inductor increases

The reactance of an inductor is given by,

${X}_{L}=\omega L$

And the reactance of a capacitor is given by,

${X}_{C}=\frac{1}{\omega C}$

Here, *ω* = 2π*f** *, where *f* is the frequency of the source. So, when *f* increases, *ω* increases.

∴ *X _{L}* will increase and

*X*will decrease.

_{C}#### Page No 329:

#### Question 2:

The reactance of a circuit is zero. It is possible that the circuit contains

(a) an inductor and a capacitor

(b) an inductor but no capacitor

(c) a capacitor but no inductor

(d) neither an inductor nor a capacitor

#### Answer:

(a) an inductor and a capacitor

(d) neither an inductor nor a capacitor

(a) The reactance of a circuit containing a capacitor and an inductor is given by,

$X={X}_{L}-{X}_{C}=\omega L-\frac{1}{\omega C}\phantom{\rule{0ex}{0ex}}$

If $\omega L=\frac{1}{\omega C}$,* X* = 0. So, the circuit contains an inductor and a capacitor.

(b) For a circuit without a capacitor, reactance is given by,

${X}^{\text{'}}={X}_{L}=\omega L$, which cannot be zero for an AC source.

(c) Similarly, for a circuit without an inductor, reactance is given by,

${X}^{\text{'}\text{'}}={X}_{C}=\frac{1}{\omega C}$, which also cannot be zero.

(d) For a circuit without any capacitor and inductor, reactance, *X ^{'}* = 0 (

*L*=

*C*= 0)

#### Page No 330:

#### Question 3:

In an AC series circuit, the instantaneous current is zero when the instantaneous voltage is maximum. Connected to the source may be a

(a) pure inductor

(b) pure capacitor

(c) pure resistor

(d) combination of an inductor and a capacitor

#### Answer:

(a) pure inductor

(b) pure capacitor

(d) combination of an inductor and a capacitor.

For a pure inductive circuit, voltage leads the current by $90\xb0$. So, the instantaneous current is zero when the instantaneous voltage is maximum.

Similar is the case with a purely capacitive circuit, in which, current leads the voltage by $90\xb0$.

Also, in a circuit containing a combination of inductor and capacitor, the current may lead or lag the voltage by $90\xb0$, depending upon whether the voltage across the inductor or the capacitor is greater. Here too, there is a phase difference of $90\xb0$ between the voltage and current. Hence, the the instantaneous current is zero when the instantaneous voltage is maximum.

#### Page No 330:

#### Question 4:

An inductor coil of some resistance is connected to an AC source. Which of the following quantities have zero average value over a cycle?

(a) Current

(b) Induced emf in the inductor

(c) Joule heat

(d) Magnetic energy stored in the inductor

#### Answer:

(a) Current

(b) Induced emf in the inductor

For a series *L*-*R* circuit, the AC current can be given by,

$i={i}_{0}\mathrm{sin}\omega t$

From the graph, we can see that the average value of current over a cycle is zero.

Since it a series *L*-*R* circuit, the phase difference between current and voltage is $\frac{\pi}{2}$. The AC voltage can be given by,

$V={V}_{0}\mathrm{cos}\omega t$

From the graph, we can see that the average value of voltage over a cycle is also zero.

Joule's heat through the resistor is given by,

${H}_{avg}={{i}_{rms}}^{2}R$, which is non zero.

Similarly, magnetic energy stored in the inductor is given by,

${U}_{avg}=\frac{1}{2}L{{i}_{rms}}^{2}$ , which is also non-zero.

#### Page No 330:

#### Question 5:

The AC voltage across a resistance can be measured using

(a) a potentiometer

(b) a hot-wire voltmeter

(c) a moving-coil galvanometer

(d) a moving-magnet galvanometer

#### Answer:

(b) a hot-wire voltmeter

Only a hot-wire voltmeter can be used to measure an AC voltage across a resistor.

#### Page No 330:

#### Question 6:

To convert mechanical energy into electrical energy, one can use

(a) DC dynamo

(b) AC dynamo

(c) motor

(d) transformer

#### Answer:

(a) DC dynamo

(b) AC dynamo

An AC or DC dynamo can be used to convert mechanical energy to electrical energy.

A motor converts electrical energy to mechanical energy and a transformer is used to step up or down the voltage or simply to transfer electrical energy.

#### Page No 330:

#### Question 7:

An AC source rated 100 V (rms) supplies a current of 10 A (rms) to a circuit. The average power delivered by the source

(a) must be 1000 W

(b) may be 1000 W

(c) may be greater than 1000 W

(d) may be less than 1000 W

#### Answer:

(b) may be 1000 W

(d) may be less than 1000 W

The average power delivered by an AC source is given by,

${P}_{avg}={V}_{rms}{I}_{rms}\mathrm{cos}\varphi $

Given:

V_{rms} = 100 V

I_{rms} = 10 A

$\Rightarrow {P}_{avg}=1000\mathrm{cos}\varphi $

$\because \varphi \in \left(-\frac{\mathrm{\pi}}{2},\frac{\mathrm{\pi}}{2}\right)$

$\Rightarrow \mathrm{cos}\varphi \in \left(0,1\right)$

$\therefore 0\le {P}_{avg}\le 1000$

#### Page No 330:

#### Question 1:

Find the time required for a 50 Hz alternating current to change its value from zero to the rms value.

#### Answer:

Frequency of alternating current, *f* = 50 Hz

Alternation current $\left(i\right)$ is given by,

*i* = *i*_{0}sinωt ...(1)

Here, *i*_{0} = peak value of current

Root mean square value of current $\left({i}_{rms}\right)$ is given by,

${i}_{\mathrm{rms}}=\frac{{i}_{0}}{\sqrt{2}}...\left(1\right)\phantom{\rule{0ex}{0ex}}$

On substituting the value of the root mean square value of current in place of alternating current in equation (1), we get:

$\frac{{i}_{0}}{\sqrt{2}}={i}_{0}\mathrm{sin}\omega t\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{1}{\sqrt{2}}=\mathrm{sin}\omega t\mathit{}=\mathrm{sin}\frac{\mathrm{\pi}}{4}\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{\mathrm{\pi}}{4}=\mathrm{\omega}t\phantom{\rule{0ex}{0ex}}\Rightarrow t\mathit{}\mathit{=}\frac{\mathrm{\pi}}{4\omega}\mathit{}=\frac{\mathrm{\pi}}{4\times 2\mathrm{\pi}f}\left(\because \omega =2\mathrm{\pi}f\right)\phantom{\rule{0ex}{0ex}}=\frac{1}{8f}=\frac{1}{8\times 50}\phantom{\rule{0ex}{0ex}}=\frac{1}{400}=0.0025\mathrm{s}\phantom{\rule{0ex}{0ex}}=2.5\mathrm{ms}$

#### Page No 330:

#### Question 2:

The household supply of electricity is at 220 V (rms value) and 50 Hz. Find the peak voltage and the least possible time in which the voltage can change from the rms value to zero.

#### Answer:

RMS value of voltage, *E*_{rms} = 220 V,

Frequency of alternating current, *f *= 50 Hz

(a) Peak value of voltage $\left({E}_{0}\right)$ is given by,

${E}_{0}={E}_{\mathrm{rms}}\sqrt{2}\phantom{\rule{0ex}{0ex}}$,

where *E*_{rms}_{ }= root mean square value of voltage

${E}_{0}={E}_{\mathrm{rms}}\sqrt{2}\phantom{\rule{0ex}{0ex}}\Rightarrow {E}_{0}=\sqrt{2}\times 220\phantom{\rule{0ex}{0ex}}\Rightarrow {E}_{0}=311.08\mathrm{V}=311\mathrm{V}$

(b) Voltage $\left(E\right)$ is given by,

$E={E}_{0}\mathrm{sin}\omega t$,

where *E*_{0} = peak value of voltage

Time taken for the current to reach zero from the rms value = Time taken for the current to reach the rms value from zero

In one complete cycle, current starts from zero and again reaches zero.

So, first we need to find the time taken for the current to reach the rms value from zero.

$\mathrm{As}\mathit{}E=\frac{{E}_{0}}{\sqrt{2}},\phantom{\rule{0ex}{0ex}}\frac{{E}_{0}}{\sqrt{2}}={E}_{0}\mathrm{sin}\omega t\phantom{\rule{0ex}{0ex}}\Rightarrow \omega t\mathit{}=\frac{\mathrm{\pi}}{4}\phantom{\rule{0ex}{0ex}}\Rightarrow t=\frac{\mathrm{\pi}}{4\omega}=\frac{\mathrm{\pi}}{4\times 2\mathrm{\pi}f}\phantom{\rule{0ex}{0ex}}\Rightarrow t=\frac{\mathrm{\pi}}{8\mathrm{\pi}50}=\frac{1}{400}\phantom{\rule{0ex}{0ex}}\Rightarrow t=2.5\mathrm{ms}\phantom{\rule{0ex}{0ex}}$

Thus, the least possible time in which voltage can change from the rms value to zero is 2.5 ms.

#### Page No 330:

#### Question 3:

A bulb rated 60 W at 220 V is connected across a household supply of alternating voltage of 220 V. Calculate the maximum instantaneous current through the filament.

#### Answer:

Power of the bulb, *P* = 60 W

Voltage at the bulb, V = 220 V

RMS value of alternating voltage, *E*_{rms} = 220 V

P = *V*^{2}*R**,*

where *R* = resistance of the bulb

$\therefore R\mathit{=}\frac{{V}^{\mathit{2}}}{P}=\frac{220\times 220}{60}\phantom{\rule{0ex}{0ex}}=806.67\phantom{\rule{0ex}{0ex}}$

Peak value of voltage $\left({E}_{0}\right)$ is given by,

${E}_{0}={E}_{\mathrm{rms}}\sqrt{2}$

=$220\times \sqrt{2}$

= 311.08

Now, maximum current through the filament $\left({i}_{0}\right)$ is,

${i}_{0}=\frac{{E}_{0}}{R}$

$\Rightarrow {i}_{0}=\frac{311.08}{806.67}=0.39\mathrm{A}$

#### Page No 330:

#### Question 4:

An electric bulb is designed to operate at 12 volts DC. If this bulb is connected to an AC source and gives normal brightness, what would be the peak voltage of the source?

#### Answer:

Voltage across the electric bulb, *E* = 12 volts

Let *E*_{0} be the peak value of voltage.

We know that heat produced by passing an alternating current$\left(i\right)$ through a resistor is equal to heat produced by passing a constant current$\left({i}_{rms}\right)$ through the same resistor. If *R* is the resistance of the electric bulb and *T* is the temperature, then

${i}^{\mathit{2}}RT\mathit{=}{i}_{\mathit{rms}}^{\mathit{2}}RT\phantom{\rule{0ex}{0ex}}\Rightarrow \frac{{E}^{\mathit{2}}}{{R}^{\mathit{2}}}\mathit{=}\frac{{E}_{\mathit{rms}}^{\mathit{2}}}{{R}^{\mathit{2}}}\phantom{\rule{0ex}{0ex}}\Rightarrow {E}^{\mathit{2}}\mathit{=}\frac{{E}_{\mathit{0}}^{\mathit{2}}}{\mathit{2}}\mathit{}\left(\mathit{\because}{{E}^{\mathit{2}}}_{rms}\mathit{=}\frac{{{E}^{\mathit{2}}}_{\mathit{0}}}{\mathit{2}}\right)\phantom{\rule{0ex}{0ex}}\Rightarrow {E}_{0}^{2}=2{E}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow {E}_{0}^{2}=2\times {\left(12\right)}^{2}=2\times 144\phantom{\rule{0ex}{0ex}}\Rightarrow {E}_{0}=\sqrt{2\times 144}\phantom{\rule{0ex}{0ex}}=16.97=17\mathrm{V}$

Thus , peak value of voltage is 17 V.

#### Page No 330:

#### Question 5:

The peak power consumed by a resistive coil, when connected to an AC source, is 80 W. Find the energy consumed by the coil in 100 seconds, which is many times larger than the time period of the source.

#### Answer:

Peak power of the resistive coil, ${P}_{0}=80\mathrm{W}$

Time,* t* = 100 s

RMS value of power $\left({P}_{rms}\right)$ is given by,

${P}_{\mathrm{rms}}=\frac{{P}_{0}}{2}$,

where *P*_{0} = Peak value of power

$\therefore {P}_{\mathrm{rms}}=\frac{{P}_{0}}{2}=40\mathrm{W}$

Energy consumed $\left(E\right)$ is given by,

*E* = *P*_{rms} × *t*

= 40 × 100

= 4000 J = 4.0 kJ

#### Page No 330:

#### Question 6:

The dielectric strength of air is 3.0 × 10^{6} V/m. A parallel-plate air-capacitor has area 20 cm^{2} and plate separation 0.10 mm. Find the maximum rms voltage of an AC source that can be safely connected to this capacitor.

#### Answer:

Given:

Area of parallel-plate air-capacitor, A = 20 cm^{2}

Separation between the plates, *d* = 0.1 mm

Dielectric strength of air, *E*= 3 × 10^{6} V/m

*E* = $\frac{V}{d}$,

where *V* = potential difference across the capacitor

$\therefore $ V = *Ed*

= 3 × 10^{6 }× 0.1 × 10^{−3}

= 3 × 10^{2} = 300 V

Thus, peak value of voltage is 300 V.

Maximum rms value of voltage $\left({V}_{\mathrm{rms}}\right)$ is given by,

${V}_{\mathrm{rms}}=\frac{{V}_{0}}{\sqrt{2}}\phantom{\rule{0ex}{0ex}}=\frac{300}{\sqrt{2}}=212\mathrm{V}$

#### Page No 330:

#### Question 7:

The current in a discharging LR circuit is given by* i* = *i*_{0}_{ }*e*^{−t}^{/}τ , where τ is the time constant of the circuit. Calculate the rms current for the period *t *= 0 to *t* = τ.

#### Answer:

As per the question,

$i={i}_{0}{e}^{\frac{-t}{\tau}}$

We need to find the rms current. So, taking the average of i within the limits 0 to τ and then dividing by the given time period τ, we get:

${{i}_{\mathrm{rms}}}^{2}=\frac{1}{\tau}{\int}_{\mathit{0}}^{\tau}{i}_{0}^{2}\mathit{}{\mathrm{e}}^{\mathit{-}\mathit{2}t\mathit{/}\tau}\mathit{}dt\phantom{\rule{0ex}{0ex}}\Rightarrow {{i}_{rms}}^{2}=\frac{{i}_{0}^{2}}{\tau}{\int}_{0}^{\mathrm{\tau}}{\mathrm{e}}^{\mathit{-}\mathit{2}t\mathit{/}\tau}\mathit{}dt\phantom{\rule{0ex}{0ex}}=\frac{{i}_{0}^{2}}{\tau}\times {\left[\frac{-\tau}{2}\mathit{}{\mathrm{e}}^{\mathit{-}\mathit{2}t\mathit{/}\tau}\right]}_{0}^{\tau}\phantom{\rule{0ex}{0ex}}=-\frac{{i}_{0}^{2}}{\tau}\times \frac{\tau}{2}\times \left[{\mathrm{e}}^{-2}-1\right]\phantom{\rule{0ex}{0ex}}=\frac{{{i}_{0}}^{2}}{2}(1-\frac{1}{{\mathrm{e}}^{2}})\phantom{\rule{0ex}{0ex}}{i}_{\mathrm{rms}}=\frac{{i}_{0}}{e}\sqrt{\frac{{e}^{2}-1}{2}}$

#### Page No 330:

#### Question 8:

A capacitor of capacitance 10 μF is connected to an oscillator with output voltage ε = (10 V) sin ωt. Find the peak currents in the circuit for ω = 10 s^{−1}, 100 s^{−1}, 500 s^{−1} and 1000 s^{−1}.

#### Answer:

Capacitance of the capacitor, *C* = 10 μF = 10 × 10^{−6} F = 10^{−5} F

Output voltage of the oscillator, $\epsilon $= (10 V)sinωt

On comparing the output voltage of the oscillator with $\epsilon ={\epsilon}_{0}\mathrm{sin}\omega t$, we get:

Peak voltage ${\epsilon}_{0}$ = 10 V

For a capacitive circuit,

Reactance, ${X}_{c}=\frac{1}{\omega C}$

Here, $\omega $ = angular frequency

*C* = capacitor of capacitance

Peak current, ${I}_{0}$ = $\frac{{\epsilon}_{0}}{{X}_{c}}$

(a) At *ω *= 10 s^{−1}:

Peak current,

*I*_{0} = $\frac{{\epsilon}_{0}}{{X}_{c}}$

$=\frac{{\epsilon}_{0}}{1\mathit{/}\omega C}\phantom{\rule{0ex}{0ex}}=\frac{10}{1/10\times {10}^{-5}}\mathrm{A}$

= 1 × 10^{−3} A

(b) At *ω* = 100 s^{−1}:

Peak current, *I*_{0} = $\frac{{\epsilon}_{0}}{1/\omega C}$

$\Rightarrow {I}_{0}=\frac{10}{1/100\times {10}^{-5}}\phantom{\rule{0ex}{0ex}}\Rightarrow {I}_{0}=\frac{10}{{10}^{3}}=1\times {10}^{-2}\mathrm{A}\phantom{\rule{0ex}{0ex}}=0.01\mathrm{A}$

(c) At ω = 500 s^{−1}:

Peak current, *I*_{0} = $\frac{{\epsilon}_{0}}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\omega C$}\right.}}$

${I}_{0}\mathit{}=\frac{{\epsilon}_{0}}{1/\omega C}\phantom{\rule{0ex}{0ex}}\Rightarrow {I}_{0}=\frac{10}{1/500\times {10}^{-5}}\phantom{\rule{0ex}{0ex}}\Rightarrow {I}_{0}=10\times 500\times {10}^{-5}\phantom{\rule{0ex}{0ex}}=5\times {10}^{-2}\mathrm{A}=0.05\mathrm{A}$

(d) At ω = 1000 s^{−1}:

Peak current,* **I*_{0} = $\frac{{\epsilon}_{0}}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\omega C$}\right.}}$

$\Rightarrow {I}_{0}=\frac{10}{1/1000\times {10}^{-5}}\phantom{\rule{0ex}{0ex}}\Rightarrow {I}_{0}=10\times 1000\times {10}^{-5}\phantom{\rule{0ex}{0ex}}\Rightarrow {I}_{0}={10}^{-1}\mathrm{A}=0.1\mathrm{A}$

#### Page No 330:

#### Question 9:

A coil of inductance 5.0 mH and negligible resistance is connected to the oscillator of the previous problem. Find the peak currents in the circuit for ω = 100 s^{−1}, 500 s^{−1}, 1000 s^{−1}.

#### Answer:

Given:

Inductance of the coil,* L * = 5.0 mH = 0.005 H

(a) At *ω* = 100 s^{−1}:

Reactance of coil $\left({X}_{L}\right)$ is given by,

*X*_{L} = $\omega L$

Here, $\omega $ = angular frequency

$\therefore $ *X*_{L} = 100 $\times $ 0.005 = 0.5 $\mathrm{\Omega}$

Peak current, *I*_{0} = $\frac{10}{0.5}=20\mathrm{A}$

(b) At *ω *= 500 s^{−1}:

$\mathrm{Reactance},{X}_{\mathrm{L}}=500\times \frac{5}{1000}\phantom{\rule{0ex}{0ex}}=2.5\mathrm{\Omega}\phantom{\rule{0ex}{0ex}}$

Peak current, *I*_{0} = $\frac{10}{2.5}=4\mathrm{A}$

(c) ω = 1000 s^{−1}:

Reactance, *X*_{L}_{ }= 1000 $\times 0.005$ = 5 $\mathrm{\Omega}$

Peak current, *I*_{0} = $\frac{10}{5}$ = 2 A

#### Page No 330:

#### Question 10:

A coil has a resistance of 10 Ω and an inductance of 0.4 henry. It is connected to an AC source of 6.5 V, $\frac{30}{\mathrm{\pi}}\mathrm{Hz}$. Find the average power consumed in the circuit.

#### Answer:

Given:

Resistance of coil, *R* = 10 Ω

Inductance of coil, *L *= 0.4 Henry

Voltage of AC source,* **E*rms = 6.5 V

Frequency of AC source, $f$ = $\frac{30}{\mathrm{\pi}}\mathrm{Hz}$

Reactance of resistance-inductance circuit $\left(Z\right)$ is given by,

Z = $\sqrt{{R}^{2}+{{X}_{L}}^{2}}$

Here, *R* = resistance of the circuit

*X*_{L} = Reactance of the pure inductive circuit

*Z* = $\sqrt{{R}^{2}+{\left(2\mathrm{\pi}fL\right)}^{2}}$

= $\sqrt{{\left(10\right)}^{2}+{\left(2\times \mathrm{\pi}\times \frac{30}{\mathrm{\pi}}\times 0.4\right)}^{2}}$

Average power consumed in the circuit (*P*) is given by,

*P *= *E*_{rms}*I*_{rms}cos$\varphi $

cos$\varphi $ = $\frac{R}{Z}$,* **I*_{rms} = $\frac{{E}_{rms}}{Z}$

$\phantom{\rule{0ex}{0ex}}\therefore P=6.5\times \frac{6.5}{Z}\times \frac{R}{Z}\phantom{\rule{0ex}{0ex}}\Rightarrow P=\frac{6.5\times 6.5\times 10}{{\left(\sqrt{{R}^{2}+{\left(\omega L\right)}^{2}}\right)}^{2}}\phantom{\rule{0ex}{0ex}}\Rightarrow P=\frac{6.5\times 6.5\times 10}{100+576}\phantom{\rule{0ex}{0ex}}\Rightarrow P=\frac{6.5\times 6.5\times 10}{676}\phantom{\rule{0ex}{0ex}}\Rightarrow P=0.625=\frac{5}{8}\mathrm{W}$

#### Page No 330:

#### Question 11:

A resistor of resistance 100 Ω is connected to an AC source ε = (12 V) sin (250 π *s*^{−1})*t*. Find the energy dissipated as heat during *t* = 0 to *t* = 1.0 ms.

#### Answer:

Given:

Peak voltage of AC source,* **E*_{0} = 12 V

Angular frequency, *ω* = 250 $\mathrm{\pi}$ s^{−1}

Resistance of resistor, *R* = 100 Ω

Energy dissipated as heat $\left(H\right)$ is given by,

$H=\frac{{{E}_{\mathrm{rms}}}^{2}}{R}T$

Here, *E*_{rms} = RMS value of voltage

* R* = Resistance of the resistor

*T* = Temperature

Energy dissipated as heat during *t* = 0 to *t* = 1.0 ms,

$H={\int}_{\mathit{0}}^{t}dH\phantom{\rule{0ex}{0ex}}=\int \frac{{E}_{0}^{2}{\mathrm{sin}}^{2}\mathrm{\omega}t}{R}dt\left(\because {E}_{rms}={E}_{0}\mathrm{sin}\omega t\right)\phantom{\rule{0ex}{0ex}}=\frac{144}{100}{\int}_{0}^{{10}^{-3}}{\mathrm{sin}}^{2}\omega t\mathit{}dt\phantom{\rule{0ex}{0ex}}=1.44{\int}_{0}^{{10}^{-3}}\left(\frac{1-\mathrm{cos}2\omega t}{2}\right)dt\phantom{\rule{0ex}{0ex}}=\frac{1.44}{2}\left[{\int}_{0}^{{10}^{-3}}dt+{\int}_{0}^{{10}^{-3}}\mathrm{cos}2\omega tdt\right]\phantom{\rule{0ex}{0ex}}=0.72\left[{10}^{-3}-{\left\{\frac{\mathrm{sin}2\omega t}{2\omega}\right\}}_{0}^{{10}^{-3}}\right]\phantom{\rule{0ex}{0ex}}=0.72\left[\frac{1}{1000}-\frac{1}{500\mathrm{\pi}}\right]\phantom{\rule{0ex}{0ex}}=0.72\left[\frac{1}{1000}-\frac{2}{1000\mathrm{\pi}}\right]\phantom{\rule{0ex}{0ex}}=\left(\frac{\mathrm{\pi}-2}{1000\mathrm{\pi}}\right)\times 0.72\phantom{\rule{0ex}{0ex}}=0.0002614=2.61\times {10}^{-4}\mathrm{J}$

#### Page No 330:

#### Question 12:

In a series RC circuit with an AC source, R = 300 Ω, C = 25 μF, ε_{0} = 50 V and ν = 50/π Hz. Find the peak current and the average power dissipated in the circuit.

#### Answer:

Given:

Resistance of the series RC circuit, *R* = 300 Ω

Capacitance of the series RC circuit,* C* = 25 μF

Peak value of voltage, *ε*_{0} = 50 V

Frequency of the AC source, ν = 50/$\mathrm{\pi}$ Hz

Capacitive reactance $\left({X}_{C}\right)$ is given by,

${X}_{C}=\frac{1}{\omega C}$

Here, $\omega $ = angular frequency of AC source

*C* = capacitive reactance of capacitance

$\therefore {X}_{C}=\frac{1}{2\mathrm{\pi}\times {\displaystyle \frac{50}{\mathrm{\pi}}}\times 25\times {10}^{-6}}\phantom{\rule{0ex}{0ex}}\Rightarrow {X}_{C}=\frac{{10}^{4}}{25}\mathrm{\Omega}$

Net reactance of the series RC circuit $\left(Z\right)$ = $\sqrt{{R}^{2}+{\left({X}_{C}\right)}^{2}}$

$\Rightarrow $ *Z = *$\sqrt{{\left(300\right)}^{2}+{\left(\frac{{10}^{4}}{25}\right)}^{2}}$

= $\sqrt{{\left(300\right)}^{2}+{\left(400\right)}^{2}}$ = 500 $\mathrm{\Omega}$

(a) Peak value of current $\left({I}_{0}\right)$ is given by,

${I}_{0}=\frac{{\epsilon}_{0}}{Z}\phantom{\rule{0ex}{0ex}}\Rightarrow {I}_{0}=\frac{50}{500}=0.1\mathrm{A}$

(b) Average power dissipated in the circuit $\left(P\right)$ is given by,

* P* = ${\epsilon}_{\mathrm{rms}}$*I*_{rms} cosϕ.

${\epsilon}_{rms}$ = $\frac{{\epsilon}_{0}}{\sqrt{2}}$

and ${I}_{\mathrm{rms}}=\frac{{I}_{0}}{\sqrt{2}}$

$\therefore P=\frac{{E}_{0}}{\sqrt{2}}\times \frac{{I}_{0}}{\sqrt{2}}\times \frac{R}{Z}\phantom{\rule{0ex}{0ex}}\Rightarrow P=\frac{50\times 0.1\times 300}{2\times 500}\phantom{\rule{0ex}{0ex}}\Rightarrow P=\frac{3}{2}=1.5\mathrm{W}$

#### Page No 330:

#### Question 13:

An electric bulb is designed to consume 55 W when operated at 110 volts. It is connected to a 220 V, 50 Hz line through a choke coil in series. What should be the inductance of the coil for which the bulb gets correct voltage?

#### Answer:

Power consumed by the electric bulb,* P* = 55 W

Voltage at which the bulb is operated, V= 110 V

Voltage of the line, *V* = 220 V

Frequency of the source, *v* = 50 Hz

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

where *R* = resistance of electric bulb

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

$=\frac{110\times 110}{55}=220\mathrm{\Omega}$

If *L* is the inductance of the coil, then total reactance of the circuit (*Z*) is given by,

* Z* = $\sqrt{{R}^{2}+{\left(\omega L\right)}^{2}}$

=$\sqrt{{\left(220\right)}^{2}+{\left(100\mathrm{\pi}L\right)}^{2}}$ $\left(\because \omega =2\mathrm{\pi}f\right)$

Here, $\omega $ = angular frequency of the circuit

Now, current through the bulb, $I=\frac{V}{Z}$

$\therefore $ Voltage drop across the bulb, V = $\frac{V}{Z}\times R$

As per question,

$110=\frac{220\times 220}{\sqrt{{\left(220\right)}^{2}+{\left(100\mathrm{\pi}L\right)}^{2}}}\phantom{\rule{0ex}{0ex}}110=\frac{{\left(220\right)}^{2}}{\sqrt{{\left(220\right)}^{2}+{\left(100\mathrm{\pi}L\right)}^{2}}}\phantom{\rule{0ex}{0ex}}\Rightarrow 220\times 2=\sqrt{{\left(220\right)}^{2}+{\left(100\mathrm{\pi}L\right)}^{2}}\phantom{\rule{0ex}{0ex}}\Rightarrow {\left(220\right)}^{2}+{\left(100\mathrm{\pi}L\right)}^{2}={\left(440\right)}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow 48400+{10}^{4}{\mathrm{\pi}}^{2}{L}^{2}=193600\phantom{\rule{0ex}{0ex}}\Rightarrow {10}^{4}{\mathrm{\pi}}^{2}{L}^{2}=193600-48400\phantom{\rule{0ex}{0ex}}\Rightarrow {L}^{2}=\frac{145200}{{\mathrm{\pi}}^{2}\times {10}^{4}}\phantom{\rule{0ex}{0ex}}=1.4726\phantom{\rule{0ex}{0ex}}\Rightarrow L=1.2135\cong 1.2\mathrm{H}$

#### Page No 330:

#### Question 14:

In a series LCR circuit with an AC source, R = 300 Ω, C = 20 μF, L = 1.0 henry, ε_{rms} = 50 V and ν = 50/π Hz. Find (a) the rms current in the circuit and (b) the rms potential difference across the capacitor, the resistor and the inductor. Note that the sum of the rms potential differences across the three elements is greater than the rms voltage of the source.

#### Answer:

Given:

Resistance in series LCR circuit, *R *= 300 Ω

Capacitance in series LCR circuit, *C* = 20 μF= 20 × 10^{−6} F

Inductance in series LCR circuit, *L* = 1 Henry

RMS value of voltage, *ε*_{rms} = 50 V

Frequency of source, *f* = 50/$\mathrm{\pi}$ Hz

Reactance of the inductor (*X*_{L}) is given by,

*X*_{L}= $\omega L$ = 2$\mathrm{\pi}$*fL** *

$\Rightarrow $*X*_{L} = 2$\times \mathrm{\pi}\times \frac{50}{\mathrm{\pi}}\times 1$ = 100 $\mathrm{\Omega}$

Reactance of the capacitance $\left({X}_{C}\right)$ is given by,

${X}_{C}=\frac{1}{\omega C}$ = $\frac{1}{2\mathrm{\pi}fC}$

*$\Rightarrow $**X*_{C} = $\frac{1}{2\mathrm{\pi}\times {\displaystyle \frac{50}{\mathrm{\pi}}}\times 20\times {10}^{-6}}$

$\Rightarrow $ *X*_{C} = $500$ $\mathrm{\Omega}$

(a) Impedance of an LCR circuit $\left(Z\right)$ is given by,

$Z=\sqrt{{R}^{2}+{\left({X}_{\mathrm{C}}-{X}_{\mathrm{L}}\right)}^{2}}\phantom{\rule{0ex}{0ex}}\Rightarrow Z=\sqrt{{\left(300\right)}^{2}+{\left(500-100\right)}^{2}}\phantom{\rule{0ex}{0ex}}\Rightarrow Z=\sqrt{{\left(300\right)}^{2}+{\left(400\right)}^{2}}\phantom{\rule{0ex}{0ex}}\Rightarrow Z=500\phantom{\rule{0ex}{0ex}}$

RMS value of current $\left({I}_{rms}\right)$ is given by,

${I}_{rms}=\frac{{\epsilon}_{rms}}{Z}$

$\Rightarrow $*I*_{rms} = $\frac{50}{500}$

$\Rightarrow $*I*_{rms} = $0.1\mathrm{A}$

(b) Potential across the capacitor $\left({V}_{C}\right)$ is given by,

*V*_{C} = *I*_{rms} × X_{C}

$\Rightarrow $*V _{C}* = 0.1 × 500 = 50 V

Potential difference across the resistor $\left({V}_{R}\right)$ is given by,

*V*

_{R = }

*I*

_{rms}×

*R*

$\Rightarrow $

*V*= 0.1 × 300 = 30 V

_{R}Potential difference across the inductor $\left({V}_{L}\right)$ is given by,

*V*

_{L}=

*I*

_{rms}× X

_{L}

$\Rightarrow $

*V*

_{L}= 0.1 × 100 = 10 V

R.M.S potential = 50 V

Net sum of all the potential drops = 50 V + 30 V + 10 V = 90 V

Sum of the potential drops > RMS potential applied

#### Page No 330:

#### Question 15:

Consider the situation of the previous problem. Find the average electric field energy stored in the capacitor and the average magnetic field energy stored in the coil.

#### Answer:

Given:

Resistance in the LCR circuit, *R* = 300 Ω

Capacitance in the LCR circuit, *C* = 20 μF = 20 × 10^{−6} F

Inductance in the LCR circuit, *L* = 1 henry

Net impedance of the LCR circuit, *Z *= 500 ohm

RMS value of voltage, ${\epsilon}_{\mathrm{rms}}$ = 50 V

RMS value of current,* **I*_{rms} = 0.1 A

Peak current $\left({I}_{0}\right)$ is given by,

${I}_{0}=\frac{{E}_{\mathrm{rms}}\sqrt{\mathit{2}}}{Z}=\frac{50\times 1.414}{500}=0.1414\mathrm{A}$

Electrical energy stored in capacitor$\left({U}_{C}\right)$ is given by,

${U}_{C}=\frac{1}{2}C{V}^{2}$

$\Rightarrow {U}_{C}=\frac{1}{2}\times 20\times {10}^{-6}\times 50\times 50\phantom{\rule{0ex}{0ex}}\Rightarrow {U}_{C}=25\times {10}^{-3}\mathrm{J}=25\mathrm{mJ}$

Magnetic field energy stored in the coil $\left({U}_{L}\right)$ is given by,

${U}_{L}=\frac{1}{2}L{I}_{0}^{2}$

$\Rightarrow {U}_{L}=\frac{1}{2}\times 1\times {\left(0.1414\right)}^{2}\phantom{\rule{0ex}{0ex}}\Rightarrow {U}_{L}\approx 5\times {10}^{-3}\mathrm{J}\phantom{\rule{0ex}{0ex}}\Rightarrow {U}_{L}=5\mathrm{mJ}$

#### Page No 330:

#### Question 16:

An inductance of 2.0 H, a capacitance of 18μF and a resistance of 10 kΩ are connected to an AC source of 20 V with adjustable frequency. (a) What frequency should be chosen to maximise the current in the circuit? (b) What is the value of this maximum current?

#### Answer:

Given:

Inductance of inductor, *L =* 2.0 H

Capacitance of capacitor, *C* = 18 μF

Resistance of resistor, *R* = 10 kΩ

Voltage of AC source, *E = *20 V

(a) In an LCR circuit, current is maximum when reactance is minimum, which occurs at resonance, i.e. when capacitive reactance becomes equal to the inductive reactance,i.e.

* **X*_{L} =* **X*_{C}

$\Rightarrow \omega L=\frac{1}{\omega C}\phantom{\rule{0ex}{0ex}}\Rightarrow {\omega}^{2}=\frac{1}{LC}=\frac{1}{2\times 18\times {10}^{-6}}\phantom{\rule{0ex}{0ex}}\Rightarrow {\omega}^{2}=\frac{{10}^{6}}{36}\phantom{\rule{0ex}{0ex}}\Rightarrow \omega =\frac{{10}^{3}}{6}\phantom{\rule{0ex}{0ex}}\Rightarrow 2\mathrm{\pi}f=\frac{{10}^{3}}{6}\phantom{\rule{0ex}{0ex}}\Rightarrow f=\frac{1000}{6\times 2\mathrm{\pi}}=26.539\mathrm{Hz}\phantom{\rule{0ex}{0ex}}=27\mathrm{Hz}$

(b) At resonance, reactance is minimum.

Minimum Reactance*, Z* = *R*

Maximum current (*I*) is given by,

$I=\frac{E}{R}\phantom{\rule{0ex}{0ex}}\Rightarrow I\mathit{=}\frac{20}{10\times {10}^{3}}$

$\Rightarrow I=\frac{2\mathrm{A}}{{10}^{3}}=2\mathrm{mA}$

#### Page No 330:

#### Question 17:

An inductor-coil, a capacitor and an AC source of rms voltage 24 V are connected in series. When the frequency of the source is varied, a maximum rms current of 6.0 A is observed. If this inductor coil is connected to a battery of emf 12 V and internal resistance 4.0 Ω, what will be the current?

#### Answer:

RMS value of voltage*, **E*_{rms} = 24 V

Internal resistance of battery, *r* = 4 Ω

RMS value of current, *I _{rms}* = 6 A

Reactance $\left(R\right)$ is given by,

$R=\frac{E}{{I}_{\mathrm{rms}}}\phantom{\rule{0ex}{0ex}}\Rightarrow R=\frac{24}{6}=4\mathrm{\Omega}$

Let

*R*' be the total resistance of the circuit. Then,

*R*' =

*R*+

*r*

$\Rightarrow $ R'

$\Rightarrow $ R

*=*4 Ω + 4 Ω

$\Rightarrow $

*R*' = 8 Ω

Current,

*I*= $\frac{12}{8}\mathrm{A}$

= 1.5 A

#### Page No 330:

#### Question 18:

Figure (39-E1) shows a typical circuit for a low-pass filter. An AC input V_{i }= 10 mV is applied at the left end and the output V_{0} is received at the right end. Find the output voltage for ν = 10 k Hz, 1.0 MHz and 10.0 MHz. Note that as the frequency is increased the output decreases and, hence, the name low-pass filter.

Figure

#### Answer:

Here,

Input voltage to the filter, *V*i = 10 × 10^{−3}^{ }V

Resistance of the circuit, *R* = 1 × 10^{3} Ω

Capacitance of the circuit, *C* = 10 × 10^{−9} F

(a) When frequency, *f = *10 kHz

A low pass filter consists of resistance and capacitance. Voltage across the capacitor is taken as the output.

Capacitive reactance $\left({X}_{C}\right)$ is given by,

${X}_{\mathrm{C}}=\frac{\mathit{1}}{\mathit{\omega}\mathit{C}}=\frac{1}{2\mathrm{\pi}fC}$

$\Rightarrow {X}_{C}=\frac{1}{2\mathrm{\pi}\times 10\times {10}^{3}\times 10\times {10}^{-9}}\phantom{\rule{0ex}{0ex}}\Rightarrow {X}_{C}=\frac{1}{2\mathrm{\pi}\times {10}^{-4}}\phantom{\rule{0ex}{0ex}}\Rightarrow {X}_{C}=\frac{{10}^{4}}{2\mathrm{\pi}}=\frac{5000}{\mathrm{\pi}}\mathrm{\Omega}$

Net impedence of the resistance-capacitance circuit (Z) is given by,

$Z=\sqrt{{R}^{2}+{X}_{\mathrm{C}}^{2}}\phantom{\rule{0ex}{0ex}}\Rightarrow Z=\sqrt{\left(1+{10}^{3}\right)+{\left(5000/\mathrm{\pi}\right)}^{2}}\phantom{\rule{0ex}{0ex}}\Rightarrow Z=\sqrt{{10}^{6}+{\left(5000/\mathrm{\pi}\right)}^{2}}\phantom{\rule{0ex}{0ex}}$

Current (*I*_{0}) is given by,

${I}_{0}=\frac{{V}_{i}}{Z}\phantom{\rule{0ex}{0ex}}\Rightarrow {I}_{0}=\frac{10\times {10}^{-3}}{\sqrt{{10}^{6}+{\left(5000/\mathrm{\pi}\right)}^{2}}}\phantom{\rule{0ex}{0ex}}$

Output across the capacitor $\left({V}_{0}\right)$ is given by,

* *${V}_{0}=\frac{{10}^{2}}{\sqrt{{10}^{5}+{\left(50/\mathrm{\pi}\right)}^{2}}}\times \frac{500}{\mathrm{\pi}}\phantom{\rule{0ex}{0ex}}\Rightarrow {V}_{0}=1.6124\mathrm{V}=1.6\mathrm{mV}$

(b)When frequency,* f *= 1 MHz = $1\times $10^{6} Hz

Capacitive reactance $\left({X}_{C}\right)$ is given by,

${X}_{c}=\frac{1}{\omega C}\phantom{\rule{0ex}{0ex}}\Rightarrow {X}_{C}=\frac{\mathit{1}}{\mathit{2}\pi fC}\phantom{\rule{0ex}{0ex}}\Rightarrow {X}_{C}=\frac{1}{2\mathrm{\pi}\times {10}^{6}\times {10}^{-9}\times 10}\phantom{\rule{0ex}{0ex}}\Rightarrow {X}_{C}=\frac{1}{2\mathrm{\pi}\times {10}^{-2}}\phantom{\rule{0ex}{0ex}}\Rightarrow {X}_{C}=\frac{100}{2\mathrm{\pi}}\phantom{\rule{0ex}{0ex}}\Rightarrow {X}_{C}=\frac{500}{\mathrm{\pi}}\mathrm{\Omega}\phantom{\rule{0ex}{0ex}}\mathrm{Total}\mathrm{impedence}\left(Z\right)=\sqrt{{R}^{2}+{{\mathrm{X}}_{\mathrm{C}}}^{2}}\phantom{\rule{0ex}{0ex}}\Rightarrow Z=\sqrt{{\left({10}^{3}\right)}^{2}+{\left(50/\mathrm{\pi}\right)}^{2}}\phantom{\rule{0ex}{0ex}}\mathrm{Current}\left({I}_{0}\right)=\frac{{V}_{\mathit{1}}}{Z}\phantom{\rule{0ex}{0ex}}\Rightarrow {I}_{0}=\frac{10\times {10}^{-3}}{\sqrt{{10}^{6}+{\left(50/\mathrm{\pi}\right)}^{2}}}\phantom{\rule{0ex}{0ex}}\mathrm{Output}\mathrm{voltage}\left({V}_{0}\right)\mathit{=}{I}_{\mathit{0}}{X}_{C}\phantom{\rule{0ex}{0ex}}\Rightarrow {V}_{0}=\frac{{10}^{-2}}{\sqrt{{10}^{5}+{\left(50/\mathrm{\pi}\right)}^{2}}}\times \frac{50}{\mathrm{\pi}}\phantom{\rule{0ex}{0ex}}\Rightarrow {V}_{0}=0.16\mathrm{mV}$

(c) When frequency, *f* = 10 MHz = 10 × 10^{6} Hz = 10^{7} Hz

Capacitive reactance $\left({X}_{C}\right)$ is given by,

${X}_{c}=\frac{1}{\omega C}\phantom{\rule{0ex}{0ex}}\Rightarrow {X}_{C}=\frac{1}{2\pi fC}\phantom{\rule{0ex}{0ex}}\Rightarrow {X}_{C}=\frac{1}{2\mathrm{\pi}\times {10}^{7}\times 10\times {10}^{-9}}\phantom{\rule{0ex}{0ex}}\Rightarrow {X}_{C}=\frac{5}{\mathrm{\pi}}\mathrm{\Omega}\phantom{\rule{0ex}{0ex}}\mathrm{Impedence}\left(Z\right)=\sqrt{{R}^{2}+{X}_{c}^{2}}\phantom{\rule{0ex}{0ex}}\Rightarrow Z=\sqrt{{\left({10}^{3}\right)}^{2}+{\left(5/\mathrm{\pi}\right)}^{2}}\phantom{\rule{0ex}{0ex}}\mathrm{Current}\left({I}_{0}\right)=\frac{{V}_{1}}{Z}\phantom{\rule{0ex}{0ex}}\Rightarrow {I}_{0}=\frac{10\times {10}^{-3}}{\sqrt{{10}^{6}+{\left(5/\mathrm{\pi}\right)}^{2}}}\phantom{\rule{0ex}{0ex}}{V}_{0}={I}_{0}{X}_{\mathrm{C}}\phantom{\rule{0ex}{0ex}}\Rightarrow {V}_{0}=\frac{{10}^{-2}}{\sqrt{{10}^{6}+{\left(5/\mathrm{\pi}\right)}^{2}}}\times \frac{5}{\mathrm{\pi}}\phantom{\rule{0ex}{0ex}}\Rightarrow {V}_{o}=16\mathrm{\mu V}$

#### Page No 331:

#### Question 19:

A transformer has 50 turns in the primary and 100 in the secondary. If the primary is connected to a 220 V DC supply, what will be the voltage across the secondary?

#### Answer:

A transformer works on the principle of electromagnetic induction, which is only possible in case of AC.

Hence, when DC (zero frequency) is supplied to it, the primary coil blocks the current supplied to it. Thus, the induced current in the secondary coil is zero. So, the output voltage will be zero.

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