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#### Question 1:

Given:
Charge on the positive plane = +Q1
Charge on the negative plate = $-$Q2

To calculate: Charge on the capacitor

Let ABCD be the Gaussian surface such that faces AD and BC lie inside plates X and Y, respectively.
Let q be the charge appearing on surface II. Then, the distribution of the charges on faces I, III and IV will be in accordance with the figure.

Let the area of the plates be A and the permittivity of the free space be${\in }_{0}$.

Now, to determine qâ€‹ in terms of Q1 and Q2, we need to apply Gauss's law to calculate the electric field due to all four faces of the capacitor at point P. Also, we know that the electric field inside a capacitor is zero.

Electric field due to face I at point P, E1$\frac{{Q}_{1}-q}{2{\in }_{0}A}$
Electric field due to face II at point P, E2$\frac{+q}{2{\in }_{0}A}$
Electric field due to face III at point P, E3$\frac{-q}{2{\in }_{0}A}$
Electric field due to face IV at point P, E4$-\left(\frac{-{Q}_{2}+q}{2{\in }_{0}A}\right)$ (Negative sign is used as point P lies on the LHS of face IV.)
Since point P lies inside the conductor,
E1 + E2 + E3 + E40

Q1 $-$q + q $-$q $-$ ($-$Qâ€‹2 + q ) = 0
$⇒$q = $\frac{{Q}_{1}+{Q}_{2}}{2}$
Thus, the charge on the capacitor is $\frac{{Q}_{1}+{Q}_{2}}{2}$, which is the charge on faces II and III.

#### Question 2:

Given:
Charge on the positive plane = +Q1
Charge on the negative plate = $-$Q2

To calculate: Charge on the capacitor

Let ABCD be the Gaussian surface such that faces AD and BC lie inside plates X and Y, respectively.
Let q be the charge appearing on surface II. Then, the distribution of the charges on faces I, III and IV will be in accordance with the figure.

Let the area of the plates be A and the permittivity of the free space be${\in }_{0}$.

Now, to determine qâ€‹ in terms of Q1 and Q2, we need to apply Gauss's law to calculate the electric field due to all four faces of the capacitor at point P. Also, we know that the electric field inside a capacitor is zero.

Electric field due to face I at point P, E1$\frac{{Q}_{1}-q}{2{\in }_{0}A}$
Electric field due to face II at point P, E2$\frac{+q}{2{\in }_{0}A}$
Electric field due to face III at point P, E3$\frac{-q}{2{\in }_{0}A}$
Electric field due to face IV at point P, E4$-\left(\frac{-{Q}_{2}+q}{2{\in }_{0}A}\right)$ (Negative sign is used as point P lies on the LHS of face IV.)
Since point P lies inside the conductor,
E1 + E2 + E3 + E40

Q1 $-$q + q $-$q $-$ ($-$Qâ€‹2 + q ) = 0
$⇒$q = $\frac{{Q}_{1}+{Q}_{2}}{2}$
Thus, the charge on the capacitor is $\frac{{Q}_{1}+{Q}_{2}}{2}$, which is the charge on faces II and III.

No. Since capacitance is a proportionality constant, it depends neither on the charge on the plates nor on the potential. It only depends upon the size and shape of the capacitor and on the dielectric used between the plates.
The formula that shows its dependence on the size and shape of the capacitor is as follows:
C$\frac{{\in }_{0}A}{d}$
Here, A is the area of the plates of the capacitor and d is the distance between the plates of the capacitor.

#### Question 3:

No. Since capacitance is a proportionality constant, it depends neither on the charge on the plates nor on the potential. It only depends upon the size and shape of the capacitor and on the dielectric used between the plates.
The formula that shows its dependence on the size and shape of the capacitor is as follows:
C$\frac{{\in }_{0}A}{d}$
Here, A is the area of the plates of the capacitor and d is the distance between the plates of the capacitor.

The potential of a metal sphere is directly proportional to the charge q given to it and inversely proportional to its radius r.
i.e.
$V=\frac{q}{4\mathrm{\pi }{\in }_{0}r}$
Since both the spheres are conductors with the same radius and charge, the charge given to them appears on the surface evenly. Thus, the potential on the surface or within the sphere will be the same, no matter the sphere is hollow or solid.

#### Question 4:

The potential of a metal sphere is directly proportional to the charge q given to it and inversely proportional to its radius r.
i.e.
$V=\frac{q}{4\mathrm{\pi }{\in }_{0}r}$
Since both the spheres are conductors with the same radius and charge, the charge given to them appears on the surface evenly. Thus, the potential on the surface or within the sphere will be the same, no matter the sphere is hollow or solid.

It is given that the plates of the capacitor have the same charges. In other words, they are at the same potential, so the potential difference between them is zero.

Let us consider that the charge on face II is q so that the induced charge on face III is $-$q and the distribution is according to the figure.
Now, if we consider Gaussian surface ABCD, whose faces lie inside the two plates, and calculate the field at point P due to all four surfaces, it will be
E1$=\frac{Q-q}{2{\in }_{0}A}$
E2 =  $\frac{q}{2{\in }_{0}A}$
E3 =  _$\frac{q}{2{\in }_{0}A}$
E4 =_ $\frac{Q+q}{2{\in }_{0}A}$      (It is $-$ve because point P is on the left side of face IV.)
Now, as point P lies inside the conductor, the total field must be zero.
E1 + E2 + E3 + E4 = 0
or
Q $-$qq $-$q + Qq = 0
q = 0
Hence, on faces II and III, the charge is equal to zero; and on faces I and IV, the charge is Q.
Thus, it seems that the whole charge given is moved to the outer surfaces, with zero charge on the facing surfaces.

#### Question 5:

It is given that the plates of the capacitor have the same charges. In other words, they are at the same potential, so the potential difference between them is zero.

Let us consider that the charge on face II is q so that the induced charge on face III is $-$q and the distribution is according to the figure.
Now, if we consider Gaussian surface ABCD, whose faces lie inside the two plates, and calculate the field at point P due to all four surfaces, it will be
E1$=\frac{Q-q}{2{\in }_{0}A}$
E2 =  $\frac{q}{2{\in }_{0}A}$
E3 =  _$\frac{q}{2{\in }_{0}A}$
E4 =_ $\frac{Q+q}{2{\in }_{0}A}$      (It is $-$ve because point P is on the left side of face IV.)
Now, as point P lies inside the conductor, the total field must be zero.
E1 + E2 + E3 + E4 = 0
or
Q $-$qq $-$q + Qq = 0
q = 0
Hence, on faces II and III, the charge is equal to zero; and on faces I and IV, the charge is Q.
Thus, it seems that the whole charge given is moved to the outer surfaces, with zero charge on the facing surfaces.

No. This information is not sufficient. Since the charge is proportional to the potential difference across the capacitor, we need to know the potential difference applied across the capacitor.

$\propto$V $⇒$q = CV

Here, q is the charge, V is the potential difference applied and C is the proportionality constant, i.e. capacitance.

#### Question 6:

No. This information is not sufficient. Since the charge is proportional to the potential difference across the capacitor, we need to know the potential difference applied across the capacitor.

$\propto$V $⇒$q = CV

Here, q is the charge, V is the potential difference applied and C is the proportionality constant, i.e. capacitance.

The amount of polarisation can be understood as the extent of perfect alignment of the molecules of a dielectric with an external electric field. The more aligned the molecules are with the external magnetic field, the more is the polarisation and the more will be the dielectric constant.
But with increase in temperature, the thermal agitation of the molecules or the randomness in their alignment with the field increases.
Thus, we can say that increase in temperature results in reduced polarisation and reduced dielectric constant.

#### Question 7:

The amount of polarisation can be understood as the extent of perfect alignment of the molecules of a dielectric with an external electric field. The more aligned the molecules are with the external magnetic field, the more is the polarisation and the more will be the dielectric constant.
But with increase in temperature, the thermal agitation of the molecules or the randomness in their alignment with the field increases.
Thus, we can say that increase in temperature results in reduced polarisation and reduced dielectric constant.

As the energy of the system decreases, the change in the energy is negative.
Force is defined as a negative rate of change of energy with respect to distance.
F$-\frac{\partial U}{\partial x}$
So, as the energy decreases, the force due to the electric field of the capacitor increases when the dielectric is dragged into the capacitor.

#### Question 1:

As the energy of the system decreases, the change in the energy is negative.
Force is defined as a negative rate of change of energy with respect to distance.
F$-\frac{\partial U}{\partial x}$
So, as the energy decreases, the force due to the electric field of the capacitor increases when the dielectric is dragged into the capacitor.

(d) zero

Since the net charge enclosed by the Gaussian surface is zero, the total flux of the electric field through the closed Gaussian surface enclosing the capacitor is zero.

$\Phi =∯E.ds=\frac{q}{{\in }_{0}}$= 0
Here,
$\Phi$ = Electric flux
q = Total charge enclosed by the Gaussian surface.

#### Question 2:

(d) zero

Since the net charge enclosed by the Gaussian surface is zero, the total flux of the electric field through the closed Gaussian surface enclosing the capacitor is zero.

$\Phi =∯E.ds=\frac{q}{{\in }_{0}}$= 0
Here,
$\Phi$ = Electric flux
q = Total charge enclosed by the Gaussian surface.

(d) C/2 and 2V

Since the voltage gets added up when the capacitors are connected in series, the voltage of the combination is 2V.
Also, the capacitance of a series combination is given by
$\frac{1}{{C}_{\mathrm{net}}}=\frac{1}{{C}_{1}}+\frac{1}{{C}_{2}}$
Here,
Câ€‹netâ€‹ = Net capacitance of the combination
C1 = C2 = C
Cnet= $\frac{C}{2}$

#### Question 3:

(d) C/2 and 2V

Since the voltage gets added up when the capacitors are connected in series, the voltage of the combination is 2V.
Also, the capacitance of a series combination is given by
$\frac{1}{{C}_{\mathrm{net}}}=\frac{1}{{C}_{1}}+\frac{1}{{C}_{2}}$
Here,
Câ€‹netâ€‹ = Net capacitance of the combination
C1 = C2 = C
Cnet= $\frac{C}{2}$

(c) 2C and V

In a parallel combination of capacitors, the potential difference across the capacitors remain the same, as the right-hand-side plates and the left-hand-side plates of both the capacitors are connected to the same terminals of the battery. Therefore, the potential remains the same, that is, V.

For the parallel combination of capacitors, the capacitance is given by
Ceq = C1 + C2
Here,
C1 = C2 = C
Ceq = 2C

#### Question 4:

(c) 2C and V

In a parallel combination of capacitors, the potential difference across the capacitors remain the same, as the right-hand-side plates and the left-hand-side plates of both the capacitors are connected to the same terminals of the battery. Therefore, the potential remains the same, that is, V.

For the parallel combination of capacitors, the capacitance is given by
Ceq = C1 + C2
Here,
C1 = C2 = C
Ceq = 2C

(b) 2C

Since the potential at point A is equal to the potential at point B, no current will flow along arm AB. Hence, the capacitor on arm AB will not contribute to the circuit. Also, because the remaining two capacitors are connected in parallel, the net capacitance of the circuit is given by
Ceq = C + C = 2C

#### Question 5:

(b) 2C

Since the potential at point A is equal to the potential at point B, no current will flow along arm AB. Hence, the capacitor on arm AB will not contribute to the circuit. Also, because the remaining two capacitors are connected in parallel, the net capacitance of the circuit is given by
Ceq = C + C = 2C

(c) remain unchanged

The force between the plates is given by
F = $\frac{{q}^{2}}{2{\in }_{0}A}$
Since the capacitor is isolated, the charge on the plates remains constant.
We know that the charge is conserved in an isolated system.
Thus, the force acting between the plates remains unchanged.

#### Question 6:

(c) remain unchanged

The force between the plates is given by
F = $\frac{{q}^{2}}{2{\in }_{0}A}$
Since the capacitor is isolated, the charge on the plates remains constant.
We know that the charge is conserved in an isolated system.
Thus, the force acting between the plates remains unchanged.

(d) $\frac{1}{{r}^{4}}$

Energy density U is given by

U = $\frac{1}{2}{\in }_{0}{E}^{2}$                ...(1)

The electric field created by a point charge at a distance r is given by
E = $\frac{q}{4\mathrm{\pi }{\in }_{0}{r}^{\mathit{2}}}$

On putting the above form of E in eq. 1, we get

U = $\frac{1}{2}{\in }_{0}{\left(\frac{q}{4\mathrm{\pi }{\in }_{0}{r}^{\mathit{2}}}\right)}^{2}$

Thus, U is directly proportional to $\frac{1}{{r}^{4}}$.

#### Question 7:

(d) $\frac{1}{{r}^{4}}$

Energy density U is given by

U = $\frac{1}{2}{\in }_{0}{E}^{2}$                ...(1)

The electric field created by a point charge at a distance r is given by
E = $\frac{q}{4\mathrm{\pi }{\in }_{0}{r}^{\mathit{2}}}$

On putting the above form of E in eq. 1, we get

U = $\frac{1}{2}{\in }_{0}{\left(\frac{q}{4\mathrm{\pi }{\in }_{0}{r}^{\mathit{2}}}\right)}^{2}$

Thus, U is directly proportional to $\frac{1}{{r}^{4}}$.

(b) ${Q}_{+}={Q}_{-}$

The charge induced on the plates of a capacitor is independent of the area of the plates.
${Q}_{+}={Q}_{-}$

#### Question 8:

(b) ${Q}_{+}={Q}_{-}$

The charge induced on the plates of a capacitor is independent of the area of the plates.
${Q}_{+}={Q}_{-}$

(d) $\infty$

The thin metal plate inserted between the plates of a parallel-plate capacitor of capacitance C connects the two plates of the capacitor; hence, the distance d between the plates of the capacitor reduces to zero. It can be observed that the charges on the plates begin to overlap each other via the metallic plate and hence begin to conduct continuously.

Mathematically,
C = $\frac{{\in }_{0}A}{d}$
In this case, d = 0.
C = $\infty$

#### Question 9:

(d) $\infty$

The thin metal plate inserted between the plates of a parallel-plate capacitor of capacitance C connects the two plates of the capacitor; hence, the distance d between the plates of the capacitor reduces to zero. It can be observed that the charges on the plates begin to overlap each other via the metallic plate and hence begin to conduct continuously.

Mathematically,
C = $\frac{{\in }_{0}A}{d}$
In this case, d = 0.
C = $\infty$

(c) C1 < Câ€‹2

Region AB shows the potential difference across capacitor Câ€‹1 and region CD shows the potential difference across capacitor C2. Now, we can see from the graph that region AB is greater than region CD. Therefore, the potential difference across capacitor C1 is greater than that across capacitor C2
âˆµ Capacitance, C$\frac{Q}{V}$
∴ Câ€‹â€‹1 < C2              (Q remains the same in series connection.)

#### Question 10:

(c) C1 < Câ€‹2

Region AB shows the potential difference across capacitor Câ€‹1 and region CD shows the potential difference across capacitor C2. Now, we can see from the graph that region AB is greater than region CD. Therefore, the potential difference across capacitor C1 is greater than that across capacitor C2
âˆµ Capacitance, C$\frac{Q}{V}$
∴ Câ€‹â€‹1 < C2              (Q remains the same in series connection.)

(a) increase

Oil between the plates of the capacitor acts as a dielectric. We know that the electric field decreases by a factor of $\frac{1}{K}$ of the original field when we insert a dielectric between the plates of a capacitor (K is the dielectric constant of the dielectric). So, if the oil is pumped out, the electric field between the plates will increase, as the dielectric has been removed.

#### Question 11:

(a) increase

Oil between the plates of the capacitor acts as a dielectric. We know that the electric field decreases by a factor of $\frac{1}{K}$ of the original field when we insert a dielectric between the plates of a capacitor (K is the dielectric constant of the dielectric). So, if the oil is pumped out, the electric field between the plates will increase, as the dielectric has been removed.

(b) $\frac{{Q}_{1}}{{Q}_{2}}=\frac{{C}_{1}}{{C}_{2}}$

When the spheres are connected, charges flow between them until they both acquire the same common potential V.
The final charges on the spheres are given by
Q1 = Câ€‹1V and Q2 = C2V

$\therefore \frac{{Q}_{1}}{{Q}_{2}}=\frac{{C}_{1}V}{{C}_{2}V}$ = $\frac{{C}_{1}}{{C}_{2}}$
C1C2V

#### Question 12:

(b) $\frac{{Q}_{1}}{{Q}_{2}}=\frac{{C}_{1}}{{C}_{2}}$

When the spheres are connected, charges flow between them until they both acquire the same common potential V.
The final charges on the spheres are given by
Q1 = Câ€‹1V and Q2 = C2V

$\therefore \frac{{Q}_{1}}{{Q}_{2}}=\frac{{C}_{1}V}{{C}_{2}V}$ = $\frac{{C}_{1}}{{C}_{2}}$
C1C2V

(d) 2 µF, 18 µF

The minimum capacitance can be obtained by connecting all capacitors in series. It can be calculated as follows:

$⇒$C = 2 µF

The maximum capacitance can be obtained by connecting all capacitors in parallel. It can be calculated as follows:
C = 6 + 6 + 6 = 18 µF

#### Question 1:

(d) 2 µF, 18 µF

The minimum capacitance can be obtained by connecting all capacitors in series. It can be calculated as follows:

$⇒$C = 2 µF

The maximum capacitance can be obtained by connecting all capacitors in parallel. It can be calculated as follows:
C = 6 + 6 + 6 = 18 µF

(c) the charges on the plates

The capacitance of a capacitor is given by

C = $\frac{{\in }_{0}A}{d}$

Here, A is the area of the plates of the capacitor and d is the distance between the plates.
So, we can clearly see that the capacitance of a capacitor does depend on the size and shape of the plates and the separation between the plates; it does not depend on the charges on the plates.

#### Question 2:

(c) the charges on the plates

The capacitance of a capacitor is given by

C = $\frac{{\in }_{0}A}{d}$

Here, A is the area of the plates of the capacitor and d is the distance between the plates.
So, we can clearly see that the capacitance of a capacitor does depend on the size and shape of the plates and the separation between the plates; it does not depend on the charges on the plates.

(b) The charge on the capacitor

When we insert a dielectric between the plates of a capacitor, induced charges of opposite polarity appear on the face of the dielectric. They build an electric field inside the dielectric, directed opposite to the original field of the capacitor.
Thus, the net effect is a reduced electric field.
Also, as the potential is proportional to the field, the potential decreases and so does the stored energy U, which is given by
U$\frac{qV}{2}$
Thus, only the charge on the capacitor remains unchanged, as the charge is conserved in an isolated system.

#### Question 3:

(b) The charge on the capacitor

When we insert a dielectric between the plates of a capacitor, induced charges of opposite polarity appear on the face of the dielectric. They build an electric field inside the dielectric, directed opposite to the original field of the capacitor.
Thus, the net effect is a reduced electric field.
Also, as the potential is proportional to the field, the potential decreases and so does the stored energy U, which is given by
U$\frac{qV}{2}$
Thus, only the charge on the capacitor remains unchanged, as the charge is conserved in an isolated system.

(d) Qâ€‹' must be smaller than Q.

The relation between the induced charge Q' and the charge on the capacitor Q is given by
Q' = $Q\left(1-\frac{1}{K}\right)$
Here, K is the dielectric constant that is always greater than or equal to 1.
So, we can see that for K > 1, Q' will always be less than Q.

#### Question 4:

(d) Qâ€‹' must be smaller than Q.

The relation between the induced charge Q' and the charge on the capacitor Q is given by
Q' = $Q\left(1-\frac{1}{K}\right)$
Here, K is the dielectric constant that is always greater than or equal to 1.
So, we can see that for K > 1, Q' will always be less than Q.

(a) the facing surfaces of the capacitor have equal and opposite charges
(b) the two plates of the capacitor have equal and opposite charges
(d) the outer surfaces of the plates have equal charges

In H.C Verma the answer is (a), (c) ,(d). But according to us the answer should be (a), (b), (d) all these options are the properties of a capacitor and the option (c) is incorrect because the battery is a source of energy not charge. Moreover if a capacitor plates have equal charge on outside and equal charge on inside then one can think that the charge on the plates must be also equal so option (b) cant be incorrect.

#### Question 5:

(a) the facing surfaces of the capacitor have equal and opposite charges
(b) the two plates of the capacitor have equal and opposite charges
(d) the outer surfaces of the plates have equal charges

In H.C Verma the answer is (a), (c) ,(d). But according to us the answer should be (a), (b), (d) all these options are the properties of a capacitor and the option (c) is incorrect because the battery is a source of energy not charge. Moreover if a capacitor plates have equal charge on outside and equal charge on inside then one can think that the charge on the plates must be also equal so option (b) cant be incorrect.

(b) Potential difference across the capacitor
(c) Energy of the capacitor.

Because the charge always remains conserved in an isolated system, it will remain the same.
Now,
V=$\frac{Qd}{{\in }_{0}A}$
Here, Q, A and d are the charge, area and distance between the plates, respectively.
Thus, as d increases, V increases.
Energy is given by
E$\frac{qV}{2}$
So, it will also increase.
Energy density u, that is, energy stored per unit volume in the electric field is given by
u = $\frac{1}{2}{\in }_{0}{E}^{2}$
So, u will remain constant with increase in distance between the plates.

#### Question 6:

(b) Potential difference across the capacitor
(c) Energy of the capacitor.

Because the charge always remains conserved in an isolated system, it will remain the same.
Now,
V=$\frac{Qd}{{\in }_{0}A}$
Here, Q, A and d are the charge, area and distance between the plates, respectively.
Thus, as d increases, V increases.
Energy is given by
E$\frac{qV}{2}$
So, it will also increase.
Energy density u, that is, energy stored per unit volume in the electric field is given by
u = $\frac{1}{2}{\in }_{0}{E}^{2}$
So, u will remain constant with increase in distance between the plates.

(d) Equal and opposite charges will appear on the two faces of the metal plate.

The capacitance of the capacitor in which a dielectric slab of dielectric constant K, area A and thickness t is inserted between the plates of the capacitor of area A and separated by a distance d is given by

C = $\frac{{\in }_{0}A}{\left(d-t\right)+\left(t/K\right)}$

Since it is given that the thickness of the sheet is negligible, the above formula reduces to C = $\frac{{\in }_{0}A}{d}$. In other words, there will not be any change in the electric field, potential or charge.

Only, equal and opposite charges will appear on the two faces of the metal plate because of induction due to the presence of the charges on the plates of the capacitor.

#### Question 7:

(d) Equal and opposite charges will appear on the two faces of the metal plate.

The capacitance of the capacitor in which a dielectric slab of dielectric constant K, area A and thickness t is inserted between the plates of the capacitor of area A and separated by a distance d is given by

C = $\frac{{\in }_{0}A}{\left(d-t\right)+\left(t/K\right)}$

Since it is given that the thickness of the sheet is negligible, the above formula reduces to C = $\frac{{\in }_{0}A}{d}$. In other words, there will not be any change in the electric field, potential or charge.

Only, equal and opposite charges will appear on the two faces of the metal plate because of induction due to the presence of the charges on the plates of the capacitor.

(b) The charge appearing on the capacitor is greater after the action XWY than after the action XYZ.
(c) The electric energy stored in the capacitor is greater after the action WXY than after the action XYW.
(d) The electric field in the capacitor after the action XW is the same as that after WX.

Justification of option (b)

If the potential is held constant, that is, the battery remains attached to the circuit, then the charge on the capacitor increases by a factor of K on inserting a dielectric of a dielectric constant K between the plates of the capacitor.

Mathematically,
q = Kqâ€‹0
Here, q0 and are the charges without dielectric and with dielectric, respectively.
The amount of charge stored does not depend upon the polarity of the plates.
Thus, the charge appearing on the capacitor is greater after the action XWY than after the action XYZ.

Justification of option (c)

Since the battery is disconnected before inserting a dielectric, the amount of charge remains constant, that is, q = qâ€‹0, because after the battery is disconnected, the capacitor gets no source to store charge from. In other words, the capacitor is now an isolated system where the amount of charge is conserved and so is the energy U as $\frac{1}{2}$$\epsilon$. Hence, inserting a dielectric after disconnecting the battery will not bring any change in the amount of charge stored in the capacitor. So, the energy stored in the capacitor will also not change after the action XYW.
However, during the action WXY, the amount of charge that will get stored in the capacitor will get increased by a factor of K, as the battery is disconnected after inserting a dielectric between the plates of the capacitor and the energy stored will also get multiplied by a factor of K.
Thus, the electric energy stored in the capacitor is greater after the action WXY than after the action XYW.

Justification of option (d)

The electric field between the plates E depends on the potential across the capacitor and the distance d between the plates of the capacitor.
Mathematically,
E = $\frac{\epsilon }{d}$
In either case, that is, during actions XW and WX, the potential remains the same, that is, $\epsilon$. Thus, the electric field E remains the same.

Denial of option (a)

During the action XYZ, the battery has to do extra work equivalent to $\frac{1}{2}$CV2 to change the polarity of the plates of the capacitor. In other words, the total work to be done by the battery will be $\frac{1}{2}$CV2 + $\frac{1}{2}$CV2. This extra work done will be dissipated as heat energy. Thus, thermal energy is developed. However, the stored electric energy remains unchanged, that is, $\frac{1}{2}$CV2.

#### Question 1:

(b) The charge appearing on the capacitor is greater after the action XWY than after the action XYZ.
(c) The electric energy stored in the capacitor is greater after the action WXY than after the action XYW.
(d) The electric field in the capacitor after the action XW is the same as that after WX.

Justification of option (b)

If the potential is held constant, that is, the battery remains attached to the circuit, then the charge on the capacitor increases by a factor of K on inserting a dielectric of a dielectric constant K between the plates of the capacitor.

Mathematically,
q = Kqâ€‹0
Here, q0 and are the charges without dielectric and with dielectric, respectively.
The amount of charge stored does not depend upon the polarity of the plates.
Thus, the charge appearing on the capacitor is greater after the action XWY than after the action XYZ.

Justification of option (c)

Since the battery is disconnected before inserting a dielectric, the amount of charge remains constant, that is, q = qâ€‹0, because after the battery is disconnected, the capacitor gets no source to store charge from. In other words, the capacitor is now an isolated system where the amount of charge is conserved and so is the energy U as $\frac{1}{2}$$\epsilon$. Hence, inserting a dielectric after disconnecting the battery will not bring any change in the amount of charge stored in the capacitor. So, the energy stored in the capacitor will also not change after the action XYW.
However, during the action WXY, the amount of charge that will get stored in the capacitor will get increased by a factor of K, as the battery is disconnected after inserting a dielectric between the plates of the capacitor and the energy stored will also get multiplied by a factor of K.
Thus, the electric energy stored in the capacitor is greater after the action WXY than after the action XYW.

Justification of option (d)

The electric field between the plates E depends on the potential across the capacitor and the distance d between the plates of the capacitor.
Mathematically,
E = $\frac{\epsilon }{d}$
In either case, that is, during actions XW and WX, the potential remains the same, that is, $\epsilon$. Thus, the electric field E remains the same.

Denial of option (a)

During the action XYZ, the battery has to do extra work equivalent to $\frac{1}{2}$CV2 to change the polarity of the plates of the capacitor. In other words, the total work to be done by the battery will be $\frac{1}{2}$CV2 + $\frac{1}{2}$CV2. This extra work done will be dissipated as heat energy. Thus, thermal energy is developed. However, the stored electric energy remains unchanged, that is, $\frac{1}{2}$CV2.

As 1.0 × 1012 electrons are transferred from one conductor to another, the conductor to which the electrons are transferred becomes negatively charged and the other conductor becomes positively charged.
Now,
Magnitude of the net charge on each conductor, Q = (1.0 × 1012) × (1.6 × 10$-$19) C = 1.6 × 10$-$7 C
Magnitude of the potential difference between the conductors, V = 10 V

The capacitance C is the ratio of the magnitude of the charge on either conductor to the magnitude of the potential difference between the conductors.
$C=\frac{Q}{V}$

Hence, the value of the capacitance of the given two conductor systems is 1.6 × 10$-$8 F.

#### Question 2:

As 1.0 × 1012 electrons are transferred from one conductor to another, the conductor to which the electrons are transferred becomes negatively charged and the other conductor becomes positively charged.
Now,
Magnitude of the net charge on each conductor, Q = (1.0 × 1012) × (1.6 × 10$-$19) C = 1.6 × 10$-$7 C
Magnitude of the potential difference between the conductors, V = 10 V

The capacitance C is the ratio of the magnitude of the charge on either conductor to the magnitude of the potential difference between the conductors.
$C=\frac{Q}{V}$

Hence, the value of the capacitance of the given two conductor systems is 1.6 × 10$-$8 F.

The capacitance of a parallel-plate capacitor is given by

Here,
A = Area of the plate
d = Distance between the parallel plates
Given:

#### Question 3:

The capacitance of a parallel-plate capacitor is given by

Here,
A = Area of the plate
d = Distance between the parallel plates
Given:

The capacitance of a parallel-plate capacitor is given by

Here,
A = Area of the plate
d = Distance between the parallel plates
Now,
Let the radius of the disc be r.

Thus, the radius of the plates of the capacitor for the given configuration is 6 km.

#### Question 4:

The capacitance of a parallel-plate capacitor is given by

Here,
A = Area of the plate
d = Distance between the parallel plates
Now,
Let the radius of the disc be r.

Thus, the radius of the plates of the capacitor for the given configuration is 6 km.

The capacitance of a parallel-plate capacitor is given by

Here,
A = Area of the plate
d = Distance between the parallel plates
Given:
A = 25 cm2 = 25 × 10$-$4 m2
d = 1.00 mm = 1 × 10$-$3 m
Now,

When the battery of voltage 6 V is connected to the capacitor, the charge (Q) that flows from the battery is equal to the amount of the charge that the given capacitor can hold.
$⇒$Q = CV

The work done by the battery in charging the capacitor is calculated by taking the product of the magnitude of the charge transferred and the voltage of the battery.
Thus, we get

Thus, the charge flown through the battery is 1.33 × 10$-$10 C and the work done by the battery is 8.0 × 10$-$10 J.

#### Question 5:

The capacitance of a parallel-plate capacitor is given by

Here,
A = Area of the plate
d = Distance between the parallel plates
Given:
A = 25 cm2 = 25 × 10$-$4 m2
d = 1.00 mm = 1 × 10$-$3 m
Now,

When the battery of voltage 6 V is connected to the capacitor, the charge (Q) that flows from the battery is equal to the amount of the charge that the given capacitor can hold.
$⇒$Q = CV

The work done by the battery in charging the capacitor is calculated by taking the product of the magnitude of the charge transferred and the voltage of the battery.
Thus, we get

Thus, the charge flown through the battery is 1.33 × 10$-$10 C and the work done by the battery is 8.0 × 10$-$10 J.

Given:
Area of the plate, A = 25 cm2 = 25 × 10$-$4 m2
Separation between the plates, d = 2 mm = 2 × 10$-$3 m
Potential difference between the plates, V = 12 V
The capacitance of the given capacitor is given by

(a) Charge on the capacitor is given by
Q = CV

(b) When the separation between the plates is decreased to 1 mm, the capacitance C' can be calculated as:

Charge on the capacitor is given by
Q' = C'V

#### Question 6:

Given:
Area of the plate, A = 25 cm2 = 25 × 10$-$4 m2
Separation between the plates, d = 2 mm = 2 × 10$-$3 m
Potential difference between the plates, V = 12 V
The capacitance of the given capacitor is given by

(a) Charge on the capacitor is given by
Q = CV

(b) When the separation between the plates is decreased to 1 mm, the capacitance C' can be calculated as:

Charge on the capacitor is given by
Q' = C'V

The capacitances of three capacitors are C1= 2 μF, C2= 4 μF and C3= 6 μF and the voltage of the battery (V) is 12 V.
As the capacitors are connected in parallel, the equivalent capacitance is given by
Ceq = C1 + C2 + C3
= (2 + 4 + 6) μF = 12 μF = 12 × 10$-$6 F

Due to parallel connection, the potential difference across each capacitor is the same and is equal to 12 V.

Therefore, the charge on each capacitor can be calculated as follows:
The charge on the capacitor of capacitance C1= 2 μF is given by

Q1= C1V = (2 × 10$-$6) × 12 C = 24 × 10$-$6 C = 24 μC

Similarly, the charges on the other two capacitors are given by
Q2 = C2V = (4 × 10$-$6) × 12 C = 48 × 10$-$6 C = 48 μC
and
Q3= C3V = (6 × 10$-$6) × 12 C = 72 × 10$-$6 C = 72 μC

#### Question 7:

The capacitances of three capacitors are C1= 2 μF, C2= 4 μF and C3= 6 μF and the voltage of the battery (V) is 12 V.
As the capacitors are connected in parallel, the equivalent capacitance is given by
Ceq = C1 + C2 + C3
= (2 + 4 + 6) μF = 12 μF = 12 × 10$-$6 F

Due to parallel connection, the potential difference across each capacitor is the same and is equal to 12 V.

Therefore, the charge on each capacitor can be calculated as follows:
The charge on the capacitor of capacitance C1= 2 μF is given by

Q1= C1V = (2 × 10$-$6) × 12 C = 24 × 10$-$6 C = 24 μC

Similarly, the charges on the other two capacitors are given by
Q2 = C2V = (4 × 10$-$6) × 12 C = 48 × 10$-$6 C = 48 μC
and
Q3= C3V = (6 × 10$-$6) × 12 C = 72 × 10$-$6 C = 72 μC

When the capacitors are connected in series, the equivalent capacitance is given by

Because the capacitors are connected in series, the same charge will go to each of them and it is equal to the total charge given by the battery.
Now,
Let the charge at each capacitor be q.

The work done by the battery (W) is given by

#### Question 8:

When the capacitors are connected in series, the equivalent capacitance is given by

Because the capacitors are connected in series, the same charge will go to each of them and it is equal to the total charge given by the battery.
Now,
Let the charge at each capacitor be q.

The work done by the battery (W) is given by

Let us first find the equivalent capacitance. It can be observed from the circuit diagram that capacitors B and C are in parallel and are in series with capacitor A.

The equivalent capacitance can be calculated as follows:

Capacitors B and C are parallel and are in series with capacitor A. The equivalent capacitance of capacitors B and C is given by
(4 + 4) μF = 8 μF
It is the same as the capacitance of capacitor A. Therefore, equal potential difference will be there on capacitor A and the system of capacitors B and C.
Now,
Potential difference across capacitor A = 6 V
Thus,
Charge on capacitor A = (8 µF) × (6 V) = 48 µC
And,
Potential difference across capacitors B and C = 6 V
Thus,
Charge on capacitor B = (4 µF) × (6 V) = 24 µF
Charge on capacitor C = (4 µF) × (6 V) = 24 µF

#### Question 9:

Let us first find the equivalent capacitance. It can be observed from the circuit diagram that capacitors B and C are in parallel and are in series with capacitor A.

The equivalent capacitance can be calculated as follows:

Capacitors B and C are parallel and are in series with capacitor A. The equivalent capacitance of capacitors B and C is given by
(4 + 4) μF = 8 μF
It is the same as the capacitance of capacitor A. Therefore, equal potential difference will be there on capacitor A and the system of capacitors B and C.
Now,
Potential difference across capacitor A = 6 V
Thus,
Charge on capacitor A = (8 µF) × (6 V) = 48 µC
And,
Potential difference across capacitors B and C = 6 V
Thus,
Charge on capacitor B = (4 µF) × (6 V) = 24 µF
Charge on capacitor C = (4 µF) × (6 V) = 24 µF

(a)

For the combination of capacitors given in figure (a), the pairs of capacitors C1and C2are in parallel.
The equivalent capacitance of each parallel combination of capacitors is given by
C1+ C2 = 4 + 6 = 10 μF
The equivalent circuit can be drawn as:

The equivalent capacitance for the above series circuit is given by

(b)

For the combination of capacitors given in figure (b), the pairs of capacitors C1and C2are in parallel.
The equivalent capacitance of each parallel combination of capacitors is given by
C1+ C2 = 4 + 6 = 10 μF

The equivalent circuit can be drawn as:

In the above circuit, it can be seen that CA and CB are in series and are in parallel to the series combination of CC and CB.

The equivalent capacitance for the series combination of CA and CB is given by

Similarly, the equivalent capacitance of the series combination of CC and CD is 5 μF.

∴ Net equivalent capacitance = 5 + 5 = 10 μF

#### Question 10:

(a)

For the combination of capacitors given in figure (a), the pairs of capacitors C1and C2are in parallel.
The equivalent capacitance of each parallel combination of capacitors is given by
C1+ C2 = 4 + 6 = 10 μF
The equivalent circuit can be drawn as:

The equivalent capacitance for the above series circuit is given by

(b)

For the combination of capacitors given in figure (b), the pairs of capacitors C1and C2are in parallel.
The equivalent capacitance of each parallel combination of capacitors is given by
C1+ C2 = 4 + 6 = 10 μF

The equivalent circuit can be drawn as:

In the above circuit, it can be seen that CA and CB are in series and are in parallel to the series combination of CC and CB.

The equivalent capacitance for the series combination of CA and CB is given by

Similarly, the equivalent capacitance of the series combination of CC and CD is 5 μF.

∴ Net equivalent capacitance = 5 + 5 = 10 μF

The equivalent circuit for the given case can be drawn as:

It can be observed that capacitors C1and C2 are in parallel.
Therefore, the equivalent capacitance is given by
Ceq = C1+ C2
= 5 + 6 = 11 μF
Now,
Charge supplied by the battery = (Ceq)(V) = (11 μF) × (10 V) = 110 μC

#### Question 11:

The equivalent circuit for the given case can be drawn as:

It can be observed that capacitors C1and C2 are in parallel.
Therefore, the equivalent capacitance is given by
Ceq = C1+ C2
= 5 + 6 = 11 μF
Now,
Charge supplied by the battery = (Ceq)(V) = (11 μF) × (10 V) = 110 μC

It is given that the outer cylinders are kept in contact and the inner cylinders are connected through a wire. Thus, the equivalent circuit can be drawn as:

The left plate of the capacitors (A and B) shown in the figure represents the inner plates of the cylindrical capacitors.

As the capacitors are connected in parallel, the potential difference across them is the same.
∴ Magnitude of the charge on each capacitor = CV = (2.2 μF) × (10V) = 22 µC
As plates A and B are connected to the positive terminal of the battery, the charge on each of them is +22 µC.
∴ Net charge on the inner plates = 22 µC + 22 µC = +44 µC

#### Question 12:

It is given that the outer cylinders are kept in contact and the inner cylinders are connected through a wire. Thus, the equivalent circuit can be drawn as:

The left plate of the capacitors (A and B) shown in the figure represents the inner plates of the cylindrical capacitors.

As the capacitors are connected in parallel, the potential difference across them is the same.
∴ Magnitude of the charge on each capacitor = CV = (2.2 μF) × (10V) = 22 µC
As plates A and B are connected to the positive terminal of the battery, the charge on each of them is +22 µC.
∴ Net charge on the inner plates = 22 µC + 22 µC = +44 µC

We need to calculate the capacitance of an isolated charged sphere.
Let us assume that the charge on the sphere is Q and its radius is R.

Capacitance of the charged sphere can be found by imagining a concentric sphere of infinite radius consisting of −Q charge.
Potential difference between the spheres = $\frac{1}{4\pi {\in }_{0}}\frac{Q}{R}$ − 0 = $\frac{1}{4\pi {\in }_{0}}\frac{Q}{R}$

Capacitance is the ratio of the magnitude of the charge on each conductor to the potential difference between them.
$C=\frac{Q}{\frac{1}{4\pi {\in }_{0}}\frac{Q}{R}}=4\pi {\in }_{0}R$
Therefore, the capacitances of spheres of radii R1 and R2 are C1 and C2, respectively. They are given by

If the spheres are connected by a metal wire, the charge will flow from one sphere to another till their potentials become the same.
As there potentials become the same, the potential difference between the conductors for both the capacitors also becomes the same. Thus, it can be concluded that the capacitors are connected in parallel.

Thus, the capacitance of the combination is given by
${C}_{\mathrm{eq}}$ = C1 + C2

#### Question 13:

We need to calculate the capacitance of an isolated charged sphere.
Let us assume that the charge on the sphere is Q and its radius is R.

Capacitance of the charged sphere can be found by imagining a concentric sphere of infinite radius consisting of −Q charge.
Potential difference between the spheres = $\frac{1}{4\pi {\in }_{0}}\frac{Q}{R}$ − 0 = $\frac{1}{4\pi {\in }_{0}}\frac{Q}{R}$

Capacitance is the ratio of the magnitude of the charge on each conductor to the potential difference between them.
$C=\frac{Q}{\frac{1}{4\pi {\in }_{0}}\frac{Q}{R}}=4\pi {\in }_{0}R$
Therefore, the capacitances of spheres of radii R1 and R2 are C1 and C2, respectively. They are given by

If the spheres are connected by a metal wire, the charge will flow from one sphere to another till their potentials become the same.
As there potentials become the same, the potential difference between the conductors for both the capacitors also becomes the same. Thus, it can be concluded that the capacitors are connected in parallel.

Thus, the capacitance of the combination is given by
${C}_{\mathrm{eq}}$ = C1 + C2

There are three rows of capacitors connected in parallel in the given system. In each row, three capacitors of capacitance 2 μF are connected in series.

For each row, the equivalent capacitance is given by

As three rows are connected in parallel, their equivalent capacitance is given by
Ceq= Cr + Cr + Cr = $\frac{2}{3}+\frac{2}{3}+\frac{2}{3}$ = 2 μF

The voltage across each row is the same and is equal to 60 V.
As all capacitors have the same capacitance in each row, the potential difference across their plates is the same.

∴ Potential difference across each capacitor = 20 V

#### Question 14:

There are three rows of capacitors connected in parallel in the given system. In each row, three capacitors of capacitance 2 μF are connected in series.

For each row, the equivalent capacitance is given by

As three rows are connected in parallel, their equivalent capacitance is given by
Ceq= Cr + Cr + Cr = $\frac{2}{3}+\frac{2}{3}+\frac{2}{3}$ = 2 μF

The voltage across each row is the same and is equal to 60 V.
As all capacitors have the same capacitance in each row, the potential difference across their plates is the same.

∴ Potential difference across each capacitor = 20 V

Let the number of capacitors in series (connected in a row) be x.
The maximum voltage that the capacitors can withstand is 50 V.
The voltage across each row should be equal to 200 V.
Therefore,
x × 50 = 200
Thus,
x = 4 capacitors
Now,
Let there be y such rows.
So, the equivalent capacitance of the combination will be xy.
xy = 10
y = 10 x = 4 capacitors
Thus, to yield the required result, the combination of 4 rows, each of 4 capacitors having capacitance 10 µF and breakdown voltage 50 V, is required.

#### Question 15:

Let the number of capacitors in series (connected in a row) be x.
The maximum voltage that the capacitors can withstand is 50 V.
The voltage across each row should be equal to 200 V.
Therefore,
x × 50 = 200
Thus,
x = 4 capacitors
Now,
Let there be y such rows.
So, the equivalent capacitance of the combination will be xy.
xy = 10
y = 10 x = 4 capacitors
Thus, to yield the required result, the combination of 4 rows, each of 4 capacitors having capacitance 10 µF and breakdown voltage 50 V, is required.

(a)
The capacitance of the two rows connected in parallel is given by
and

As the two rows are in parallel, the potential difference across each row is the same and is equal to 50 V.
The charge on the branch ACB with capacitance is given by
Q = ×(50 V) =

The charge on the branch ADB with capacitance is given by

The potential at point D is given by

Similarly, the potential at point C is given by

(b) As the potential difference between points C and D is zero, the bridge remains balanced and no charge flows from C to D. If a capacitor is connected between points C and D, then the change on the capacitor will be zero.

#### Question 16:

(a)
The capacitance of the two rows connected in parallel is given by
and

As the two rows are in parallel, the potential difference across each row is the same and is equal to 50 V.
The charge on the branch ACB with capacitance is given by
Q = ×(50 V) =

The charge on the branch ADB with capacitance is given by

The potential at point D is given by

Similarly, the potential at point C is given by

(b) As the potential difference between points C and D is zero, the bridge remains balanced and no charge flows from C to D. If a capacitor is connected between points C and D, then the change on the capacitor will be zero.

Capacitors C1and C2 above and below ab are connected in series and in parallel with C3.
Thus, the equivalent capacitor between a and b is given by

#### Question 17:

Capacitors C1and C2 above and below ab are connected in series and in parallel with C3.
Thus, the equivalent capacitor between a and b is given by

The total area of the flat plate is A. The width of each stair is the same. Therefore, the area of the surface of each stair facing the flat plate is the same, that is, $\frac{A}{3}$.

From the figure, it can be observed that the capacitor assembly is made up of three capacitors. The three capacitors are connected in parallel.
For capacitor C1, the area of the plates is $\frac{A}{3}$ and the separation between the plates is d.
For capacitor C2, the area of the plates is $\frac{A}{3}$ and the separation between the plates is (d + b).
For capacitor C3, the area of the plates is $\frac{A}{3}$ and the separation between the plates is (d + 2b).
Therefore,

As the three capacitors are in parallel combination,

#### Question 18:

The total area of the flat plate is A. The width of each stair is the same. Therefore, the area of the surface of each stair facing the flat plate is the same, that is, $\frac{A}{3}$.

From the figure, it can be observed that the capacitor assembly is made up of three capacitors. The three capacitors are connected in parallel.
For capacitor C1, the area of the plates is $\frac{A}{3}$ and the separation between the plates is d.
For capacitor C2, the area of the plates is $\frac{A}{3}$ and the separation between the plates is (d + b).
For capacitor C3, the area of the plates is $\frac{A}{3}$ and the separation between the plates is (d + 2b).
Therefore,

As the three capacitors are in parallel combination,

(a) The capacitance of a cylindrical capacitor is given by

(b) When a capacitor of the same height with cylinders of radii 4 mm and 8 mm is taken, its capacitance comes to 8 pF, which is the same as above because the ratio of the radii is the same.

#### Question 19:

(a) The capacitance of a cylindrical capacitor is given by

(b) When a capacitor of the same height with cylinders of radii 4 mm and 8 mm is taken, its capacitance comes to 8 pF, which is the same as above because the ratio of the radii is the same.

Given:

Charge on the given capacitor,

Capacitance of the uncharged capacitor,
When the charged capacitor is connected with the uncharged capacitor, the net charge on the system of the capacitors becomes

The potential difference across the plates of the capacitors will be the same.
Thus,

From eqs. (i) and (ii), we get

#### Question 20:

Given:

Charge on the given capacitor,

Capacitance of the uncharged capacitor,
When the charged capacitor is connected with the uncharged capacitor, the net charge on the system of the capacitors becomes

The potential difference across the plates of the capacitors will be the same.
Thus,

From eqs. (i) and (ii), we get

Initially, when the switch S is open, the equivalent capacitance is given by

The charge supplied by the battery is given by

When the switch S is closed, no charge goes to the capacitor connected in parallel with the switch.
Thus, the equivalent capacitance is given by

The charge supplied by the battery is given by

The initial charge stored in the shorted capacitor starts discharging."?
Hence, the charge that flows from A to B is given by

#### Question 21:

Initially, when the switch S is open, the equivalent capacitance is given by

The charge supplied by the battery is given by

When the switch S is closed, no charge goes to the capacitor connected in parallel with the switch.
Thus, the equivalent capacitance is given by

The charge supplied by the battery is given by

The initial charge stored in the shorted capacitor starts discharging."?
Hence, the charge that flows from A to B is given by

The particle is balanced when the electrical force on it is balanced by its weight.
Thus,

Here,
d = Separation between the plates of the capacitor
V' = Potential difference across the capacitor containing the particle

We know that the capacitance of a capacitor is given by
$C=\frac{{\in }_{0}A}{d}\phantom{\rule{0ex}{0ex}}⇒d=\frac{{\in }_{0}A}{C}$
Thus, eq. (i) becomes

Since the values of both the capacitors are the same,
V = 2V' = 2 $×$ 21.86 $\approx$ 43 mV

#### Question 22:

The particle is balanced when the electrical force on it is balanced by its weight.
Thus,

Here,
d = Separation between the plates of the capacitor
V' = Potential difference across the capacitor containing the particle

We know that the capacitance of a capacitor is given by
$C=\frac{{\in }_{0}A}{d}\phantom{\rule{0ex}{0ex}}⇒d=\frac{{\in }_{0}A}{C}$
Thus, eq. (i) becomes

Since the values of both the capacitors are the same,
V = 2V' = 2 $×$ 21.86 $\approx$ 43 mV

Let:
Velocity of the electron = u
Mass of the electron = m
Now,
Horizontal distance, x = u × t
$⇒t=\frac{x}{v}$  ...(i)

Let the electric field inside the capacitor be E
∴ Acceleration of the electron =
Vertical distance,

Capacitance of the two capacitors:
C1 = $\frac{{\in }_{0}{a}^{2}}{{d}_{1}}$ and C2 = $\frac{{\in }_{0}{a}^{2}}{{d}_{2}}$
It is given that the capacitors are connected in series.
Thus, the equivalent capacitance is given by
${C}_{\mathrm{eq}}=\frac{{C}_{1}{C}_{2}}{{C}_{1}+{C}_{2}}$

Total charge on the system of capacitors, Q = CeqV =
As the capacitors are in series, charge on both of them is the same.

The potential difference across the capacitor containing the electron is given by

The magnitude of the electric field inside the capacitor is given by
$E=\frac{V}{{d}_{1}}=\frac{V}{{d}_{1}+{d}_{2}}$
The charge on electron q is represented by e.
On putting the values of q and E in (ii), we get

The minimum velocity of the electron is given by

#### Question 23:

Let:
Velocity of the electron = u
Mass of the electron = m
Now,
Horizontal distance, x = u × t
$⇒t=\frac{x}{v}$  ...(i)

Let the electric field inside the capacitor be E
∴ Acceleration of the electron =
Vertical distance,

Capacitance of the two capacitors:
C1 = $\frac{{\in }_{0}{a}^{2}}{{d}_{1}}$ and C2 = $\frac{{\in }_{0}{a}^{2}}{{d}_{2}}$
It is given that the capacitors are connected in series.
Thus, the equivalent capacitance is given by
${C}_{\mathrm{eq}}=\frac{{C}_{1}{C}_{2}}{{C}_{1}+{C}_{2}}$

Total charge on the system of capacitors, Q = CeqV =
As the capacitors are in series, charge on both of them is the same.

The potential difference across the capacitor containing the electron is given by

The magnitude of the electric field inside the capacitor is given by
$E=\frac{V}{{d}_{1}}=\frac{V}{{d}_{1}+{d}_{2}}$
The charge on electron q is represented by e.
On putting the values of q and E in (ii), we get

The minimum velocity of the electron is given by

Let the electric field inside the capacitor be E.

Now,
Magnitude of acceleration of the electron, ${a}_{\mathrm{e}}=\frac{{q}_{\mathrm{e}}E}{{m}_{\mathrm{e}}}$
Magnitude of acceleration of the proton, ${a}_{\mathrm{p}}=\frac{{q}_{\mathrm{p}}E}{{m}_{\mathrm{p}}}$
Let t be the time taken by the electron and proton to reach the positive and negative plates, respectively.
The initial velocities of the proton and electron are zero.
Thus, the distance travelled by the proton is given by
$x=\frac{1}{2}\frac{{q}_{\mathrm{p}}E}{{m}_{\mathrm{p}}}{t}^{2}$           ...(1)
And, the distance travelled by the electron is given by
$2-x=\frac{1}{2}\frac{{q}_{\mathrm{e}}E}{{m}_{\mathrm{e}}}{t}^{2}$       ...(2)
On dividing (1) by (2), we get

#### Question 24:

Let the electric field inside the capacitor be E.

Now,
Magnitude of acceleration of the electron, ${a}_{\mathrm{e}}=\frac{{q}_{\mathrm{e}}E}{{m}_{\mathrm{e}}}$
Magnitude of acceleration of the proton, ${a}_{\mathrm{p}}=\frac{{q}_{\mathrm{p}}E}{{m}_{\mathrm{p}}}$
Let t be the time taken by the electron and proton to reach the positive and negative plates, respectively.
The initial velocities of the proton and electron are zero.
Thus, the distance travelled by the proton is given by
$x=\frac{1}{2}\frac{{q}_{\mathrm{p}}E}{{m}_{\mathrm{p}}}{t}^{2}$           ...(1)
And, the distance travelled by the electron is given by
$2-x=\frac{1}{2}\frac{{q}_{\mathrm{e}}E}{{m}_{\mathrm{e}}}{t}^{2}$       ...(2)
On dividing (1) by (2), we get

Parts (a), (b) and (c) are identical, as all of them form a bridge circuit. In that circuit, capacitors of 1 µF and 2 µF are connected to terminal A and the 5 µF capacitor and capacitors of 3 µF and 6 µF are connected to terminal B and the 5 µF capacitor.

For the given situation, it can be observed that the bridge is in balance; thus, no current will flow through the 5 µF capacitor.
So to simplify the circuit, 5 µF capacitor can be removed from the circuit.
Now, 1 µF and 3 µF capacitors are in series.
And 2 µF and 6 µF capacitors are also in series combination.
These two combination are in parallel with each other.
The equivalent capacitance can be calculated as:

Ceq = 2.25 µF

Thus, parts (a), (b) and (c) are identical.

And,

Ceq = 2.25 µF

#### Question 25:

Parts (a), (b) and (c) are identical, as all of them form a bridge circuit. In that circuit, capacitors of 1 µF and 2 µF are connected to terminal A and the 5 µF capacitor and capacitors of 3 µF and 6 µF are connected to terminal B and the 5 µF capacitor.

For the given situation, it can be observed that the bridge is in balance; thus, no current will flow through the 5 µF capacitor.
So to simplify the circuit, 5 µF capacitor can be removed from the circuit.
Now, 1 µF and 3 µF capacitors are in series.
And 2 µF and 6 µF capacitors are also in series combination.
These two combination are in parallel with each other.
The equivalent capacitance can be calculated as:

Ceq = 2.25 µF

Thus, parts (a), (b) and (c) are identical.

And,

Ceq = 2.25 µF

(a) $q={q}_{1}+{q}_{2}$       ...(i)

On applying Kirchhoff's voltage law in the loop CabDC, we get

On applying Kirchhoff's voltage law in the loop DCBAD, we get

$\frac{q}{2}+\frac{{q}_{1}}{4}-12=0$

From eqs. (ii) and (iii), we get

(b) Let the charge in the loop be q.

Now, on applying Kirchhoff's voltage law in the loop, we get

$\frac{q}{2}+\frac{q}{4}-24+12=0$

(c)

In the loop,

(d)

Net charge flowing through all branches, q = 24 + 24 + 24 = 72 μC
Net capacitance of all branches, C = 4 + 2 + 1 = 7 μF
The total potential difference (V) between points a and b is given by

As the negative terminals of the batteries are connected to a, the net potential between points a and b is $-$10.3 V.

#### Question 26:

(a) $q={q}_{1}+{q}_{2}$       ...(i)

On applying Kirchhoff's voltage law in the loop CabDC, we get

On applying Kirchhoff's voltage law in the loop DCBAD, we get

$\frac{q}{2}+\frac{{q}_{1}}{4}-12=0$

From eqs. (ii) and (iii), we get

(b) Let the charge in the loop be q.

Now, on applying Kirchhoff's voltage law in the loop, we get

$\frac{q}{2}+\frac{q}{4}-24+12=0$

(c)

In the loop,

(d)

Net charge flowing through all branches, q = 24 + 24 + 24 = 72 μC
Net capacitance of all branches, C = 4 + 2 + 1 = 7 μF
The total potential difference (V) between points a and b is given by

As the negative terminals of the batteries are connected to a, the net potential between points a and b is $-$10.3 V.

(a)

Applying star-delta conversion in the part indicated in the diagram.

The capacitance of the C1 is given by

Thus, the equivalent circuit can be drawn as:

Therefore, the equivalent capacitance is given by
Ceq =

(b) By star-delta conversion,

(c)
It is a balanced bridge.
Therefore, the capacitor of capacitance 5 μF can be removed.

The capacitance of the two branches are:

∴ Equivalent capacitance $=\frac{4}{3}+\frac{8}{3}+4$

(d)

It is also a balanced diagram.

It can be observed that the bridges are balanced.
Therefore, the capacitors of capacitance 6 μF between the branches can be removed

The capacitances of the four branches are:

∴ Equivalent capacitance

#### Question 27:

(a)

Applying star-delta conversion in the part indicated in the diagram.

The capacitance of the C1 is given by

Thus, the equivalent circuit can be drawn as:

Therefore, the equivalent capacitance is given by
Ceq =

(b) By star-delta conversion,

(c)
It is a balanced bridge.
Therefore, the capacitor of capacitance 5 μF can be removed.

The capacitance of the two branches are:

∴ Equivalent capacitance $=\frac{4}{3}+\frac{8}{3}+4$

(d)

It is also a balanced diagram.

It can be observed that the bridges are balanced.
Therefore, the capacitors of capacitance 6 μF between the branches can be removed

The capacitances of the four branches are:

∴ Equivalent capacitance

Capacitors 5 and 1 are in series.
Their equivalent capacitance,
Ceq = 1
Now, this capacitor system is parallel to capacitor 6. Thus, the equivalent capacitance becomes
1 + 1 = 2 μF

The above capacitor system is in series with capacitor 2. Thus, the equivalent capacitance becomes

The above capacitor system is in parallel with capacitor 7. Thus, the equivalent capacitance becomes
1 + 1 = 2 μF

The above capacitor system is in series with capacitor 3. Thus, the equivalent capacitance becomes

The above capacitor system is in parallel with capacitor 8. Thus, the equivalent capacitance becomes
1 + 1 = 2 μF

The above capacitor system is in series with capacitor 4. Thus, the equivalent capacitance becomes

Hence, the equivalent capacitance between points A and B of the given capacitor system is 1 μF.

#### Question 28:

Capacitors 5 and 1 are in series.
Their equivalent capacitance,
Ceq = 1
Now, this capacitor system is parallel to capacitor 6. Thus, the equivalent capacitance becomes
1 + 1 = 2 μF

The above capacitor system is in series with capacitor 2. Thus, the equivalent capacitance becomes

The above capacitor system is in parallel with capacitor 7. Thus, the equivalent capacitance becomes
1 + 1 = 2 μF

The above capacitor system is in series with capacitor 3. Thus, the equivalent capacitance becomes

The above capacitor system is in parallel with capacitor 8. Thus, the equivalent capacitance becomes
1 + 1 = 2 μF

The above capacitor system is in series with capacitor 4. Thus, the equivalent capacitance becomes

Hence, the equivalent capacitance between points A and B of the given capacitor system is 1 μF.

Let the equivalent capacitance of the infinite ladder be C.
Because it is an infinite ladder, the change in the equivalent capacitance will be negligible if we add one more ladder at point AB, as shown in the given figure.

From the given figure, the equivalent capacitance can be calculated as:

#### Question 29:

Let the equivalent capacitance of the infinite ladder be C.
Because it is an infinite ladder, the change in the equivalent capacitance will be negligible if we add one more ladder at point AB, as shown in the given figure.

From the given figure, the equivalent capacitance can be calculated as:

The equivalent capacitance of the ladder between points A and B becomes independent of the number of sections in between when the capacitance between A and B is C.

The capacitors C and 4 µF are in series; their equivalent capacitance is given by
${C}_{1}=\frac{C×4}{C+4}$

The capacitors C1 and 2 µF are in parallel; their equivalent capacitance is given by
C = C1 + 2 µF

$⇒C=\frac{C×4}{C+4}+2\phantom{\rule{0ex}{0ex}}$
$⇒$4C + 8 + 2 C = 4C + C2
$⇒$C2 − 2C − 8 = 0
$⇒$C = −2, C = 4
Capacitance cannot be negative.
∴ C = 4 µF
The value of C is 4 µF.

#### Question 30:

The equivalent capacitance of the ladder between points A and B becomes independent of the number of sections in between when the capacitance between A and B is C.

The capacitors C and 4 µF are in series; their equivalent capacitance is given by
${C}_{1}=\frac{C×4}{C+4}$

The capacitors C1 and 2 µF are in parallel; their equivalent capacitance is given by
C = C1 + 2 µF

$⇒C=\frac{C×4}{C+4}+2\phantom{\rule{0ex}{0ex}}$
$⇒$4C + 8 + 2 C = 4C + C2
$⇒$C2 − 2C − 8 = 0
$⇒$C = −2, C = 4
Capacitance cannot be negative.
∴ C = 4 µF
The value of C is 4 µF.

The charge on the positive plate is q1and that on the negative plate is q2.
Given:

Now,
Net charge on the capacitor =

The  potential difference developed between the plates is given by

#### Question 31:

The charge on the positive plate is q1and that on the negative plate is q2.
Given:

Now,
Net charge on the capacitor =

The  potential difference developed between the plates is given by

Given:
Capacitance of the isolated capacitor = 10 µF
Charge on the positive plate = 20 µC
Effective charge on the capacitor =
The potential difference between the plates of the capacitor is given by
$V=\frac{Q}{C}$
∴ Potential difference =

#### Question 32:

Given:
Capacitance of the isolated capacitor = 10 µF
Charge on the positive plate = 20 µC
Effective charge on the capacitor =
The potential difference between the plates of the capacitor is given by
$V=\frac{Q}{C}$
∴ Potential difference =

Charges on the plates:

The effective charge on the capacitor is given by

The potential difference is given by

#### Question 33:

Charges on the plates:

The effective charge on the capacitor is given by

The potential difference is given by

Here, three capacitors are formed and each of them has area A.
Now,

The capacitance of the capacitors is given by

Because the three capacitors are arranged in series, the equivalent capacitance is given by

The total charge on the capacitor is given by
Q = CeqV

The charge on a single plate is given by

#### Question 34:

Here, three capacitors are formed and each of them has area A.
Now,

The capacitance of the capacitors is given by

Because the three capacitors are arranged in series, the equivalent capacitance is given by

The total charge on the capacitor is given by
Q = CeqV

The charge on a single plate is given by

(a) When the charge of 1 µC is introduced on plate B, the charge gets equally distributed on its surface. All sides of the plate gets 0.5 µC of charge. Due to induction, 0.5 µC charge is induced on the upper surface of plate A.

(b) Given:
Capacitance, C = 50 nF = 50 × 10$-$9 F = 5 × 10$-$8 F
Effective charge on the capacitor,
∴ Potential difference between plates B and A,

#### Question 35:

(a) When the charge of 1 µC is introduced on plate B, the charge gets equally distributed on its surface. All sides of the plate gets 0.5 µC of charge. Due to induction, 0.5 µC charge is induced on the upper surface of plate A.

(b) Given:
Capacitance, C = 50 nF = 50 × 10$-$9 F = 5 × 10$-$8 F
Effective charge on the capacitor,
∴ Potential difference between plates B and A,

When 1µC charge is given to the upper plate, the charge gets distributed. The two sides of the upper plate have +0.5 µC charge. One side of the middle plate facing the upper plate has $-$0.5 µC charge and the other side has +0.5 µC charge. Similarly, one side of the lower plate facing the middle plate has $-$0.5 µC charge and the other side has +0.5 µC charge.

(a) Effective charge on the capacitor formed by the upper and middle plates = 0.5 µC
Capacitance = 50 nF = 0.05 µF
∴ Potential difference between the plates,

(b) Effective charge on the capacitor formed by the middle and lower plates = 0.5 µC
Capacitance = 0.50 µF
∴ Potential difference between the plates,

#### Question 36:

When 1µC charge is given to the upper plate, the charge gets distributed. The two sides of the upper plate have +0.5 µC charge. One side of the middle plate facing the upper plate has $-$0.5 µC charge and the other side has +0.5 µC charge. Similarly, one side of the lower plate facing the middle plate has $-$0.5 µC charge and the other side has +0.5 µC charge.

(a) Effective charge on the capacitor formed by the upper and middle plates = 0.5 µC
Capacitance = 50 nF = 0.05 µF
∴ Potential difference between the plates,

(b) Effective charge on the capacitor formed by the middle and lower plates = 0.5 µC
Capacitance = 0.50 µF
∴ Potential difference between the plates,

Given:

When the capacitors are connected in series, their equivalent capacitance is given by
${\mathrm{C}}_{\mathrm{eq}}=\frac{{C}_{1}{C}_{2}}{{C}_{1}+{C}_{2}}$
∴ Equivalent capacitance,

(a) The charge on both capacitors is equal as they are connected in series. It is given by

(b) The energies in the capacitors are given by

#### Question 37:

Given:

When the capacitors are connected in series, their equivalent capacitance is given by
${\mathrm{C}}_{\mathrm{eq}}=\frac{{C}_{1}{C}_{2}}{{C}_{1}+{C}_{2}}$
∴ Equivalent capacitance,

(a) The charge on both capacitors is equal as they are connected in series. It is given by

(b) The energies in the capacitors are given by

Given:

Now,
The equivalent capacitance is given by

The energy supplied by the battery is given by

#### Question 38:

Given:

Now,
The equivalent capacitance is given by

The energy supplied by the battery is given by

Capacitors b and c are in parallel; their equivalent capacitance is 20 µF.
Thus, the net capacitance of the circuit is given by

The total charge of the battery is given by

For a and d,

For b and c,

#### Question 39:

Capacitors b and c are in parallel; their equivalent capacitance is 20 µF.
Thus, the net capacitance of the circuit is given by

The total charge of the battery is given by

For a and d,

For b and c,

Given:
Energy stored in the charged capacitor = 4.0 J
When the capacitors are connected, the charge flows from the charged capacitor to the uncharged capacitor. Because the capacitors are identical, the charge flows till the charge in both the capacitors becomes equal.
The energy of a capacitor is given by
$E=\frac{{q}^{2}}{2C}$
As the charge in both the capacitors is the same, their capacitance is also the same. So, the energy is equally divided between them.
Thus, the energy stored in each of the capacitors is 2.0 J.

#### Question 40:

Given:
Energy stored in the charged capacitor = 4.0 J
When the capacitors are connected, the charge flows from the charged capacitor to the uncharged capacitor. Because the capacitors are identical, the charge flows till the charge in both the capacitors becomes equal.
The energy of a capacitor is given by
$E=\frac{{q}^{2}}{2C}$
As the charge in both the capacitors is the same, their capacitance is also the same. So, the energy is equally divided between them.
Thus, the energy stored in each of the capacitors is 2.0 J.

Charge on the 2 µF capacitor when it is not connected to the 4 µF capacitor is given by

(a) On connecting the capacitors, the charge flows from the 2 µF capacitor to the 4 µF capacitor.
Now, let the charges on the 2 µF and 4 µF capacitors be q1 and q2, respectively.

As they are connected in parallel, the potential difference across them is the same.

The total charge on the capacitors will be the same as the initial charge stored on the 2 µF capacitor.

(b) Energies stored in the capacitors are given by

(c)
Initial energy stored in the 2 µF capacitor is given by

Total energy of the capacitors after they are connected in parallel is given by
Ef = E1 + E2
$⇒$Ef = 16 + 32 = 48 μJ

Heat produced during the charge transfer is given by
E = Ef$-$Ei
$⇒$E = 144 $-$ 48 = 96 μJ

#### Question 41:

Charge on the 2 µF capacitor when it is not connected to the 4 µF capacitor is given by

(a) On connecting the capacitors, the charge flows from the 2 µF capacitor to the 4 µF capacitor.
Now, let the charges on the 2 µF and 4 µF capacitors be q1 and q2, respectively.

As they are connected in parallel, the potential difference across them is the same.

The total charge on the capacitors will be the same as the initial charge stored on the 2 µF capacitor.

(b) Energies stored in the capacitors are given by

(c)
Initial energy stored in the 2 µF capacitor is given by

Total energy of the capacitors after they are connected in parallel is given by
Ef = E1 + E2
$⇒$Ef = 16 + 32 = 48 μJ

Heat produced during the charge transfer is given by
E = Ef$-$Ei
$⇒$E = 144 $-$ 48 = 96 μJ

Given:
Charge on the sphere = Q
Radius of the sphere = R
Capacitance of the sphere,
Thus, the energy of the sphere is given by

#### Question 42:

Given:
Charge on the sphere = Q
Radius of the sphere = R
Capacitance of the sphere,
Thus, the energy of the sphere is given by

Potential of the inner metallic sphere is given by

Capacitance of the capacitor formed by two concentric spheres of radii R and 2R is given by
$C=4\pi {\in }_{0}\frac{R×2R}{2R-R}\phantom{\rule{0ex}{0ex}}⇒C=4\pi {\in }_{0}×2R$

Potential of the outer sphere is given by

Electrostatic energy stored outside the sphere is given by
$E=\frac{1}{2}C{\left(V-{V}_{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒E=\frac{1}{2}×4\mathrm{\pi }{\in }_{0}×2R×\frac{{V}^{2}}{4}\phantom{\rule{0ex}{0ex}}⇒E=\mathrm{\pi }{\in }_{0}×R{V}^{2}$

#### Question 43:

Potential of the inner metallic sphere is given by

Capacitance of the capacitor formed by two concentric spheres of radii R and 2R is given by
$C=4\pi {\in }_{0}\frac{R×2R}{2R-R}\phantom{\rule{0ex}{0ex}}⇒C=4\pi {\in }_{0}×2R$

Potential of the outer sphere is given by

Electrostatic energy stored outside the sphere is given by
$E=\frac{1}{2}C{\left(V-{V}_{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒E=\frac{1}{2}×4\mathrm{\pi }{\in }_{0}×2R×\frac{{V}^{2}}{4}\phantom{\rule{0ex}{0ex}}⇒E=\mathrm{\pi }{\in }_{0}×R{V}^{2}$

Given,
Surface charge density of the plane,
Volume of the cube,

Electric field near the charged conducting plane is given as,

Energy density of electric filed,

#### Question 44:

Given,
Surface charge density of the plane,
Volume of the cube,

Electric field near the charged conducting plane is given as,

Energy density of electric filed,

Given:

Initial capacitance of the capacitor:

Final capacitance of the capacitor:

(a) Charge flown through the circuit:

(b) Energy absorbed by the battery:

(c) Energy stored before the process, Ei

Energy stored after the process, Ef

(d) Work done, W = Force × Distance
W = Change in the energy
$⇒$W = 6.35 × 10−10 J

#### Question 45:

Given:

Initial capacitance of the capacitor:

Final capacitance of the capacitor:

(a) Charge flown through the circuit:

(b) Energy absorbed by the battery:

(c) Energy stored before the process, Ei

Energy stored after the process, Ef

(d) Work done, W = Force × Distance
W = Change in the energy
$⇒$W = 6.35 × 10−10 J

(a) Before connecting to the battery of 12 volts,

After connecting to the battery of 12 volts,

(b) Charge flown through the 12 V battery is 1200 $\mathrm{\mu C}$.

(c) We know,

It is the work done on the battery.

(d) Initial electrostatic field energy:

Final electrostatic field energy:

Decrease in the electrostatic field energy:

(e) After reconnection,

Heat developed during the flow of charge after reconnection is given by

This amount of energy is developed as heat when the charge flows through the capacitor.

#### Question 46:

(a) Before connecting to the battery of 12 volts,

After connecting to the battery of 12 volts,

(b) Charge flown through the 12 V battery is 1200 $\mathrm{\mu C}$.

(c) We know,

It is the work done on the battery.

(d) Initial electrostatic field energy:

Final electrostatic field energy:

Decrease in the electrostatic field energy:

(e) After reconnection,

Heat developed during the flow of charge after reconnection is given by

This amount of energy is developed as heat when the charge flows through the capacitor.

Since the switch is opened for a long time, the capacitor is in series.
${C}_{eq}=\frac{C}{2}$

When the switch is closed, the charge flown from the battery is given by

$q=\frac{\mathrm{C}}{2}×\epsilon =\frac{\mathrm{C}\epsilon }{2}$

(b) Work done, W$=q×V$
$⇒q=\frac{\mathrm{C}\epsilon }{2}×\epsilon =\frac{\mathrm{C}{\epsilon }^{2}}{2}$

(c) The change in the energy stored in the capacitors is given by

(d) The heat developed in the system is given by

#### Question 47:

Since the switch is opened for a long time, the capacitor is in series.
${C}_{eq}=\frac{C}{2}$

When the switch is closed, the charge flown from the battery is given by

$q=\frac{\mathrm{C}}{2}×\epsilon =\frac{\mathrm{C}\epsilon }{2}$

(b) Work done, W$=q×V$
$⇒q=\frac{\mathrm{C}\epsilon }{2}×\epsilon =\frac{\mathrm{C}{\epsilon }^{2}}{2}$

(c) The change in the energy stored in the capacitors is given by

(d) The heat developed in the system is given by

Given:

(a)
Energy stored in the first capacitor:

Energy stored in the second capacitor:

(b) The capacitors are connected to each other in such a way that the positive plate of the first capacitor is connected to the negative plate of the second capacitor and vice versa.

∴ Net change in the system, Qnet = 120 $-$ 72 = 48

Now, let V be the common potential of the two capacitors.
From the conservation of charge before and after connecting, we get

New charges:

(c)

(d) The energy is dissipated as heat.

#### Question 48:

Given:

(a)
Energy stored in the first capacitor:

Energy stored in the second capacitor:

(b) The capacitors are connected to each other in such a way that the positive plate of the first capacitor is connected to the negative plate of the second capacitor and vice versa.

∴ Net change in the system, Qnet = 120 $-$ 72 = 48

Now, let V be the common potential of the two capacitors.
From the conservation of charge before and after connecting, we get

New charges:

(c)

(d) The energy is dissipated as heat.

As a capacitor of capacitance C is connected to the battery of voltage V, one of its plates gets charged to Q = CV and the other gets charged to −Q coulomb. When the polarity is reversed, charge −Q appears on the first plate and charge +Q appears on the second plate. So to maintain these charges, charge 2Q passes through the battery from the negative terminal to the positive terminal.

The work done by the battery is given by
$W=2QV=2C{E}^{2}$

In the given process, the capacitor's energy remains the same in both cases. The work done by the battery appears as heat in the connecting wires.
Now,
Heat produced:

#### Question 49:

As a capacitor of capacitance C is connected to the battery of voltage V, one of its plates gets charged to Q = CV and the other gets charged to −Q coulomb. When the polarity is reversed, charge −Q appears on the first plate and charge +Q appears on the second plate. So to maintain these charges, charge 2Q passes through the battery from the negative terminal to the positive terminal.

The work done by the battery is given by
$W=2QV=2C{E}^{2}$

In the given process, the capacitor's energy remains the same in both cases. The work done by the battery appears as heat in the connecting wires.
Now,
Heat produced:

The area of the plates of the capacitor is given by

The separation between the parallel plates is given by

Here, the thickness of the dielectric is the same as the separation between the parallel plates.
Thus, the capacitance is given by

#### Question 50:

The area of the plates of the capacitor is given by

The separation between the parallel plates is given by

Here, the thickness of the dielectric is the same as the separation between the parallel plates.
Thus, the capacitance is given by

Given:
Capacitance of the capacitor, C = 1.42 nF
Dielectric constant, k = 4
Voltage of the battery connected, V = 6 V

(a) The charge supplied by the battery is given by

(b) The charge induced on the dielectric is given by

(c) The charge appearing on one of the coated plates is given by
Qnet= 8.5 $-$ 6.4 = 2.1 nC

#### Question 51:

Given:
Capacitance of the capacitor, C = 1.42 nF
Dielectric constant, k = 4
Voltage of the battery connected, V = 6 V

(a) The charge supplied by the battery is given by

(b) The charge induced on the dielectric is given by

(c) The charge appearing on one of the coated plates is given by
Qnet= 8.5 $-$ 6.4 = 2.1 nC

Given:
Area of the plate = 100 cm2
Separation between the plates = 0.500 cm = 5 × 10$-$3 m
Thickness of the metal, t = 4 × 10$-$3 m

Here,
k = Dielectric constant of the metal
d = Separation between the plates
t = Thickness of the metal

For the metal, k = $\infty$.

Here, the capacitance is independent of the position of the metal. At any position, the net separation is (dt).

#### Question 52:

Given:
Area of the plate = 100 cm2
Separation between the plates = 0.500 cm = 5 × 10$-$3 m
Thickness of the metal, t = 4 × 10$-$3 m

Here,
k = Dielectric constant of the metal
d = Separation between the plates
t = Thickness of the metal

For the metal, k = $\infty$.

Here, the capacitance is independent of the position of the metal. At any position, the net separation is (dt).

Given:

Initial charge on the capacitor = 50 µC

Now, let the dielectric constant of the material inserted be k.

As 100 µC of extra charge flows through the battery,  the net charge on the capacitor becomes

50 + 100 = 150 µC

Now,

On dividing (i) by (ii), we get
$\frac{{C}_{1}}{{C}_{2}}=\frac{{q}_{1}}{{q}_{2}}=\frac{1}{k}\phantom{\rule{0ex}{0ex}}⇒\frac{50}{150}=\frac{1}{k}\phantom{\rule{0ex}{0ex}}⇒k=3$

Thus, the dielectric constant of the given material is 3.

#### Question 53:

Given:

Initial charge on the capacitor = 50 µC

Now, let the dielectric constant of the material inserted be k.

As 100 µC of extra charge flows through the battery,  the net charge on the capacitor becomes

50 + 100 = 150 µC

Now,

On dividing (i) by (ii), we get
$\frac{{C}_{1}}{{C}_{2}}=\frac{{q}_{1}}{{q}_{2}}=\frac{1}{k}\phantom{\rule{0ex}{0ex}}⇒\frac{50}{150}=\frac{1}{k}\phantom{\rule{0ex}{0ex}}⇒k=3$

Thus, the dielectric constant of the given material is 3.

For the given capacitor,

(a) The charge on the positive plate is calculated using q = CV.

Thus,

(b) The electric field between the plates of the capacitor is given by

(c) Separation between the plates of the capacitor, d = 2×10$-$3 m
Dielectric constant of the dielectric inserted, k = 5
Thickness of the dielectric inserted, t = 1×10$-$3 m
Now,
Area of the plates of the capacitor

When the dielectric placed in it, the capacitance becomes

(d)
The charge stored in the capacitor initially is given by

The charge on the capacitor after inserting the dielectric slab is given by

#### Question 54:

For the given capacitor,

(a) The charge on the positive plate is calculated using q = CV.

Thus,

(b) The electric field between the plates of the capacitor is given by

(c) Separation between the plates of the capacitor, d = 2×10$-$3 m
Dielectric constant of the dielectric inserted, k = 5
Thickness of the dielectric inserted, t = 1×10$-$3 m
Now,
Area of the plates of the capacitor

When the dielectric placed in it, the capacitance becomes

(d)
The charge stored in the capacitor initially is given by

The charge on the capacitor after inserting the dielectric slab is given by

The given system of the capacitor will behave as two capacitors connected in series.

Let the capacitances be C1 and C2.
Now,

Thus, the net capacitance is given by

#### Question 55:

The given system of the capacitor will behave as two capacitors connected in series.

Let the capacitances be C1 and C2.
Now,

Thus, the net capacitance is given by

The capacitance of the capacitor without the dielectric slab is given by

When the dielectric slab is inserted, the capacitance becomes

(a) Increased electrostatic energy:

(b)
Charge on the capacitor containing dielectric is,

Potential difference across the capacitor after removing battery and then dielectric is,

Increased electrostatic energy,

(c) When the battery is connected, energy is increased after insertion of the dielectric slab because of the increase in the capacitance of the capacitor. Now, capacitor will abstract more change from the battery.

When the battery is disconnected, and dielectric slab is taken out then energy stored in the will increase because of increase of potential difference across the capacitor.

#### Question 56:

The capacitance of the capacitor without the dielectric slab is given by

When the dielectric slab is inserted, the capacitance becomes

(a) Increased electrostatic energy:

(b)
Charge on the capacitor containing dielectric is,

Potential difference across the capacitor after removing battery and then dielectric is,

Increased electrostatic energy,

(c) When the battery is connected, energy is increased after insertion of the dielectric slab because of the increase in the capacitance of the capacitor. Now, capacitor will abstract more change from the battery.

When the battery is disconnected, and dielectric slab is taken out then energy stored in the will increase because of increase of potential difference across the capacitor.

The two parts of the capacitor are in series with capacitances C1and C2.
Here,

Because they are in series, the net capacitance is calculated as:

(b) Here, the capacitor has three parts. These can be taken in series.

Now,

Thus, the net capacitance is calculated as:

(c)

Here,

These two parts are in parallel.

#### Question 57:

The two parts of the capacitor are in series with capacitances C1and C2.
Here,

Because they are in series, the net capacitance is calculated as:

(b) Here, the capacitor has three parts. These can be taken in series.

Now,

Thus, the net capacitance is calculated as:

(c)

Here,

These two parts are in parallel.

Let us consider an elemental capacitor of width dx at a distance x from the left end of the capacitor. It has two capacitive elements of dielectric constants K1 and K2with plate separations (x tan θ) and (dx tan θ) in series, respectively. The areas of the plates of the capacitors are adx.

The capacitances of the capacitive elements of the elemental capacitor are:

The net capacitance of the elemental capacitor is given by

Thus, integrating the above expression to calculate the net capacitance

As we know that substituting in the expression for capacitance C.
Now,

#### Question 58:

Let us consider an elemental capacitor of width dx at a distance x from the left end of the capacitor. It has two capacitive elements of dielectric constants K1 and K2with plate separations (x tan θ) and (dx tan θ) in series, respectively. The areas of the plates of the capacitors are adx.

The capacitances of the capacitive elements of the elemental capacitor are:

The net capacitance of the elemental capacitor is given by

Thus, integrating the above expression to calculate the net capacitance

As we know that substituting in the expression for capacitance C.
Now,

When the switch is closed, both capacitors are in parallel.
$⇒$The total energy of the capacitor when the switch is closed is given by

Ei

When the switch is opened and the dielectric is induced, the capacitance of the capacitor A becomes
$C\text{'}=KC=3C$

The energy stored in the capacitor A is given by

The energy in the capacitor B is given by
${E}_{\mathrm{B}}=\frac{1}{2}×\frac{C}{3}×{V}^{2}$

∴ Total final energy

Now,
Ratio of the energies,

#### Question 59:

When the switch is closed, both capacitors are in parallel.
$⇒$The total energy of the capacitor when the switch is closed is given by

Ei

When the switch is opened and the dielectric is induced, the capacitance of the capacitor A becomes
$C\text{'}=KC=3C$

The energy stored in the capacitor A is given by

The energy in the capacitor B is given by
${E}_{\mathrm{B}}=\frac{1}{2}×\frac{C}{3}×{V}^{2}$

∴ Total final energy

Now,
Ratio of the energies,

Initial capacitance,
The energy of the capacitor before the insertion of the dielectric is given by

After inserting the dielectric slab, the capacitance becomes

and the final voltage becomes
${V}_{1}=\frac{V}{K}$

Thus, the final energy stored in the capacitor is given by

Now,

Work done = Change in energy
W = E2 $-$E1

#### Question 60:

Initial capacitance,
The energy of the capacitor before the insertion of the dielectric is given by

After inserting the dielectric slab, the capacitance becomes

and the final voltage becomes
${V}_{1}=\frac{V}{K}$

Thus, the final energy stored in the capacitor is given by

Now,

Work done = Change in energy
W = E2 $-$E1

(a) The magnitude of the charge can be calculated as:
Charge = Capacitance × Potential difference

(b) When a dielectric is introduced, the potential difference decreases.

We know,

(c) Now, the charge on the capacitance can be calculated as:
Charge = Capacitance × Potential difference

(d) The charge induced on the dielectric can be calculated as:

#### Question 61:

(a) The magnitude of the charge can be calculated as:
Charge = Capacitance × Potential difference

(b) When a dielectric is introduced, the potential difference decreases.

We know,

(c) Now, the charge on the capacitance can be calculated as:
Charge = Capacitance × Potential difference

(d) The charge induced on the dielectric can be calculated as:

We have two capacitors: one made by the shells a and c and the other made by the shells b and c.

The capacitance of the capacitor Cac is given by
${C}_{\mathrm{ac}}=\frac{4\pi {\in }_{0}acK}{\left(c-a\right)}$

The capacitance of the capacitor Ccb is given by
${C}_{\mathrm{cb}}=\frac{4\pi {\in }_{0}bcK}{K\left(b-c\right)}$

The two capacitors are in series; thus, the equivalent capacitance is given by
$\frac{1}{C}=\frac{1}{{C}_{\mathrm{ac}}}+\frac{1}{{C}_{\mathrm{cb}}}\phantom{\rule{0ex}{0ex}}⇒\frac{1}{C}=\frac{\left(c-a\right)}{4\pi {\in }_{0}acK}+\frac{\left(b-c\right)}{4\pi {\in }_{0}cb}\phantom{\rule{0ex}{0ex}}⇒\frac{1}{C}=\frac{b\left(c-a\right)+Ka\left(b-c\right)}{K4\pi {\in }_{0}abc}\phantom{\rule{0ex}{0ex}}⇒C=\frac{K4\pi {\in }_{0}abc}{b\left(c-a\right)+Ka\left(b-c\right)}$

#### Question 62:

We have two capacitors: one made by the shells a and c and the other made by the shells b and c.

The capacitance of the capacitor Cac is given by
${C}_{\mathrm{ac}}=\frac{4\pi {\in }_{0}acK}{\left(c-a\right)}$

The capacitance of the capacitor Ccb is given by
${C}_{\mathrm{cb}}=\frac{4\pi {\in }_{0}bcK}{K\left(b-c\right)}$

The two capacitors are in series; thus, the equivalent capacitance is given by
$\frac{1}{C}=\frac{1}{{C}_{\mathrm{ac}}}+\frac{1}{{C}_{\mathrm{cb}}}\phantom{\rule{0ex}{0ex}}⇒\frac{1}{C}=\frac{\left(c-a\right)}{4\pi {\in }_{0}acK}+\frac{\left(b-c\right)}{4\pi {\in }_{0}cb}\phantom{\rule{0ex}{0ex}}⇒\frac{1}{C}=\frac{b\left(c-a\right)+Ka\left(b-c\right)}{K4\pi {\in }_{0}abc}\phantom{\rule{0ex}{0ex}}⇒C=\frac{K4\pi {\in }_{0}abc}{b\left(c-a\right)+Ka\left(b-c\right)}$

The spherical shells form two spherical capacitors: one made by A and B and the other made by B and C.

The capacitance of the spherical capacitor made by the shells of radii r1 and r2 is given by
$C=\frac{4\pi {\in }_{0}}{\left[\frac{1}{{r}_{1}}-\frac{1}{{r}_{2}}\right]}=\frac{4\pi {\in }_{0}{r}_{1}{r}_{2}}{{r}_{2}-{r}_{1}}$
The capacitance of the capacitor made by A and B is given by

${C}_{\mathrm{AB}}=\frac{4\pi {\in }_{0}ab}{b-a}$

The capacitance of the capacitor made by B and C is given by

${C}_{\mathrm{BC}}=\frac{4\pi {\in }_{0}bc}{c-b}$

As the capacitors are in series, the net capacitance is given by

$\frac{1}{C}=\frac{1}{{C}_{\mathrm{AB}}}+\frac{1}{{C}_{\mathrm{BC}}}\phantom{\rule{0ex}{0ex}}⇒C=\frac{{C}_{\mathrm{AB}}{C}_{\mathrm{BC}}}{{C}_{\mathrm{AB}}+{C}_{\mathrm{BC}}}=\frac{\frac{\left(4\pi {\in }_{0}{\right)}^{2}a{b}^{2}c}{\left(b-a\right)\left(c-b\right)}}{\frac{4\pi {\in }_{0}ab}{\left(b-a\right)}+\frac{4\pi {\in }_{0}bc}{\left(c-b\right)}}\phantom{\rule{0ex}{0ex}}⇒C=\frac{\frac{4\pi {\in }_{0}a{b}^{2}c}{\left(b-a\right)\left(c-b\right)}}{\left(\frac{ab\left(c-b\right)+bc\left(b-a\right)}{\left(b-a\right)\left(c-b\right)}\right)}\phantom{\rule{0ex}{0ex}}⇒C=\frac{4\pi {\in }_{0}a{b}^{2}c}{\left[ab\left(c-b\right)+bc\left(b-a\right)\right]}\phantom{\rule{0ex}{0ex}}⇒C=\frac{4\pi {\in }_{0}a{b}^{2}c}{{b}^{2}\left(c-a\right)}=\frac{4\pi {\in }_{0}ac}{\left(c-a\right)}$

#### Question 63:

The spherical shells form two spherical capacitors: one made by A and B and the other made by B and C.

The capacitance of the spherical capacitor made by the shells of radii r1 and r2 is given by
$C=\frac{4\pi {\in }_{0}}{\left[\frac{1}{{r}_{1}}-\frac{1}{{r}_{2}}\right]}=\frac{4\pi {\in }_{0}{r}_{1}{r}_{2}}{{r}_{2}-{r}_{1}}$
The capacitance of the capacitor made by A and B is given by

${C}_{\mathrm{AB}}=\frac{4\pi {\in }_{0}ab}{b-a}$

The capacitance of the capacitor made by B and C is given by

${C}_{\mathrm{BC}}=\frac{4\pi {\in }_{0}bc}{c-b}$

As the capacitors are in series, the net capacitance is given by

$\frac{1}{C}=\frac{1}{{C}_{\mathrm{AB}}}+\frac{1}{{C}_{\mathrm{BC}}}\phantom{\rule{0ex}{0ex}}⇒C=\frac{{C}_{\mathrm{AB}}{C}_{\mathrm{BC}}}{{C}_{\mathrm{AB}}+{C}_{\mathrm{BC}}}=\frac{\frac{\left(4\pi {\in }_{0}{\right)}^{2}a{b}^{2}c}{\left(b-a\right)\left(c-b\right)}}{\frac{4\pi {\in }_{0}ab}{\left(b-a\right)}+\frac{4\pi {\in }_{0}bc}{\left(c-b\right)}}\phantom{\rule{0ex}{0ex}}⇒C=\frac{\frac{4\pi {\in }_{0}a{b}^{2}c}{\left(b-a\right)\left(c-b\right)}}{\left(\frac{ab\left(c-b\right)+bc\left(b-a\right)}{\left(b-a\right)\left(c-b\right)}\right)}\phantom{\rule{0ex}{0ex}}⇒C=\frac{4\pi {\in }_{0}a{b}^{2}c}{\left[ab\left(c-b\right)+bc\left(b-a\right)\right]}\phantom{\rule{0ex}{0ex}}⇒C=\frac{4\pi {\in }_{0}a{b}^{2}c}{{b}^{2}\left(c-a\right)}=\frac{4\pi {\in }_{0}ac}{\left(c-a\right)}$

Since the space between the two inner shells is filled with a dielectric, capacitance CAB becomes ${C}_{\mathrm{AB}}=\frac{4\pi {\in }_{0}abK}{\left(b-a\right)}$ and capacitance CBCbecomes ${C}_{\mathrm{BC}}=\frac{4\pi {\in }_{0}bc}{\left(c-b\right)}$.
Now, as the capacitors are in series, the equivalent capacitance is given by

$\frac{1}{C}=\frac{1}{{C}_{\mathrm{AB}}}+\frac{1}{{C}_{\mathrm{BC}}}\phantom{\rule{0ex}{0ex}}⇒C=\frac{{C}_{\mathrm{AB}}{C}_{\mathrm{BC}}}{{C}_{\mathrm{AB}}+{C}_{\mathrm{BC}}}\phantom{\rule{0ex}{0ex}}⇒C=\frac{\frac{\left(4\pi {\in }_{0}{\right)}^{2}a{b}^{2}Kc}{\left(b-a\right)\left(c-b\right)}}{\frac{4\pi {\in }_{0}abk}{\left(b-a\right)}+\frac{4\pi {\in }_{0}bc}{\left(c-b\right)}}\phantom{\rule{0ex}{0ex}}⇒C=\frac{\frac{4\pi {\in }_{0}Ka{b}^{2}c}{\left(b-a\right)\left(c-b\right)}}{\left(\frac{abk\left(c-b\right)+bc\left(b-a\right)}{\left(b-a\right)\left(c-b\right)}\right)}\phantom{\rule{0ex}{0ex}}⇒C=\frac{4\pi {\in }_{0}Ka{b}^{2}c}{\left[abk\left(c-b\right)+bc\left(b-a\right)\right]}\phantom{\rule{0ex}{0ex}}⇒C=\frac{4\pi {\in }_{0}Kabc}{\left[ak\left(c-b\right)+c\left(b-a\right)\right]}$

#### Question 64:

Since the space between the two inner shells is filled with a dielectric, capacitance CAB becomes ${C}_{\mathrm{AB}}=\frac{4\pi {\in }_{0}abK}{\left(b-a\right)}$ and capacitance CBCbecomes ${C}_{\mathrm{BC}}=\frac{4\pi {\in }_{0}bc}{\left(c-b\right)}$.
Now, as the capacitors are in series, the equivalent capacitance is given by

$\frac{1}{C}=\frac{1}{{C}_{\mathrm{AB}}}+\frac{1}{{C}_{\mathrm{BC}}}\phantom{\rule{0ex}{0ex}}⇒C=\frac{{C}_{\mathrm{AB}}{C}_{\mathrm{BC}}}{{C}_{\mathrm{AB}}+{C}_{\mathrm{BC}}}\phantom{\rule{0ex}{0ex}}⇒C=\frac{\frac{\left(4\pi {\in }_{0}{\right)}^{2}a{b}^{2}Kc}{\left(b-a\right)\left(c-b\right)}}{\frac{4\pi {\in }_{0}abk}{\left(b-a\right)}+\frac{4\pi {\in }_{0}bc}{\left(c-b\right)}}\phantom{\rule{0ex}{0ex}}⇒C=\frac{\frac{4\pi {\in }_{0}Ka{b}^{2}c}{\left(b-a\right)\left(c-b\right)}}{\left(\frac{abk\left(c-b\right)+bc\left(b-a\right)}{\left(b-a\right)\left(c-b\right)}\right)}\phantom{\rule{0ex}{0ex}}⇒C=\frac{4\pi {\in }_{0}Ka{b}^{2}c}{\left[abk\left(c-b\right)+bc\left(b-a\right)\right]}\phantom{\rule{0ex}{0ex}}⇒C=\frac{4\pi {\in }_{0}Kabc}{\left[ak\left(c-b\right)+c\left(b-a\right)\right]}$

Charge that the capacitor can hold = 12 μC
Operating voltage = 1200 V
Breakdown strength, b =
The separation between the plates of the capacitor is given by

The capacitance of the capacitor is given by

The capacitance can be expressed in terms of the area of the plates (A) and the separation of the plates (d) as:

Thus, the area of the plates is given by

#### Question 65:

Charge that the capacitor can hold = 12 μC
Operating voltage = 1200 V
Breakdown strength, b =
The separation between the plates of the capacitor is given by

The capacitance of the capacitor is given by

The capacitance can be expressed in terms of the area of the plates (A) and the separation of the plates (d) as:

Thus, the area of the plates is given by

Area of the plates of the capacitor, A = 100 cm2 = 10$-$2 m2
Separation between the plates, d = 1 cm = 10$-$2 m
Emf of battery, V = 24 V

Therefore,
Capacitance,

Energy stored in the capacitor,

Force of attraction between the plates,

#### Question 66:

Area of the plates of the capacitor, A = 100 cm2 = 10$-$2 m2
Separation between the plates, d = 1 cm = 10$-$2 m
Emf of battery, V = 24 V

Therefore,
Capacitance,

Energy stored in the capacitor,

Force of attraction between the plates,

Let the potential of the battery connected to the capacitor be V.
Let the length of the part of the slab inside the capacitor be x.

The capacitor can be considered to be two capacitors in parallel.

The capacitances of the two capacitors in parallel are:

C1is the part of the capacitor having the dielectric inserted in it and C2 is the capacitance of the part of the capacitor without dielectric.

As, C1 and C2 are in parallel.

Therefore, the net capacitance is given by
C = C1 + C2

The dielectric slab is attracted by the electric field of the capacitor and applies a force in left direction.
Let us consider electric force of magnitude F pulls the slab in left direction.
Let there be an infinitesimal displacement dx in left direction by the force F.
The work done by the force = F.dx
Let the potential of the battery connected to the left capacitor be V1 and that of the battery connected with the right capacitor be V2.

With the displacement of slab, the capacitance will increase, therefore the energy of the capacitor will also increase. In order to maintain constant voltage, the battery will supply extra charges, therefore the battery will do work.

Work done by the battery = change in energy of capacitor + work done by the force F on the capacitor
dWB= dU  + dWF
Let the charge dq is supplied by the battery, and the change in the capacitor be dC
dWB = (dq).V = (dC).V2
dU = $\frac{1}{2}$(dC).V2
(dC).V2 = $\frac{1}{2}$(dC).V2 + F.dx
$\frac{1}{2}$(dC).V2 = F.dx

To keep the dielectric slab in equilibrium, the electrostatic force on it must be counteracted by the weight of the block attached.

Therefore,

$\frac{{\in }_{0}b{V}^{2}}{2d}\left(K-1\right)=Mg\phantom{\rule{0ex}{0ex}}⇒M=\frac{{\in }_{0}b{V}^{2}}{2dg}\left(K-1\right)$

#### Question 67:

Let the potential of the battery connected to the capacitor be V.
Let the length of the part of the slab inside the capacitor be x.

The capacitor can be considered to be two capacitors in parallel.

The capacitances of the two capacitors in parallel are:

C1is the part of the capacitor having the dielectric inserted in it and C2 is the capacitance of the part of the capacitor without dielectric.

As, C1 and C2 are in parallel.

Therefore, the net capacitance is given by
C = C1 + C2

The dielectric slab is attracted by the electric field of the capacitor and applies a force in left direction.
Let us consider electric force of magnitude F pulls the slab in left direction.
Let there be an infinitesimal displacement dx in left direction by the force F.
The work done by the force = F.dx
Let the potential of the battery connected to the left capacitor be V1 and that of the battery connected with the right capacitor be V2.

With the displacement of slab, the capacitance will increase, therefore the energy of the capacitor will also increase. In order to maintain constant voltage, the battery will supply extra charges, therefore the battery will do work.

Work done by the battery = change in energy of capacitor + work done by the force F on the capacitor
dWB= dU  + dWF
Let the charge dq is supplied by the battery, and the change in the capacitor be dC
dWB = (dq).V = (dC).V2
dU = $\frac{1}{2}$(dC).V2
(dC).V2 = $\frac{1}{2}$(dC).V2 + F.dx
$\frac{1}{2}$(dC).V2 = F.dx

To keep the dielectric slab in equilibrium, the electrostatic force on it must be counteracted by the weight of the block attached.

Therefore,

$\frac{{\in }_{0}b{V}^{2}}{2d}\left(K-1\right)=Mg\phantom{\rule{0ex}{0ex}}⇒M=\frac{{\in }_{0}b{V}^{2}}{2dg}\left(K-1\right)$

Let the potential of the battery connected to the left capacitor be V1 and that of the battery connected to the right capacitor be V2.

Considering the left capacitor,
Let the length of the part of the slab inside the capacitor be x.

The left capacitor can be considered to be two capacitors in parallel.

The capacitances of the two capacitors in parallel are:
${C}_{1}=\frac{{K}_{1}{\in }_{0}bx}{d},{C}_{2}=\frac{{\in }_{0}b\left({l}_{1}-x\right)}{d}$
C1is the part of the capacitor having the dielectric inserted in it and C2 is the capacitance of the part of the capacitor without dielectric.

As, C1 and C2 are in parallel.

Therefore, the net capacitance is given by
C = C1 + C2

Therefore, the potential energy stored in the left capacitor will be

The dielectric slab is attracted by the electric field of the capacitor and applies a force in left direction.
Let us consider electric force of magnitude F pulls the slab in left direction.
Let there be an infinitesimal displacement dx in left direction by the force F.
The work done by the force = F.dx

Let us consider a small displacement dx of the slab in the inward direction. The capacitance will increase, therefore the energy of the capacitor will also increase. In order to maintain constant voltage, the battery will supply extra charges, therefore the battery will do work.

Work done by the battery = change in energy of capacitor + work done by the force F on the capacitor
dWB= dU  + dWF
Let the charge dq is supplied by the battery, and the change in the capacitor be dC
dWB = (dq).V = (dC).V2
dU = $\frac{1}{2}$(dC).V2

(dC).V2 = $\frac{1}{2}$(dC).V2 + F.dx
$\frac{1}{2}$(dC).V2 = F.dx

Similarly, for the right side the voltage of the battery is given by
${V}_{2}=\sqrt{\frac{F×2d}{{\in }_{0}b\left({K}_{2}-1\right)}}$

Thus, the ratio of the voltages is given by
$\frac{{V}_{1}}{{V}_{2}}=\frac{\sqrt{\frac{F×2d}{{\in }_{0}b\left({\mathrm{K}}_{1}-1\right)}}}{\sqrt{\frac{F×2d}{{\in }_{0}b\left({\mathrm{K}}_{2}-1\right)}}}\phantom{\rule{0ex}{0ex}}⇒\frac{{V}_{1}}{{V}_{2}}=\frac{\sqrt{{K}_{2}-1}}{\sqrt{{K}_{1}-1}}$

Thus, the ratio of the emfs of the left battery to the right battery is $\frac{{V}_{1}}{{V}_{2}}=\frac{\sqrt{{K}_{2}-1}}{\sqrt{{K}_{1}-1}}$

#### Question 68:

Let the potential of the battery connected to the left capacitor be V1 and that of the battery connected to the right capacitor be V2.

Considering the left capacitor,
Let the length of the part of the slab inside the capacitor be x.

The left capacitor can be considered to be two capacitors in parallel.

The capacitances of the two capacitors in parallel are:
${C}_{1}=\frac{{K}_{1}{\in }_{0}bx}{d},{C}_{2}=\frac{{\in }_{0}b\left({l}_{1}-x\right)}{d}$
C1is the part of the capacitor having the dielectric inserted in it and C2 is the capacitance of the part of the capacitor without dielectric.

As, C1 and C2 are in parallel.

Therefore, the net capacitance is given by
C = C1 + C2

Therefore, the potential energy stored in the left capacitor will be

The dielectric slab is attracted by the electric field of the capacitor and applies a force in left direction.
Let us consider electric force of magnitude F pulls the slab in left direction.
Let there be an infinitesimal displacement dx in left direction by the force F.
The work done by the force = F.dx

Let us consider a small displacement dx of the slab in the inward direction. The capacitance will increase, therefore the energy of the capacitor will also increase. In order to maintain constant voltage, the battery will supply extra charges, therefore the battery will do work.

Work done by the battery = change in energy of capacitor + work done by the force F on the capacitor
dWB= dU  + dWF
Let the charge dq is supplied by the battery, and the change in the capacitor be dC
dWB = (dq).V = (dC).V2
dU = $\frac{1}{2}$(dC).V2

(dC).V2 = $\frac{1}{2}$(dC).V2 + F.dx
$\frac{1}{2}$(dC).V2 = F.dx

Similarly, for the right side the voltage of the battery is given by
${V}_{2}=\sqrt{\frac{F×2d}{{\in }_{0}b\left({K}_{2}-1\right)}}$

Thus, the ratio of the voltages is given by
$\frac{{V}_{1}}{{V}_{2}}=\frac{\sqrt{\frac{F×2d}{{\in }_{0}b\left({\mathrm{K}}_{1}-1\right)}}}{\sqrt{\frac{F×2d}{{\in }_{0}b\left({\mathrm{K}}_{2}-1\right)}}}\phantom{\rule{0ex}{0ex}}⇒\frac{{V}_{1}}{{V}_{2}}=\frac{\sqrt{{K}_{2}-1}}{\sqrt{{K}_{1}-1}}$

Thus, the ratio of the emfs of the left battery to the right battery is $\frac{{V}_{1}}{{V}_{2}}=\frac{\sqrt{{K}_{2}-1}}{\sqrt{{K}_{1}-1}}$

Given that the area of the plates of the capacitors is A.
As ''a'' length of the dielecric slab is inside the capacitor.
Therefore, the area of the plate covered with dielectric is = $\frac{A}{l}a$
The capacitance of the portion with dielectric is given by
${C}_{1}=\frac{K{\in }_{0}Aa}{ld}$
The capacitance of the portion without dielectric is given by
${C}_{1}=\frac{{\in }_{0}A\left(l-a\right)}{ld}$

The two parts can be considered to be in parallel.
Therefore, the net capacitance is given by

Let us consider a small displacement da of the slab in the inward direction. The capacitance will increase, therefore the energy of the capacitor will also increase. In order to maintain constant voltage, the battery will supply extra charges, therefore the battery will do work.

Work done by the battery = change in energy of capacitor + work done by the force F on the capacitor
dWB= dU  + dWF
Let the charge dq is supplied by the battery, and the change in the capacitor be dC
dWB = (dq).V = (dC).V2
dU = $\frac{1}{2}$(dC).V2

(dC).V2 = $\frac{1}{2}$(dC).V2 + F.da
$\frac{1}{2}$(dC).V2 = F.da

The acceleration of the dielectric is given by a0= $\frac{1}{2}\frac{{\in }_{0}A}{ldm}\left(K-1\right)$
As, the force is in inward direction, it will tend to make the dielectric to completely fill the space inside the capacitors. As, the dielectric completely fills the space inside the capacitor at this instant its velocity is not zero. The dielectric slab tends to move outside the capacitor. As the slab tends to move out, the direction of the force due to the capacitor will reverse its direction. Thus, the dielectric slab will have a periodic motion.

The time taken to move distance (l$-$a) can be calculated as:

$\left(l-a\right)=\frac{1}{2}{a}_{0}{t}^{2}\phantom{\rule{0ex}{0ex}}t=\sqrt{\frac{2\left(l-a\right)}{{a}_{0}}}\phantom{\rule{0ex}{0ex}}t=\sqrt{2\left(l-a\right)×\frac{2ldm}{{\in }_{0}A{V}^{2}\left(K-1\right)}}\phantom{\rule{0ex}{0ex}}t=\sqrt{\frac{4m\left(l-a\right)ld}{{\in }_{0}A{V}^{2}\left(K-1\right)}}$
For the complete cycle the time period will be four times the time taken for covering distance (l$-$a).
It is given by :
$T=4t=4×2\sqrt{\frac{m\left(l-a\right)ld}{{\in }_{0}A{V}^{2}\left(K-1\right)}}=8\sqrt{\frac{m\left(l-a\right)ld}{{\in }_{0}A{V}^{2}\left(K-1\right)}}$

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