NCERT Solutions for Class 12 Physics Chapter 2 Electrostatic Potential and Capacitance

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NCERT Solutions for Class 12 Physics Chapter 2 Electrostatic Potential and Capacitance includes the usage of many complicated equations and formulas that students learn in their Class 12. A PDF file of the NCERT Solutions for Class 12 Physics Electrostatic Potential and Capacitance is available here for free download. The PDF includes important questions, answers to questions from the textbook, worksheets and other assignments.

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Class 12 Physics NCERT Solutions Electrostatic Potential and Capacitance Important Questions

Q 2.1) Two charges 5 × 10^-8 C and -3 x 10^-8 C are located 16 cm apart from each other. At what point (s) on the line joining the two charges is the electric potential zero? Take the potential at infinity to be zero.

Solution:Given,

q₁ = 5 x 10^-8 C

q₂ = -3 x 10^-8 C

The two charges are at a distance, d = 16cm = 0.16m from each other.

Let us consider a point “P” over the line joining charges q₁ and q₂.

Let the distance of the considered point P from q₁ be ‘r’

Let us consider point P to have zero electric potential (V)..

The electric potential at point P is the summation of potentials due to charges q₁ and q₂.

Therefore,  $V = \frac{1}{4\pi \epsilon _{o}} . \frac{q_{1}}{r} + \frac{1}{4\pi \epsilon _{o}}.\frac{q_{2}}{d – r}$  ……….(1)

Here,

Putting  V = 0, in eqn. (1), we get,

0 = $\frac{1}{4\pi \epsilon _{o}} . \frac{q_{1}}{r} + \frac{1}{4\pi \epsilon _{o}}.\frac{q_{2}}{d – r}$

5(0.16 – r)  = 3r

0.8 = 8r

r = 0.1m = 10 cm.

Therefore, at a distance of 10 cm from the positive charge, the potential is zero between the two charges.

Let us assume a point P at a distance ‘s’  from the negative charge be outside the system, having potential zero.

So, for the above condition, the potential is given by –

Here,

For V = 0, eqn. (2) can be written as :

0 = $\frac{1}{4\pi \epsilon _{o}} . \frac{q_{1}}{s} + \frac{1}{4\pi \epsilon _{o}}.\frac{q_{2}}{s – d}$

5(s – 0.16) = 3s

0.8  = 2s

S = 0.4 m = 40 cm.

Therefore, at a distance of 40 cm from the positive charge outside the system of charges, the potential is zero.

Q 2.2)  A regular hexagon of side 10 cm has a charge 5 µC at each of its vertices. Calculate the potential at the centre of the hexagon.

Solution:

The figure shows regular hexagon containing charges q, at each of its vertices.

Here,

q =  5 µC = 5 × 10^-6 C.

Length of each side of hexagon, AB =BC = CD = DE = EF = FA = 10 cm.

The distance of the vertices from the centre O, d = 10 cm.

The electric potential at point O,

V =  $\frac{1}{4\pi \epsilon _{o}}.\frac{6q}{d}$

Here,

V =  $\frac{9 × 10^{9} × 6 × 5 × 10^{-6}}{0.1}$ = 2.7 × 10⁶ V.

Q 2.3) Two charges 2 µC and -2 µC are placed at points A and B, 6 cm apart.

(1) Identify the equipotential surface of the system.

(2) What is the direction of the electric field at every point on this surface?

Solution:

(1) An equipotential surface is defined as the surface over which the total potential is zero. In the given question the plane is normal to line AB. The plane is located at the mid – point of the line AB as the magnitude of the charges are same.

(2) At every point on this surface the direction of the electric field is normal to the plane in the direction of AB.

Q 2.4) A spherical conductor of radius 12 cm has a charge of 1.6 x 10^-7C distributed uniformly on its surface. What is the electric field

(1) inside the sphere.

(2) just outside the sphere.

(3) at a point 18 cm from the centre of sphere.

Solution:

(1) Given,

Radius of spherical conductor, r = 12cm = 0.12m

Charge is distributed uniformly over the surface, q = 1.6 x 10^-7 C.

The electric field inside a spherical conductor is zero.

(2) Electric field E, just outside the conductor is given by the relation,

E =  $\frac{1}{4\pi \epsilon _{o}}.\frac{q}{r^{2}}$

Here,

ε₀ = permittivity of free space and $\frac{1}{4\pi \epsilon _{o}}$ = 9 x 10⁹ Nm²C^-2

Therefore,

E = $\frac{9 \times 10^{9} \times 1.6 \times 10^{-7}}{(0.12)^{2}}=10^5\,NC^{-1}$

Therefore, just outside the sphere the electric field is 10⁵ NC^-1.

(3) From the centre of the sphere the electric field at a point 18m = E₁.

From the centre of the sphere, the distance of point d = 18 cm = 0.18m.

E₁ = $\frac{1}{4\pi \epsilon _{o}} . \frac{q}{d^{2}}$ = $\frac{9 \times 10^{9} \times 1.6 \times 10^{-7}}{(1.8 \times 10^{-2})^{2}}$ = 4.4 x 10⁴ NC^-1

So, from the centre of sphere the electric field at a point 18 cm away is 4.4 x 10⁴ NC^-1.

Q 2.5) A parallel plate capacitor with air between the plates has a capacitance of 8pF (1pF = 10^-12 F. What will be the capacitance if the distance between the plates is reduced by half and the space between them is filled with a substance of dielectric constant 6?

Solution: Given,

Capacitance, C = 8pF.

In the first case, the parallel plates are at a distance ‘d’  and is filled with air.

Air has dielectric constant, k = 1

Capacitance, C = $\frac{k \times \epsilon _{o} \times A}{d}$ = $\frac{\epsilon _{o} \times A}{d}$     … eq(1)

Here,

A = area of each plate

Now, if the distance between the parallel plates is reduced to half, then d₁ = d/2

Given, dielectric constant of the substance, k₁ = 6

Hence, the capacitance of the capacitor,

C₁ = $\frac{k_{1} \times \epsilon _{o} \times A}{d_{1}}$ = $\frac{6 \epsilon _{o} \times A}{d/2}$ = $\frac{12 \epsilon _{o} A}{d}$  … (2)

Taking ratios of eqns. (1) and (2), we get,

C₁ = 2 x 6 C = 12 C = 12 x  8 pF = 96pF.

Hence, the capacitance between the plates is 96pF.

Q 2.6)  Three capacitors connected in series have capacitance of 9pF each.

(1)  What is the total capacitance of the combination?

(2) What is the potential difference across each capacitor if the combination is connected to a 120 V supply?

Solution:

(1)  Given,

The capacitance of the three capacitors, C = 9 pF

Equivalent capacitance (c_eq) is the capacitance of the combination of the capacitors given by

Therefore, the total capacitance = 3pF.

(2) Given, supply voltage, V = 100V

The potential difference (V₁) across the capacitors will be equal to one – third of the supply voltage.

Therefore, V₁ =  $\frac{V}{3}$ = $\frac{120}{3}$ = 40V.

Hence, the potential difference across each capacitor is 40V.

Q 2.7)  Three capacitors of capacitances 2 pF, 3 pF and 4 pF are connected in parallel.

(1) What is the total capacitance of the combination?

(2) Determine the charge on each capacitor if the combination is connected to a 100 V supply.

Solution:

(1) Given, C₁ = 2pF, C₂ = 3pF and C₃ = 4pF.

Equivalent capacitance for the parallel combination is given by C_eq .

Therefore, C_eq = C₁ + C₂ + C₃ = 2 + 3 + 4 = 9pF

Hence, the total capacitance of the combination is 9pF.

(2) Supply voltage, V = 100V

The three capacitors are having the same voltage, V = 100v

q = VC

where,

q = charge

C = capacitance of the capacitor

V = potential difference

for capacitance, c = 2pF

q = 100 x 2 = 200pC = 2 x 10^-10C

for capacitance, c = 3pF

q = 100 x 3 = 300pC = 3 x 10^-10C

for capacitance, c = 4pF

q = 100 x 4 = 400pC = 4 x 10^-10 C

Q 2.8) In a parallel plate capacitor with air between the plates, each plate has an area of 6 x 10^-3 m ² and the distance between the plates is 3 mm. Calculate the capacitance of the capacitor. If this capacitor is connected to a 100 V supply, what is the charge on each plate of the capacitor?

Solution:  Given,

The area of plate of the capacitor, A = 6 x 10^-3 m²

Distances between the plates, d = 3mm = 3 x 10^-3 m

Voltage supplied, V = 100V

Capacitance of a parallel plate capacitor is given by, C =  $\frac{\epsilon \times A}{d}$

Here,

ε_o = permittivity of free space = 8.854 × 10^-12 N^-1 m ^-2 C^-2

C = $\frac{8.854 \times 10^{-12} \times 6 \times 10^{-3}}{3 \times 10^{-3}}$ = 17.71 x 10^-12 F = 17.71 pF.

Therefore, each plate of the capacitor is having a charge of

q = VC = 100 × 17.71 x 10^-12 C = 1.771 x 10^-9 C

Q 2.9: Explain what would happen if, in the capacitor given in Exercise 2.8, a 3 mm thick mica sheet (of dielectric constant = 6) were inserted between the plates,

(a)while the voltage supply remained connected.

(b)after the supply was disconnected.

Answer 2.9:

(a) Dielectric constant of the mica sheet, k = 6

If voltage supply remained connected, the voltage between two plates will be

constant.

Supply voltage, V = 100 V

Initial capacitance, C = 1.771 × 10⁻¹¹ F

New capacitance, C₁ = kC = 6 × 1.771 × 10⁻¹¹ F = 106 pF

New charge, q₁ = C₁V = 106 × 100 pC = 1.06 × 10^–8 C

Potential across the plates remain 100 V.

(b) Dielectric constant, k = 6

Initial capacitance, C = 1.771 × 10⁻¹¹ F

New capacitance, C₁ = kC = 6 × 1.771 × 10⁻¹¹ F = 106 pF

If the supply voltage is removed, then there will be a constant amount of charge

in the plates.

Charge = 1.771 × 10⁻⁹ C

Potential across the plates is given by,

V₁ = q/C₁ = $\frac{1.771 \times 10^{-9}}{106 \times 10^{-12}}$

= 16.7 V

Q 2.10) A 12pF capacitor is connected to a 50V battery. How much electrostatic energy is stored in the capacitor?

Solution: Given,

Capacitance of the capacitor, C = 12pF = 12 x 10^-12 F

Potential difference, V = 50 V

Electrostatic energy stored in the capacitor is given by the relation,

E = $\frac{1}{2}$ CV² = $\frac{1}{2}$ x 12 x 10^-12 x (50)² J = 1.5 x 10^-8 J

Therefore, the electrostatic energy stored in the capacitor is 1.5 x 10^-8 J.

Q 2.11) A 600pF capacitor is charged by a 200V supply. It is then disconnected from the supply and is connected to another uncharged 600 pF capacitor. How much electrostatic energy is lost in the process?

Solution: Given,

Capacitance, C = 600pF

Potential difference, V = 200v

Electrostatic energy stored in the capacitor is given by :

E₁ = $\frac{1}{2}$CV² = $\frac{1}{2}$ x (600 x 10^-12) x (200)²J = 1.2 x 10^-5 J

According to the question, the source is disconnected to the 600pF and connected to another capacitor of 600pF, then equivalent capacitance (C_eq) of the combination is given by,

= $\frac{2}{600}$ = $\frac{1}{300}$

C_eq = 300pF

New electrostatic energy can be calculated by:

E₂ = $\frac{1}{2}$CV² = $\frac{1}{2}$ x 300 x 10^-12 x (200)² J = 0.6 x 10^-5 J

Loss in electrostatic energy,

E = E₁ – E₂

E = 1.2 x 10^-5 – 0.6 x 10^-5 J = 0.6 x 10^-5 J = 6 x 10^-6 J

Therefore, the electrostatic energy lost in the process is 6 x 10^-6 J.

Q2.12) A charge of 8 mC is located at the origin. Calculate the work done in taking a small charge of –2 × 10^–9 C from a point P (0, 0, 3 cm) to a point Q (0, 4 cm, 0), via a point R (0, 6 cm, 9 cm).

Solution:

Charge located at the origin, q = 8 mC = 8 x 10^-3 C

The magnitude of the charge taken from the point P to R and then to Q, q₁ = – 2 x 10^-9 C

Here OP= d₁= 3 cm = 3 x 10^-2 m

OQ = d₂= 4 cm = 4 x 10^-2 m

Potential at the point P, $V_{1}=\frac{q}{4\pi \epsilon _{0}d_{1}}$

Potential at the point Q, $V_{2}=\frac{q}{4\pi \epsilon _{0}d_{2}}$

The work done (W) is independent of the path

Therefore, W = q₁[V₁ – V₂]

Where, $\frac{1}{4\pi \epsilon _{0}}=9 \times 10^{9}Nm^{2}C^{-2}$

Therefore,

= -144 x 10^-3 x (-100/12)

= 1.2  Joule

Therefore, the work done during the process is 1.2 J

Q2.13) A cube of side b has a charge q at each of its vertices. Determine the potential and electric field due to this charge array at the centre of the cube.

Solution:

Sides of the cube = b

Charge at the vertices = q

Diagonal of one of the sides of the cube

d = b√2

Length of the diagonal of the cube

l = b√3

The distance between one of the vertices and the centre of the cube is

r = l/2 =(b√3/2)

The electric potential (V) at the centre of the cube is due to the eight charges at the vertices

V = 8q/4πε₀

Therefore, potential at the centre of the cube is $\frac{4q}{\sqrt{3}\pi \epsilon _{0}b}$

The electric field intensity at the centre of the cube, due to the eight charges is zero.  The charges are distributed symmetrically with respect to the centre of the cube. Therefore, they get cancelled.

Q2.14) Two tiny spheres carrying charges 1.5 µC and 2.5 µC are located 30 cm apart. Find the potential and electric field:
(a) at the mid-point of the line joining the two charges, and
(b) at a point 10 cm from this midpoint in a plane normal to the
line and passing through the mid-point.

Solution:

Two tiny spheres carrying charges are located at points A and B

The charge at point A, q₁= 1.5 µC

The charge at point B, q₂ = 2.5 µC

The distance between the two charges = 30 cm  = 0.3 m

(a) Let O be the midpoint. Let V₁ and E₁ be the potential and electric field respectively at the midpoint.

V₁ = Potential due to charge at A + Potential due to charge at B

Here,

ε₀ = Permittivity of the free space

Therefore,

Electric field at O, E₁ = Electric field due to q₂ – Electric field due to q₁

= 400 x 10³ V/m

Therefore, the potential at midpoint is 2.4 x 10⁵ V and the electric field at the midpoint is 400 x 10³ V/m.

(b) Consider a point Z such that the distance OZ = 10 cm = 0.1 m as shown in the figure.

Let V₂ and E₂ be the potential and electric field respectively at point Z.  The distance

The potential at V₂ = Potential due to the charge at A + Potential due to the charge at B

= 2 x 10⁵ V

The electric field due to q₁ at Z

= 416 x 10³ V/m

The electric field due to q₂ at Z

= 694 x 10³ V/m

The resultant field intensity at Z

From the figure we get cos θ = (0.10/0.18) = 5/9 = 0.5556

θ = cos ^-1 (0.5556) = 56.25

2θ = 2 x 56.25 = 112.5⁰

cos 2θ = – 0.38

= 6.6 x 10⁵ V/m

Therefore the potential at the point Z is 694 x 10³ V/m and the electric field is 6.6 x 10⁵ V/m.

Q2.15) A spherical conducting shell of inner radius r₁ and outer radius r₂ has a charge Q.
(a) A charge q is placed at the centre of the shell. What is the surface charge density on the inner and outer surfaces of the shell?
(b) Is the electric field inside a cavity (with no charge) zero, even if the shell is not spherical, but has any irregular shape? Explain.

Solution:

(a) If a charge +q is placed at the centre of the shell, a charge of magnitude -q is induced in the inner surface of the shell. Therefore, the surface charge density at the inner surface of the shell is given by the relation,

σ₁ = Total charge / Inner surface area = – q/4πr₁²——–(1)

A charge +q is induced on the outer surface of the shell. The total charge on the outer surface of the shell is Q+q. Surface charge density at the outer surface of the shell

σ₂ = Total charge / Outer surface area = (Q+q)/4πr₁²————(2)

(b) Yes. The electric field inside a cavity (with no charge) will be zero, even if the shell is not spherical but has any irregular shape. Take a closed-loop, part of which is inside the cavity along a field line and the rest inside the conductor. Since the field inside the conductor is zero, this gives a net work done by the field in carrying a test charge over a closed loop. We know this is impossible for an electrostatic field. Hence, there are no field lines inside the cavity (i.e., no field), and no charge on the inner surface of the conductor, whatever be its shape.

Q2.16) (a) Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by
$\left ( E_{2}-E_{1} \right ).\hat{n}=\frac{\sigma }{\epsilon _{0}}$
where $\hat{n}$ is a unit vector normal to the surface at a point and σ is the surface charge density at that point. (The direction of  $\hat{n}$ is from side 1 to side 2.) Hence, show that just outside a conductor, the electric field is $\sigma \hat{n}/\epsilon _{0}$. 

(b) Show that the tangential component of the electrostatic field is continuous from one side of a charged surface to another. [Hint: For (a), use Gauss’s law. For, (b) use the fact that work done by electrostatic field on a closed loop is zero.].

Solution:

(a) Let E₁ be the electric field at one side of the charged body and E₂ is the electric field at the other side of the charged body. If the infinite plane charged body has a uniform thickness, the electric field due to one of the surface of the charged body is

here,

σ = surface charge density at that point

The electric field due to the other surface of the charged body is

The electric field at any point due to the charge surfaces

Since inside the conductor, $\vec{E_{1}}= 0$

Therefore, the electric field just outside the conductor is $\frac{\sigma }{\epsilon _{0}}\hat{n}$

(b)  When a charged particle is moved from one point to the other on a closed-loop, the work done by the electrostatic field is zero. Hence, the tangential component of the electrostatic field is continuous from one side of a charged surface to the other.

Q2.17) A long charged cylinder of linear charged density λ is surrounded by a hollow co-axial conducting cylinder. What is the electric field in the space between the two cylinders?

Solution:

Let the length of the charged cylinder and the hollow co-axial conducting cylinder be L

Charge density of the long charged cylinder is λ

Let E be the electric field in the space between the two cylinders

According to Gauss theorem, the electric flux through the Gaussian surface is given as Φ = E (2πd)L

Here,

d is the distance between the common axis of the cylinders

Therefore , Φ = E (2πd)L = q/ε₀

here, q is the charge on the inner surface of the outer cylinder

ε₀  is the permittivity of the free space

E (2πd)L = λL/ε₀

E = λ/2πdε₀

Therefore, the electric field in the space between the two cylinders is λ/2πdε₀

Q2.18) In a hydrogen atom, the electron and proton are bound at a distance of about 0.53 Å:
(a) Estimate the potential energy of the system in eV, taking the zero of the potential energy at infinite separation of the electron from proton.
(b) What is the minimum work required to free the electron, given that its kinetic energy in the orbit is half the magnitude of potential energy obtained in (a)?
(c) What are the answers to (a) and (b) above if the zero of potential
energy is taken at 1.06 Å separation?

Solution:

The distance between the proton and electron of hydrogen atom, d = 0.53 Å

Charge of the electron, q₁ = -1.6 x 10^-19 C

Charge of the proton,  q₂ = +1.6 x 10^-19 C

(a) At infinite separation of electron and proton, potential energy is zero

Potential energy of the system = Potential energy at infinity – Potential energy at distance d

= 0 – $\frac{q_{1}q_{2}}{4\pi \epsilon _{0}d}$

here,

ε₀ is the permittivity of the free space

Potential energy= 0 – $\frac{9\times 10^{9}\times (1.6 \times 10^{-19})^{2}}{0.53\times 10^{10}} = -43.7 x 10^{-19}J$

Potential energy = -43.7 x 10^-19/1.6 x 10^-19 = -27.2 eV [Since 1.6 x 10^-19 J = 1 eV]

Therefore, the potential energy of the system is -27.2 eV

(b) Half of the magnitude of the potential energy is equal to the kinetic energy

Kinetic energy = |V|/2 = (1/2) x (27.2) = 13. 6 eV

Total energy = Kinetic energy + potential energy

= 13.6 eV – 27.2 eV

Total energy = – 13.6 eV

Therefore, the minimum work required to free an electron is – 13.6 eV

(c) When the zero of the potential energy is taken as, d₁= 1.06 Å

Potential energy of the system = Potential energy at d₁ – Potential energy at d

= $\frac{q_{1}q_{2}}{4\pi \epsilon _{0}d_{1}}$ – 27.2 eV

= $\frac{9\times 10^{9}\times (1.6 \times 10^{-19})^{2}}{1.06\times 10^{10}}$ -27.2 eV

= 21.73 x 10^-19 J – 27.2 eV

= 13.58 eV -27.2 eV   [Since 1.6 x 10^-19 J = 1 eV]

= -13.6 eV

Q2.19) If one of the two electrons of the H₂ molecule is removed, we get a hydrogen molecular ion H_2⁺. In the ground state of an H_2⁺, the two protons are separated by roughly 1.5 Å, and the electron is roughly 1 Å from each proton. Determine the potential energy of the system. Specify your choice of the zero of potential energy.

Solution:

Charge of the 1^st proton, q₁ = 1.6 x 10^-19 C

Charge of the 2^nd proton, q₂ = 1.6 x 10^-19 C

Charge of the electron, q₃ = -1.6 x 10^-19 C

Distance between the 1^st and the 2^nd proton, d₁ = 1.5 x 10^-10 m

Distance between the 1^st proton and the electron, d₂ = 1 x 10^-10 m

Distance between the 2nd proton and the electron, d₃ = 1 x 10^-10 m

The potential energy at infinity is zero

Therefore, the potential energy of the system is

Substituting (1/4πε₀) = 9 x 10⁹ Nm²C^-2 we get

=-30.7 x 10^-19 J

=-19.2 eV (1eV = 1.6 x 10^-19 J)

Therefore, the potential energy of the system is -19.2 eV.

Q2.20) Two charged conducting spheres of radii a and b are connected to each other by a wire. What is the ratio of electric fields at the surfaces of the two spheres? Use the result obtained to explain why the charge density on the sharp and pointed ends of a conductor is higher than on its flatter portions.

Solution:

Let A be the sphere of radius a, Charge Q_A and capacitance C_A

Let B be the sphere of radius b, Charge Q_B and capacitance C_B

The conducting spheres are connected by a wire, therefore the potential of both the capacitors will be V

The electric field due to a, $E_{A}=\frac{Q_{A}}{4\pi \epsilon _{0}a^{2}}$

The electric field due to b, $E_{B}=\frac{Q_{B}}{4\pi \epsilon _{0}b^{2}}$

The ratio of electric fields at the surface of the spheres is

Putting equation (2) in equation (1) we get

Therefore, the ratio of the electric field at the surface is b/a

A sharp and pointed end is like a sphere of very small radius and the flat portion is like a sphere of large radius. Therefore, the charge density is of pointed ends is higher than the flat portion.

Q2.21) Two charges –q and +q are located at points (0, 0, –a) and (0, 0, a), respectively.
(a) What is the electrostatic potential at the points (0, 0, z) and (x, y, 0) ?
(b) Obtain the dependence of potential on the distance r of a point from the origin when r/a >> 1.
(c) How much work is done in moving a small test charge from the point (5,0,0) to (–7,0,0) along the x-axis? Does the answer change if the path of the test charge between the same points is not along the x-axis?

Solution:

(a)  Two charges –q and +q are located at points (0, 0, –a) and (0, 0, a), respectively. They will form a dipole. The point (0, 0, z) is on the axis of the dipole and (x,y,0) is normal to the dipole. The electrostatic potential at (x,y,0) is zero. The electrostatic potential at (0,0,z) is given by

ε₀ = Permittivity of free space

p = dipole moment of the system= q x 2a

(b) The distance “r” is much larger than half of the distance between the two charges. Therefore, the potential at the point r is inversely proportional to the square of the distance.i.e., V∝(1/r²).

(c) x,y plane is a equipotential surface and x-axis is a equipotential line. Therefore, the change in potential (dV) along x-axis will be zero. The work done in moving a small test charge from the point (5,0,0) to (–7,0,0) along the x-axis is given by

Potential at (5,0,0)

Potential at (-7,0,0)

V₂ – V₁ = 0

Work done = Charge (q) x Change in Potential (V₂ – V₁)

Since the change in potential is zero, the work done is also zero.

The change in potential is independent of the path taken between the two points. Therefore, the work done in moving a point charge will remain zero.

Q2. 22) Figure below shows a charge array known as an electric quadrupole. For a point on the axis of the quadrupole, obtain the dependence of potential on r for r/a >> 1, and contrast your results with that due to an electric dipole, and an electric monopole (i.e., a single charge).

Solution:

Four charges are placed at points A,B,B and C respectively.

Let us consider a point P located at the axis of the quadrupole.

It can be considered that the electric quadrupole has three charges.

The charge +q is placed at A

The charge -2q is placed at B

The charge +q is placed at C

AB = BC = a

BP = r

PA = r + a

PZ = r – a

Therefore, the electrostatic potential due to the system of three charges is

Since r/a >>1,

a/r<< 1

Therefore, a²/r² is negligible

So we get,

Therefore we get,

V∝ 1/r³

However, for a dipole, V ∝ 1/r²

And for a monopole, V ∝ 1/r

Q2.23) An electrical technician requires a capacitance of 2 µF in a circuit across a potential difference of 1 kV. A large number of 1 µF capacitor are available to him each of which can withstand a potential difference of not more than 400 V. Suggest a possible arrangement that requires the minimum number of capacitors.

Solution:

Required Capacitance, C= 2μF

Potential difference, V = 1 kV = 1000 V

Capacitance of each capacitor, C₁ = 1μF

Potential difference that the capacitors can withstand, V₁ = 400 V

Suppose a number of capacitors are connected in series and then connected parallel to each other. Then the number of capacitors in each row is given by

1000/400 = 2.5

Therefore, the number of capacitors connected in series is three.

So the capacitance of each row is $\frac{1}{1+1+1}= \frac{1}{3}\mu F$

Let there be n parallel rows. Each of these rows will have 3 capacitors. Therefore, the equivalent capacitance of the circuit is given as

The required capacitance of the circuit is 2μF

Therefore, n/3 = 2

n = 6

Therefore, there are 6 rows of three capacitors in the circuit. A minimum of 6 x 3 = 18 capacitors are required.

Q2.24) What is the area of the plates of a 2 F parallel plate capacitor, given that the separation between the plates is 0.5 cm? [You will realise from your answer why ordinary capacitors are in the range of µF or less. However, electrolytic capacitors do have a much larger capacitance (0.1 F) because of the very minute separation between the conductors.]

Solution:

Capacitance of the parallel plate capacitor, C = ε₀A/d

Capacitance of the capacitor, C=2 F

Separation between the plates, d= 0.5 cm = 0.5 x 10^-2 m

ε₀ = permittivity of the free space = 8.85 x 10^-12 C²N^-1m^-2

Area of the plates , A = Cd/ε₀

A = [2 x 0.5 x 10^-2]/8.85 x 10^-12

= 1130 x 10⁶ m²

Q2.25) Obtain the equivalent capacitance of the network in Figure. For a 300 V supply, determine the charge and voltage across each capacitor.

Solution:

The above figure can be redrawn as below

The capacitors C₂ and C₃ are connected in series. The equivalent capacitance C’

C’ = 100 pF

The capacitors C’ and C₁ are parallel. The equivalent capacitance is C” = C’ + C₁

C” = 100 + 100 = 200 pF

C” and C₄ are connected in series. Let the equivalent capacitance be C

C = 200/3 pF

Hence the equivalent capacitance of the circuit is 200/3 pF

Total charge, Q = CV = $\frac{200}{3}\times 10^{-12}\times 300 = 2 \times 10^{-8}C$

Q = Q₄ = 2 x 10^-8 C

Potential difference across C_4, V₄ = Q/C₄

= (2 x 10 ^-8)/(100 x 10^-12) = 200 V

Potential difference across C”,  V” = 300 V – 200 V = 100 V

Potential difference across C_1,  V₁ = V” = 100 V

Charge across C₁, Q₁ = C₁V₁ = (100 x 10^-12) x 100 = 10^-8 C

The charge across C₂ and C₃, Q₂ = Q – Q₁ = 2 x 10^-8 – 10^-8 

= 10^-8 C

Potential across C_{2 ,} V₂ = Q₂/C₂ = 10^-8/ (200 x 10^-12) = 50 V

Potential across C₃, V₃ = Q₂/C₃ = 10^-8/(200 x 10^-12) = 50 V

Therefore,

Q₁ = 10^-8 C , V₁ = 100 V

Q₂ =  10^-8 C, V₂ = 50 V

Q₂ = Q₃ = 10^-8 C, V₃ = 50 V

Q₄ = 2 x 10^-8 C , V₄ = 200 V

Q2.26) The plates of a parallel plate capacitor have an area of 90 cm² each and are separated by 2.5 mm. The capacitor is charged by connecting it to a 400 V supply.
(a) How much electrostatic energy is stored by the capacitor?
(b) View this energy as stored in the electrostatic field between the plates, and obtain the energy per unit volume u. Hence arrive at a relation between u and the magnitude of electric field E between the plates

Solution:

Area of the plates of a parallel plate capacitor, A = 90 cm² = 90 x 10^-4 m²

Separation between the plates, d = 2.5 mm = 2.5 x 10^-3 m

Potential difference across the plates, V = 400 V

Capacitance of the capacitor, C = ε₀A/d

Electrostatic energy stored in the capacitor, E = (1/2) CV²

Therefore, the electrostatic energy stored by the capacitor is 2.55 x 10^-6 J

Volume of the capacitor, V = A x d

= 90 x 10^-4  x 25 x 10^-3

= 2.25 x 10^-4 m³

Energy stored in the capacitor per unit volume is

u = E/V = $\frac{\frac{1}{2}CV^{2}}{Ad} = \frac{\frac{\epsilon _{0}A}{2d}V^{2}}{Ad}= \frac{1}{2}\epsilon _{0}(\frac{V}{d})^{2}$

Here, V/d = Electric intensity = E

Therefore, $u = \frac{1}{2}\epsilon _{0}E^{2}$

Q2.27) A 4 µF capacitor is charged by a 200 V supply. It is then disconnected from the supply and is connected to another uncharged 2 µF capacitors. How much electrostatic energy of the first capacitor is lost in the form of heat and electromagnetic radiation?

Solution:

Capacitance of the capacitor, C₁= 4 μF

Supply voltage, V₁ = 200 V

Capacitance of the uncharged capacitor, C₂= 2 μF

Electrostatic energy stored in C₁ is given as

E₁ = (1/2)C₁V₁²

= (1/2) x 4 x 10^-6 x (200)²

= 8 x 10^-2 J

When C₁ is disconnected from the power supply and connected to C₂, the voltage acquired by it is V₂.

According to the law of conservation of energy, the initial charge on the capacitor C₁ is equal to the final charge on the capacitors C₁ and C₂.

V₂ (C₁ + C₂) = C₁V₁

V₂ (4+ 2) x 10^-6 =  4 x 10^-6 x 200

V₂ = (400/3) V

Electrostatic energy of the combination is

E₂ =  (1/2)(C₁+C₂) V₁²

= (1/2) x (2+4) x 10^-6 x (400/3)²

= 5.33 x 10^-2 J

Hence, amount of electrostatic energy lost by capacitor C₁ = E₁ – E₂

= 0.08 – 0.0533 = 0.0267

= 2.67 x 10^-2 J

Q2.28) Show that the force on each plate of a parallel plate capacitor has a magnitude equal to (½) QE, where Q is the charge on the capacitor, and E is the magnitude of the electric field between the plates. Explain the origin of the factor ½.

Solution:

Let F be the force required to separate the plates of the parallel plate capacitors.

Let Δx be the distance between the two plates.

Therefore, the work done to separate the plates, W = F Δx. As a result, the potential energy of the capacitor increases by an amount equal to uAΔx.

here, u = energy density

A = area of each plate

d = distance between the plates

V = potential difference across the plates

The work done will be equal to the increase in potential energy

FΔx = uAΔx

F = uA

Substituting $u = \frac{1}{2}\epsilon _{0}E^{2}$ in above equation

The electric intensity, E = V/d

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

Therefore, F = (1/2) (CV) E

Charge on the capacitor is given as Q = CV

Therefore, F = (1/2)QE

The electric field just outside the conductor is E and inside the conductor is zero. Hence, the average of the electric field E/2 is equal to the force.

Q2.29) A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (as shown in the figure). Show that the capacitance of a spherical capacitor is given by

$C = \frac{4\pi \epsilon _{0}r_{1}r_{2}}{r_{1}-r_{2}}$