# NCERT Solutions for class 12th Physics Chapter 2 Electrostatic Potential and Capacitance

### Exercise

Question 1. Two charges 5 x 10-8c and-3 x 10-8c are located 16 cm apart. At what point (s) on the line joining the two charges is the electric potential zero? Take the potential at infinity to be zero.

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

Question 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.

Question 3. Two charges 2 µC and -2 µC are placed at points A and B 6 cm apart.
(a) Identify an equipotential surface of the system.
(b) What is the direction of the electric field at every point on this surface?

Question 4. A spherical conductor of radius 12cm has a charge of 1.6 x 10-7c
distributed uniformly on its surface. What is the electric field
(a) inside the sphere
(b) just outside the sphere
(c) at a point 18 cm from the centre of the sphere?

Note: Electric field inside a conductor is zero.

Question 5. A parallel plate capacitor with air between the plates has a capacitance of 8 pF (1 pF = 10-12 F). What will be the capacitance ifthe distance between the plates is reduced by half, and the space between them is filled with a substance of the dielectric constant 6?

Question 6. Three capacitors each of capacitance 9 pF are connected in series.
(a) What is the total capacitance of the combination?
(b) What is the potential difference across each capacitor if the combination is connected to a 120V supply?

Question 7. Three capacitors of capacitances 2 pF, 3 pF and 4 pF are connected in parallel.
(a) What is the total capacitance of the combination?
(b) Determine the charge on each capacitance if the combination is connected to a 100 V supply.

Question 8. In a parallel plate capacitor with air between the plates, each plate has an
area of 6 x 10-3 m2 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?

Question 9. Explain what would happen if in the capacitor given in question 2.8, a 3 mm thick mica sheet (of dielectric constant= 6) were inserted between the plates,
(a) while the voltage supply remained connected and,
(b) after the supply was disconnected.

Question 10. A 12pF capacitor is connected to a S0V battery. How much electrostatic energy is stored in the capacitor?

Question 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?

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

Question 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.

Question 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.

Note: Electric field is vector quantity and potential is scaler quantity. The, resultant field
is sum of fields due to all the charges.

Question 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.

Sol.(a) Surface charge density on the inner and outer shell. Taking a Gaussian surface of radius r > r I but r < r2. Since the Gaussian surface is inside the conductor, therefore electric field is zero everywhere.

The conducting shell has no net charge, yet its inner shell has – q surface charge. Because the net charge on the shell is zero and no charge can be internal to the conductor, there must be + q charge on the outer surface of the conductor, other than + Q.

(b) By Gauss’s law, the net charge on the inner surface enclosing the cavity (not having any charge) must be zero. For a cavity of arbitrary shape, this is not enough to claim that the electric field inside must be zero. The cavity may have positive and negative charges with total charge zero. To dispose of this possibility, take a closed loop, part of which is inside the cavity along a field line and the rest 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.

Question 16. (a) Show that the normal component of electrostatic field has a discontinuity from one side of a charged surface to another given by (E2 – E1).n = where n is a unit vector normal to the surface So at a point and CT is the surface charge density at that point. (The direction of n is from side 1 to side 2.) Hence show that just outside a conductor, the electric field is CT n / s0.
(b) Show that the tangential component of 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.

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

Sol. The charge + q spreads uniformly on the outer surface of A and – q
uniformly spreads on the inner surface of B. An electric field E is produced between the two shells which will be directed radially outwards as shown in Fig.

Question 18. In a hydrogen atom, the electron and proton are bound at a distance of about 0.53A.
(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 A separation?

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

Question 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 charge density on the sharp and pointed ends of a conductor is higher than that of its flatter portions.

Question 21. Two charges -q and +q are located at points (0, 0, -a) and (0, 0, a), respectively.
(a) Why is the electrostatic potential at the points (0, 0, z) and (x, y, 0)?
(b) Obtain the dependence of potential on the distance r ofa 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?

Question 22. Figure 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),

Question 23. An electrical technician requires a capacitance of 2 µFin a circuit across a potential difference of 1 kV. A large number of 1 µF capacitors are available to him each of which can withstand a potential difference ofnot more than 400 V. Suggest a possible arrangement that requires the minimum number of capacitors

Sol. Since each capacitor can withstand a potential difference of 400 V, at least three must be used in series to share 1 kV potential difference, 3 in series
will have capacitance= (1/3) µ F.
To have 2 µ F combination, we must connect six such series combinations, in parallel.
Total capacitors required = 3 in series x 6 in parallel = 18

Note: Refer Chapter at a Glance (19)

Question 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 µFor less. However, electrolytic capacitors do have a much larger capacitance (0.1 F) because of very minute separation between the conductors.]

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

Question 26. The plates of a parallel plate capacitor have an area of 90 cm2 each and area 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.

Question 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 capacitor. How much electrostatic energy of the first capacitor is lost in the form of heat and electromagnetic radiation?

Question 28. Show that the force on each plate of parallel plate capacitor has a magnitude equal to (1/2) QE, where Q is the charge on the capacitor, and E is the magnitude of electric field between the plates. Explain the origin of the factor 1/2.

Question 29. A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports Fig.

Question 30. A spherical capacitor has an inner sphere of radius 12 cm and an outer sphere of radius 13 cm. The outer sphere is earthed and the inner sphere
is given a charge of 2.5 µC . The space between the concentric spheres is filled with a liquid of dielectric constant 32.

(a) Determine the capacitance of the capacitor.
(b) What is the potential of the inner sphere?
(c) Compare the capacitance of this capacitor with that of an isolated sphere of radius 12 cm. Explain why the latter is much smaller

• (a) Two large conducting spheres carrying charges Q1 and Q2 are brought close to each other. Is the magnitude of electrostatic force between them exactly given by Q1Q2 I 4rci;0r2 , where r is the distance between their centres?
• (b) If Coulomb’s law involved 1/r1 dependence (instead of 1/r2), would Gauss’s law be still true ?
• (c) A small test charge is released at rest at a point in an electrostatic field configuration. Will it travel along the field line passing through that point?
• (d) What is the work done by the field of a nucleus in a complete circular orbit of the electron? What if the orbit is elliptical?
• (e) We know that electric field is discontinuous across the surface of a charged conductor. Is electric potential also discontinuous there?
• (f) What meaning would you give to the capacitance of a single conductor?
• (g) Guess a possible reason why water has much greater diaelectric constant(= 80) than say, mica(= 6)

Sol.

(a) No, because charge distributions on the spheres will not be uniform.
(b) No.
(c) Not necessarily. (True only if the field line is a straight line.) The field line gives the direction of acceleration, not that of velocity, in general.
(d) Zero, no matter what the shape of the complete orbit is.
(e) No, potential is continuous.
(f) A single conductor is a capacitor with one of the ‘plates’ at infinity.
(g) A water molecule has permanent dipole moment. However, detailed explanation of the value of dielectric constant requires microscopic theory and is beyond the scope of the book.

Question 32. A cylindrical capacitor has two co-axial cylinders of length 15 cm and radii 1.5 cm and 1.4 cm. The outer cylinder is earthed and the inner cylinder is given a charge of 3.5 µC. Determine the capacitance of the system and the potential of the inner cylinder. Neglect end effects (i.e., bending of field lines at the ends).

Question 33. A parallel plate capacitor is to be designed with a voltage rating 1 kV, using a material of dielectric constant 3 and dielectric strength about 107 vm-1. (Dielectric strength is the maximum electric field a material can tolerate without breakdown, i.e., without starting to conduct electricity through partial ionisation.) For safety, we should like the field never to exceed, say 10% of the dielectric strength. What minimum area of the plates is required to have a capacitance of 50 pF?

Question 34. Describe schematically the equipotential surfaces corresponding to
(a) a constant electric field in the Z-direction,
(b) a field that uniformly increases in magnitude but remains in a constant (say, Z) direction,
(c) a single positive charge at the origin, and
(d) a uniform grid consisting of long equally spaced parallel charged wires in a plane.

Note: Electric field is normal to the equipotential surface.

Question 35. In a Van De Graaff generator, a spherical metal shell is to be a 15 x 106 V electrode. The dielectric strength of the gas surrounding the electrode is 5 x 10-7 V/m. What is the minimum radius of the spherical shell required? (You will learn from this exercise why one cannot build an electrostatic generator using a very small shell which requires a small charge to acquire a high potential.)

Question 36. A small sphere of radius r1 and charge q1 is enclosed by a spherical shell of radius r2 and charge qr Show that if q1 is positive, charge will necessarily flow from the sphere to the shell (when the two are connected by a wire) no matter what the charge q2 on the shell is.

(a) The top of the atmosphere is at about 400 kV with respect to the surface of the earth, corresponding to an electric field that decreases with altitude. Near the surface of the earth, the field is about 100 vm-1. Why then do we not get an electric shock as we step out of our house into the open? (Assume the house to be a steel cage so
there is no field inside!)

(b) A man fixes outside his house one evening a two metre high insulating slab carrying on its top a large aluminium sheet of area 1 m2. Will he get an electric shock if he touches the metal sheet next morning?

(c) The discharging current in the atmosphere due to the small conductivity of air is known to be 1800 A on an average over the globe. Why then does the atmosphere not discharge itself completely in due course and become electrically neutral? In other words, what keeps the atmosphere charged?

(d) What are the forms of energy into which the electrical energy of
the atmosphere is dissipated during a lightning? (Hint: The earth
has an electric field of about 100 Vm-1 at its surface in the downward direction, corresponding to a surface charge density= -1 o-9 C m- 2• Due to the slight conductivity of the atmosphere up to about 50 km (beyond which it is good conductor), about+ 1800 C is pumped every second into the earth as a whole. The earth, however, does not get discharged since thunderstorms and lightning occurring continually all over the globe pump an equal amount of negative charge on the earth.)

Sol. (a) Our body and the ground form an equipotential surface. As we step out into the open, the original equipotential surfaces of open, the original equipotential surfaces of open air change, keeping our head and the ground at the same potential.

(b) Yes. The steady discharging current in the atmosphere charges up the aluminium sheet gradually and raises its voltage to an extent depending on the capacitance of the capacitor (formed by the sheet, slab and the ground).

(c) The atmosphere is continually being charged by thunderstorms and lightning all over the globe and discharged through regions of ordinary weather. The two opposing currents are, on an average, in equilibrium.

(d) Light energy involving in lightning; heat and sound energy in the accompanying thunder.

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