NCERT Solutions Class 12 Physics Chapter 2 Electrostatic Potential And Capacitance Download In Pdf

Chapter 2 Electrostatic Potential and Capacitance Download in pdf

**Question 2.1** Two charges 5 × 10–8 C and –3 × 10–8 C 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.

**Question 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.

**Question 2.3 **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 2.4** A spherical conductor of radius 12 cm has a charge of 1.6 × 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?

**Question 2.5** A parallel plate capacitor with air between the plates has a
capacitance of 8 pF (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?

**Question 2.6** 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 120 V supply?

**Question 2.7** 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 capacitor if the combination is
connected to a 100 V supply.

**Question 2.8** In a parallel plate capacitor with air between the plates, each plate
has an area of 6 × 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 2.9 **Explain what would happen if in the capacitor given in Exercise2.8, 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.

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

**Question 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?

ADDITIONAL EXERCISES QUESTIONS

**Question 2.12** 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).

**Question 2.13** 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 2.14** 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.

**Question 2.15 **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.

**Question 2.16** (a) Show that the normal component of electrostatic field has a
discontinuity from one side of a charged surface to another
given by
2 1
0
( ) ˆ
σ
ε
E − E n =
where ˆn is a unit vector normal to the surface at a point and
σ is the surface charge density at that point. (The direction of
ˆn is from side 1 to side 2.) Hence show that just outside a
conductor, the electric field is σ ˆn /ε0.

(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 2.17** 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?

**Question 2.18** 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?

**Question 2.19** If one of the two electrons of a 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.

**Question 2.20** 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 on its flatter portions.

**Question 2.21** 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?

**Question 2.22 **Figure2.34 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 2.23 **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 capacitors
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.

**Question 2.24** 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 very minute separation between the
conductors.]

**Question 2.25 **Obtain the equivalent capacitance of the network in Fig. 2.35. For a
300 V supply, determine the charge and voltage across each capacitor.

**Question 2.26** 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.

**Question 2.27** 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 2.28 **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 electric field between the plates. Explain
the origin of the factor ½.

**Question 2.29 **A spherical capacitor consists of two concentric spherical conductors,
held in position by suitable insulating supports (Fig2.36). Show that the capacitance of a spherical capacitor is given by
0 1 2
1 2
4
–
r r
C
r r
πε
=
where r1 and r2 are the radii of outer and inner spheres,
respectively.

**Question 2.30** 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.s the magnitude of electrostatic
force between them exactly given by Q1 Q2/4πε0r 2, where r is
the distance between their centres?

(b) If Coulomb’s law involved 1/r3 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 a much greater
dielectric constant (= 80) than say, mica (= 6).

**Question 2.32** 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 2.33 **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 2.34** 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
.

**Question 2.35** In a Van de Graaff type generator a spherical metal shell is to be a
15 × 106 V electrode. The dielectric strength of the gas surrounding
the electrode is 5 × 107 Vm–1. 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 2.36 **A small sphere of radius r1 and charge q1 is enclosed by a spherical
shell of radius r2 and charge q2. 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.

**Question 2.37 **Answer the following:

(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 1m 2.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 = –10–9 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.)

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- Chapter 1 Electric Charges and Fields
- Chapter 2 Electrostatic Potential and Capacitance
- Chapter 3 Current Electricity
- Chapter 4 Moving Charges and Magnetism
- Chapter 5 Magnetism and Matter
- Chapter 6 Electromagnetic Induction
- Chapter 7 Alternating Current
- Chapter 8 Electromagnetic Waves
- Chapter 9 Ray Optics and Optical Instruments
- Chapter 10 Wave Optics
- Chapter 11 Dual Nature of Radiation and Matter
- Chapter 12 Atoms
- Chapter 13 Nuclei
- Chapter 14 Semiconductor Electronics: Materials, Devices and Simple Circuits
- Chapter 15 Communication Systems

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