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**Exercise : ** Solutions of Questions on Page Number : **87**

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Q1 :
**
**

Two charges 5 x
10^{-8} C
and -3
x
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.

**Answer :**

There are two charges,

Distance between the two charges, *d* = 16 cm = 0.16 m

Consider a point P on the line joining the two charges, as shown in the given figure.

*r* = Distance of point P from charge *q*_{1}

Let the electric potential (*V*) at point P be zero.

Potential at point P is the sum of potentials caused by charges
*q*_{1} and *q*_{2} respectively.

Where,

= Permittivity of free space

For *V* = 0, equation (i) reduces to

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

Suppose point P is outside the system of two charges at a
distance *s* from the negative charge, where potential is
zero, as shown in the following figure.

For this arrangement, potential is given by,

For *V* = 0, equation (ii) reduces to

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

Answer needs Correction? Click Here

Q2 :
**
**

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.

**Answer :**

The given figure shows six equal amount of charges, *q*, at
the vertices of a regular hexagon.

Where,

Charge, *q* = 5 µC = 5
×
10^{ - 6} C

Side of the hexagon, *l* = AB = BC = CD = DE = EF = FA = 10
cm

Distance of each vertex from centre O, *d* = 10 cm

Electric potential at point O,

Where,

= Permittivity of free space

Therefore, the potential at the centre of the hexagon is 2.7
× 10^{6} V.

Answer needs Correction? Click Here

Q3 :
**
**

Two charges 2 ÃŽÂ¼C 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?

**Answer :**

Q4 :
**
**

A spherical conductor of radius 12 cm has a charge of 1.6
x
10^{-7}C
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?

**Answer :**

Q5 :
**
**

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?

**Answer :**

Q6 :
**
**

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?

**Answer :**

Q7 :
**
**

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.

**Answer :**

Q8 :
**
**

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?

**Answer :**

Q9 :
**
**

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 :**

Q10 :
**
**

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

**Answer :**

Q11 :
**
**

A 600 pF capacitor is charged by a 200 V 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?

**Answer :**

Q12 :
**
**

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, 3 cm) to a point Q (0, 4 cm, 0), via a
point R (0, 6 cm, 9 cm).

**Answer :**

Q13 :
**
**

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.

**Answer :**

Q14 :
**
**

Two tiny spheres carrying charges 1.5 ÃŽÂ¼C and 2.5 ÃŽÂ¼C 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.

**Answer :**

Q15 :
**
**

A spherical conducting shell of inner radius *r*1 and outer
radius *r*2 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.

**Answer :**

Q16 :
**
**

**(a)** Show that the normal component of electrostatic field
has a discontinuity from one side of a charged surface to another
given by

Where is a unit vector normal to the surface at a point and ÃÆ’ is the surface charge density at that point. (The direction of is from side 1 to side 2.) Hence show that just outside a conductor, the electric field is ÃÆ’

**(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.]

**Answer :**

Q17 :
**
**

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

**Answer :**

Q18 :
**
**

In a hydrogen atom, the electron and proton are bound at a distance of about 0.53 Ãƒâ€¦:

**(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
Ãƒâ€¦ separation?

**Answer :**

Q19 :
**
**

If one of the two electrons of a H_{2} molecule is
removed, we get a hydrogen molecular ion. In the ground state of
an, the two protons are
separated by roughly 1.5 Ãƒâ€¦, and the
electron is roughly 1 Ãƒâ€¦ from each
proton. Determine the potential energy of the system. Specify
your choice of the zero of potential energy.

**Answer :**

Q20 :
**
**

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.

**Answer :**

Q21 :
**
**

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?

**(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?

**Answer :**

Q22 :
**
**

Figure 2.34 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).

**Answer :**

Q23 :
**
**

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.

**Answer :**

Q24 :
**
**

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 realize 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.]

**Answer :**

Q25 :
**
**

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

**Answer :**

Q26 :
**
**

The plates of a parallel plate capacitor have an area of 90
cm^{2} 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.

**Answer :**

Q27 :
**
**

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?

**Answer :**

Q28 :
**
**

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.

**Answer :**

Q29 :
**
**

A spherical capacitor consists of two concentric spherical conductors, held in position by suitable insulating supports (Fig. 2.36). Show

that the capacitance of a spherical capacitor is given by

where *r*_{1} and *r*_{2} are the radii
of outer and inner spheres, respectively.

**Answer :**

Q30 :
**
**

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.

**Answer :**

Q31 :
**
**

Answer carefully:

**(a)** Two large conducting spheres carrying charges
*Q*_{1} and *Q*_{2} are brought close
to each other. Is the magnitude of electrostatic force between
them exactly given by
*Q*_{1}*Q*_{2}/4π*r* ^{2},
where *r* is the distance between their centres?

**(b)** If
Coulomb's law
involved 1/*r*^{3} dependence (instead of
1/*r*^{2}), 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).

**Answer :**

Q32 :
**
**

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).

**Answer :**

Q33 :
**
**

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 10^{7}
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?

**Answer :**

Q34 :
**
**

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.

**Answer :**

Q35 :
**
**

In a Van de Graaff type generator a spherical metal shell is to
be a 15 x 10^{6} V
electrode. The dielectric strength of the gas surrounding the
electrode is 5 x 10^{7}
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.)

**Answer :**

Q36 :
**
**

A small sphere of radius *r*_{1} and charge
*q*_{1} is enclosed by a spherical shell of radius
*r*_{2} and charge *q*_{2}. Show that
if *q*_{1} 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 *q*_{2} on the shell
is.

**Answer :**

Q37 :
**
**

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.)

**Answer :**

Physics : CBSE ** NCERT ** Exercise Solutions for Class 12th for ** Electrostatic Potential And Capacitance ** will be available online in PDF book form soon. The solutions are absolutely Free. Soon you will be able to download the solutions.

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- Chapter 1 - Electric Charges And Fields Class 12
- Chapter 3 - Current Electricity Class 12
- Physics Part 2 : Chapter 1 - Ray Optics And Optical Instruments Class 12
- Chapter 4 - Moving Charges And Magnetism Class 12
- Chapter 6 - Electromagnetic Induction Class 12
- Chapter 5 - Magnetism And Matter Class 12
- Physics Part 2 : Chapter 2 - Wave Optics Class 12
- Physics Part 2 : Chapter 5 - Nuclei Class 12
- Physics Part 2 : Chapter 3 - Dual Nature Of Radiation And Matter Class 12