EXERCISES

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.


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.


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?

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?


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?


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?

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.

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? A cavity inside a conductor is shielded from outside electrical influences.
It is worth noting that electrostatic shielding does not work the other
way round; that is, if you put charges inside the cavity, the exterior of
the conductor is not shielded from the fields by the inside charges.

    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.


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


    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

    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).


    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.


    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.


    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.


    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


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

    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?

    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?


    2.19 If one of the two electrons of a H2 molecule is removed, we get ahydrogen 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.


    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.


    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?

    2.22 Figure 2.32 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).

    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.


    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.]


    2.25 Obtain the equivalent capacitance of the network in Fig. 2.33. For a
    300 V supply, determine the charge and voltage across each
    capacitor.

    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.

    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?


    2.28 Show that the force on each plate of a 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.

    2.29 A spherical capacitor consists of two concentric spherical conductors,
    held in position by suitable insulating supports (Fig. 2.34). Show



    are the radii of outer and inner spheres,

    respectively.


    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.

    2.31 Answer carefully:
    (a) Two large conducting spheres carrying charges Q1 and Q2arebrought close to each other. Is the magnitude of electrostatic force between them exactly given by Q1 Q2/4πε0r2, 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).

    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).


    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?


    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.


    2.35 A small sphere of radius r1and charge q1is enclosed by a sphericalshell of radius r2and charge q2. Show that if q1is positive, chargewill necessarily flow from the sphere to the shell (when the two areconnected by a wire) no matter what the charge q2

    on the shell is.

    2.36 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 1m2
    . 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.)