478
11 Electrostatics
change (i) V , (ii) E, (iii) q, (iv) C and (v) U , when (a) the battery remains
connected and (b) the battery is disconnected?
11.79 In prob. (11.78) if the plate separation is increased, how would the following
quantities change (i) V , (ii) E, (iii) C, (iv) q and (v) U when (a) the battery
remains connected and (b) the battery is disconnected?
11.80 Show that the force of attraction between the plates of a parallel plate capacitor is given by F =
1
2
ε 0 AV 2
d
, where A is the area, d the distance of separation, V the voltage to which the plates are charged and ε 0 the permittivity.
11.81 Let n identical droplets, each of radius r and charge q, coalesce to form a
large drop of radius R and charge Q. Assuming that the droplets are incompressible, show that (a) the radius R = n 1/3 r ; (b) the capacitance C of the
large drop is C = n 1/3 C, where C is the capacitance of the droplet; (c)
the potential V of the large drop is given by V = n 2/3 V , where V is the
potential of the droplet; (d) the surface charge density σ = n 1/3 σ ; (e) the
energy U stored in the large drop is given by U = n 5/3 U where U is the
energy stored in the droplet.
11.82 A cylindrical capacitor has radii a and b. Show that half of the stored electrical potential energy lies within a cylinder whose radius is
√
ab.
11.83 A capacitor of capacitance C 1 = 3.0 μF withstands the maximum voltage
V 1 = 4.0 kV, while a capacitor of capacitance C 2 = 6.0 μF the maximum
voltage V 2 = 3.0 kV. If they are connected in series what maximum voltage
can the system withstand?
11.84 A Geiger–Muller tube consists of a thin uniform wire of radius ‘a’ of length
L surrounded by a concentric hollow metal cylinder of radius b with a gas
of dielectric constant K between them. Apply Gauss’ law to calculate the
capacitance of the tube.
11.85 Two spherical metallic shells of radii a and b (b > a) constitute a capacitor
with the outer shell grounded and contact is made with the inner one through
a hole in the outer one. Show that the capacitance is given by C =
4πε 0 ab
b − a
.
11.86 Show that for two concentric shells of radii a and b (b ≈ a), the capacitance
reduces to that of a parallel plate capacitor
11.87 In an R–C circuit the emf supplied by the battery is 120 V, R = 1 × 10 6
and C = 10 μF. The switch S is closed at t = 0. Find
(i) the time taken for the charge to reach 90% of its final value;
(ii) the energy stored in the capacitor at one time constant;
(iii) the Joule heating in the resistor at one time constant.
11.88 After how many time constants will the energy stored in the capacitor in
Fig. 11.13 reach one-half of its equilibrium value?
11 Electrostatics
change (i) V , (ii) E, (iii) q, (iv) C and (v) U , when (a) the battery remains
connected and (b) the battery is disconnected?
11.79 In prob. (11.78) if the plate separation is increased, how would the following
quantities change (i) V , (ii) E, (iii) C, (iv) q and (v) U when (a) the battery
remains connected and (b) the battery is disconnected?
11.80 Show that the force of attraction between the plates of a parallel plate capacitor is given by F =
1
2
ε 0 AV 2
d
, where A is the area, d the distance of separation, V the voltage to which the plates are charged and ε 0 the permittivity.
11.81 Let n identical droplets, each of radius r and charge q, coalesce to form a
large drop of radius R and charge Q. Assuming that the droplets are incompressible, show that (a) the radius R = n 1/3 r ; (b) the capacitance C of the
large drop is C = n 1/3 C, where C is the capacitance of the droplet; (c)
the potential V of the large drop is given by V = n 2/3 V , where V is the
potential of the droplet; (d) the surface charge density σ = n 1/3 σ ; (e) the
energy U stored in the large drop is given by U = n 5/3 U where U is the
energy stored in the droplet.
11.82 A cylindrical capacitor has radii a and b. Show that half of the stored electrical potential energy lies within a cylinder whose radius is
√
ab.
11.83 A capacitor of capacitance C 1 = 3.0 μF withstands the maximum voltage
V 1 = 4.0 kV, while a capacitor of capacitance C 2 = 6.0 μF the maximum
voltage V 2 = 3.0 kV. If they are connected in series what maximum voltage
can the system withstand?
11.84 A Geiger–Muller tube consists of a thin uniform wire of radius ‘a’ of length
L surrounded by a concentric hollow metal cylinder of radius b with a gas
of dielectric constant K between them. Apply Gauss’ law to calculate the
capacitance of the tube.
11.85 Two spherical metallic shells of radii a and b (b > a) constitute a capacitor
with the outer shell grounded and contact is made with the inner one through
a hole in the outer one. Show that the capacitance is given by C =
4πε 0 ab
b − a
.
11.86 Show that for two concentric shells of radii a and b (b ≈ a), the capacitance
reduces to that of a parallel plate capacitor
11.87 In an R–C circuit the emf supplied by the battery is 120 V, R = 1 × 10 6
and C = 10 μF. The switch S is closed at t = 0. Find
(i) the time taken for the charge to reach 90% of its final value;
(ii) the energy stored in the capacitor at one time constant;
(iii) the Joule heating in the resistor at one time constant.
11.88 After how many time constants will the energy stored in the capacitor in
Fig. 11.13 reach one-half of its equilibrium value?