11.2 Problems
475
is removed. Show that the total electrostatic energy stored in the spheres
decreases by an amount U , given by
U =
1
16πε 0 Q
(q 1 − q 2 )
2
What happens to this energy?
11.61 Two spherical conductors of radii R 1 and R 2 and charges Q 1 and Q 2 , respectively, are brought in contact and separated. Show that their charge densities
will be inversely proportional to their radii.
11.62 A light spherical balloon is made of conducting material. It is suggested that
it could be kept spherical simply by connecting it to a high-voltage source.
The balloon has a diameter of 100 mm.
(a) What is the voltage of the source if the electric field on the balloon surface is 5 × 10 6 V/m?
(b) What gas pressure inside the balloon would produce the same effect?
(c) The voltage source is removed and the balloon remains at the same voltage. Calculate the total electrostatic energy of the balloon.
11.63 A soap bubble of radius R 1 is given a charge q. Due to mutual repulsion
of the surface charges the radius is increased to R 2 , the pressure remaining
constant. Show that the charge is given by
q =
32
3
π
2
ε 0 p R 1 R 2
R 1
2
+ R 1 R 2 + R 2
2
1/2
11.64 An insulating spherical shell of inner radius r 1 and outer radius r 2 is charged
so that its volume charge density is given by
ρ(r ) = 0 for 0 ≤ r ≤ r 1
ρ(r ) =
A
r
for r 1 ≤ r ≤ r 2
ρ(r ) = 0 for r > r 2
where A is a constant and r is the radial distance from the centre of the shell.
Find the electric field due to the shell throughout all space.
11.65 (a) Show that electrostatic field is conservative.
(b) An isolated soap bubble of radius 1 cm is at a potential of 10 V (assume
the bubble material is a conductor); calculate the total charge on the
bubble. If it collapses to a drop of radius 1 mm (no charge loss or gain
during the process), what is the change of the electrostatic energy of the
system?
[University of Aberystwyth, Wales 2008]
475
is removed. Show that the total electrostatic energy stored in the spheres
decreases by an amount U , given by
U =
1
16πε 0 Q
(q 1 − q 2 )
2
What happens to this energy?
11.61 Two spherical conductors of radii R 1 and R 2 and charges Q 1 and Q 2 , respectively, are brought in contact and separated. Show that their charge densities
will be inversely proportional to their radii.
11.62 A light spherical balloon is made of conducting material. It is suggested that
it could be kept spherical simply by connecting it to a high-voltage source.
The balloon has a diameter of 100 mm.
(a) What is the voltage of the source if the electric field on the balloon surface is 5 × 10 6 V/m?
(b) What gas pressure inside the balloon would produce the same effect?
(c) The voltage source is removed and the balloon remains at the same voltage. Calculate the total electrostatic energy of the balloon.
11.63 A soap bubble of radius R 1 is given a charge q. Due to mutual repulsion
of the surface charges the radius is increased to R 2 , the pressure remaining
constant. Show that the charge is given by
q =
32
3
π
2
ε 0 p R 1 R 2
R 1
2
+ R 1 R 2 + R 2
2
1/2
11.64 An insulating spherical shell of inner radius r 1 and outer radius r 2 is charged
so that its volume charge density is given by
ρ(r ) = 0 for 0 ≤ r ≤ r 1
ρ(r ) =
A
r
for r 1 ≤ r ≤ r 2
ρ(r ) = 0 for r > r 2
where A is a constant and r is the radial distance from the centre of the shell.
Find the electric field due to the shell throughout all space.
11.65 (a) Show that electrostatic field is conservative.
(b) An isolated soap bubble of radius 1 cm is at a potential of 10 V (assume
the bubble material is a conductor); calculate the total charge on the
bubble. If it collapses to a drop of radius 1 mm (no charge loss or gain
during the process), what is the change of the electrostatic energy of the
system?
[University of Aberystwyth, Wales 2008]