Question

A spherical capacitor is composed of two thin, concentric conducting shells of radii R1 = 4.0cm and R2 = 8.0cm. The plates are connected to a 12.0 V battery and are fully charged.
a. Derive the equation for the capacitance and use it to determine the capacitance.
b. Determine the total charge on the capacitor.
c. The space between the plates is now filled with neoprene, increasing the total charge to 7.1E-10 C. What is the dielectric constant of neoprene?
d. Determine the change in the potential energy of the capacitor.

Answer 1

(a) For a spherical capacitor with inner radius a and outer radius b, we can write the potential difference created as,

Now, for Q charge inside, we can write the Electric Field (for a<r<b) as,

On substituting we get,

Therefore we can write the capacitance as,

For the above spherical capacitor, a = 4 x 10^(-2) m and b = 8 x 10^(-2) m, Therefore, the capacitance is,

(b) The total charge on the capacitor when it is full charged (V = 12), is

(c) When a dielectric of dielectric constant k is introduced into capacitor, the capacitance becomes kC.

Now for this case, the capacitance will become,

Now, given,

So substituting in basic equation of capacitor we get,

(d) The potential energy of capacitor is given by,

On introducing the dielectric, the potential energy becomes,

(The potential 12 V is constant)

Therefore the change in potential energy is,