Question

A concentric spherical capacitor is charged to -Q on the outer sphere and +Q on the inner sphere. The inner sphere is solid, with radius=a, while the outer sphere is hollow, with radius=b. Between the two spheres is a vaccum.

a) Find the potential difference between r=a and r=b and thus find the capacitance of this system directly.

b) Assuming the outer shell is grounded (ie, at zero potential) use to find the capacirtance.

c) Using the energy density in the electric field find the capacitance for this pair of conductors.

Answer 1

PART A)

Potential difference is given by

$V = KQ(\frac{1}{a}-\frac{1}{b})$

now the capacitance is given by

$C = \frac{Q}{V}$

$C = \frac{Q}{KQ(\frac{1}{a}-\frac{1}{b})}$

$C = \frac{4\pi \epsilon_0}{(\frac{1}{a}-\frac{1}{b})}$

PART B)

Energy is given by

$U = \frac{1}{2}CV^2$

$C = \frac{2U}{V^2}$

PART C)

$U = \frac{1}{2}CV^2$

$U = \frac{1}{2}*\frac{\epsilon_0 A}{d}*{E^2.d^2}$

$U = \frac{1}{2}*{\epsilon_0 E^2 A.d}$

$U = \frac{1}{2}*{\epsilon_0 E^2 }*Volume$

so energy density is

$U_v = \frac{1}{2}*{\epsilon_0 E^2 }$