Spherical Capacitor

A spherical capacitor is a kind of capacitor which have one or more thin hollow spherical plate/s conductors as shown on the figure below:

A spherical capacitors can be of various types namely Isolated Spherical Capacitor , Concentric Spherical Capacitors with two spheres etc. The capacitance of the spherical capacitors can be measured or calculated as following:

Isolated Spherical Capacitor:

Consider a perfectly insulated spherical conductor with a radius of ‘r’ meters. Let the relative permittivity of the dielectric of the medium where it is placed be \epsilon_r, the spherical capacitor thus formed is known as isolated spherical capacitor.
isolated spherical capacitor

Now if a charge of Q Coulomb is supplied to the outer conductor shell of the capacitor then, we know the voltage or potential built at the surface of the capacitor is given by :
V = \dfrac{Q}{4 \pi \epsilon_0 \epsilon_r r}

We also know that capacitance C = \dfrac{Q}{V}

Now, comparing the two equations above gives us:
C = \dfrac{Q}{\frac{Q}{4 \pi \epsilon_0 \epsilon_r r}} = 4 \pi \epsilon _0 \epsilon _r r=\dfrac{\epsilon_r \times r}{9 \times 10^9} farads.

Thus for an isolated spherical capacitor,
C =\dfrac{\epsilon_r \times r}{9 \times 10^9} farads

and if the dielectric used is air then the relative permittivity is one so, C =\dfrac{r}{9 \times 10^9} farads.

Note: At first glimpse it seems bizarre that a single surface spherical conductor can act as a capacitor, question arises that; where is the second terminal of the capacitor? or the capacitance of the isolated spherical capacitor is with respect to which point or potential? the answer is that the surface potential V is with respect to the earth or ground of the circuit supplying Q coulombs of charges, so the capacitance is also with respect to the ground and the second terminal of the capacitor is the ground.

Concentric Spherical Capacitor:

When two thin spherical plates are placed concentric each with r1 and r2 distance from the common center as shown in the figure below then a concentric spherical capacitor is formed:
concentric spherical capacitor

There are two ways we can use a concentric spherical capacitor, first by grounding or earthing the outer surface and supplying charge to the inner surface and second by earthing the inner surface and supplying charge to the outer surface, we can calculate the capacitance of the spherical capacitor in each case as given below:

When outer surface is earthed:
If we earth or ground the outer sphere and supply a positive charge of Q coulombs to the inner sphere then a charge of -Q will be induced at the inner surface of outer sphere and a charge of +Q induced on the outer surface of the outer sphere will be sent to the ground.

Now, the surface potential of the inner sphere due to it’s own charge is:
\dfrac{Q}{4 \pi \epsilon _0 \epsilon _r r_1} V

and the potential of the inner sphere due to the -Q charge on the outer sphere is:
\dfrac{-Q}{4 \pi \epsilon _0 \epsilon _r r_2} V

And we know the surface potential built at the outer sphere due to the charge is zero because the charge lies all within the outer sphere.

Thus, The potential difference between the two spheres is:

V = \dfrac{Q}{4 \pi \epsilon _0 \epsilon _r r_1}-\dfrac{Q}{4 \pi \epsilon _0 \epsilon _r r_2}V

Or,V = \dfrac{Q}{4 \pi \epsilon _0 \epsilon _r }\left( \dfrac{1}{r_1}-\dfrac{1}{r_2} \right)V

Or,V = \dfrac{Q}{4 \pi \epsilon _0 \epsilon _r }\left( \dfrac{r_2 - r_1}{r_1 r_2} \right)V

Or, \dfrac{Q}{V}=\dfrac{4\pi \epsilon_0 \epsilon_r r_1 r_2}{r_2 - r_1}

Thus the capacitance C is:
C = 4 \pi \epsilon_0 \epsilon_r \frac{r_1 r_2}{r_2 - r_1} farad.

When inner surface is earthed:

If we now connect the inner sphere of the capacitor to the ground and supply a positive Q charges through the outer surface of the outer sphere, then a charge of -Q will be induced at the inner sphere. Now there will be two capacitance effects as discussed below:
i. The outer surface of the outer sphere will build a capacitance C1 with the ground where C_1 = 4 \pi \epsilon_0 \epsilon_r r_2 farads.

ii. The outer surface of the inner sphere will build a capacitance C2 with the inner surface of the outer sphere, where C_2 = 4 \pi \epsilon_0 \epsilon_r \frac{r_1 r_2}{r_2 - r_1}

Thus the total capacitance of the capacitor will be:

C=C_1 + C_2

Or, C = 4 \pi \epsilon_0 \epsilon_r r_2 +4 \pi \epsilon_0 \epsilon_r \frac{r_1 r_2}{r_2 - r_1}

Thus, C = 4 \pi \epsilon_0 \epsilon_r \frac{r_2^2}{r_2 - r_1}