Capacitance of a Spherical Capacitor (outer sphere earthed)

Let A and B be two concentric metal spheres of radii a and b respectively with air as the intervening sphere. The induced charge on the inner surface of the outer sphere is – q. P is a point at a distance r from the common centre O.



Electric field at P = E = 1/4πε₀ ( q / r²) r
where r is the unit vector along OP.

The potential difference between the spheres A and B is given by

    V = - ∫a E. d l

Here, dl is the difference vector displacement along a path from B to A.

But     E. dl = E dl cos 180 = - E dl.

Further, in moving a distance dl in the direction of motion, we are moving in the direction of r decreasing, so that dl = - dr. Hence,

    E. dl = Edr.

Eq. (2) becomes V = - ∫^a  E dr.

Putting the value of E from Eq. (1), we get

    V = - ___q___     ∫a  dr = - ___q___ { - 1 }a
        ____4πε₀       __  r²      __4πε₀     {    r }b

    = ___q___ {1 – 1} = ___q___ b - a
           4πε₀      a    b         4πε₀      ab

. : Capacitance of the spherical capacitor

    C = q = ________q_________    = 4πε₀ _____ab_____ ______.
           V         (___q___) (b – a)           _________( b – a)
                  ____4πε₀     __ ab

C = 4πε₀ ____ab____ = ____4πε₀___
     ________ b – a  _____   (1 – 1)
            ________________ (a    b)

When    b     ∞, C = 4πε₀a.
This is a capacitance of an isolated conducting sphere of radius a.

For more help in Capacitance of a Spherical Capacitor (outer sphere earthed) please click the button below to submit your homework assignment.