A spherical capacitor consists of a hollow conducting sphere A of radius r surrounded by another concentric spherical shell B of radius r when a charge + Q is given to inner sphere A, it spreads uniformly over the outer surface of A.
A charge – Q is induced on the inner surface of B and + Q is induced on the inner surface of B and + Q is induced on the outer surface of B. As the outer surface of B is earthed, the induced charge + Q flows to earth – Q charge spreads over the inner surface of B.
Due to electrostatic shielding electric field within A is zero, E = 0 for r < r also, electric field outside B is zero E = 0, for r > r.
An electric filed E exists between the two spheres and is directed radically outwards.
Potential of inner sphere A is
VA = Q / 4π∈0r – Q / 4π ∈0 r
VA = Q / 4π ∈0 [1 / r – 1 / r] = Q / 4π∈0(rb – ra) / ra rb
As outer sphere B is earthed therefore, its potential VB – 0
Potential difference between the two spheres
A and B = VA – VB = Q / 4π ∈0 (rb – ra) / ra rb
As C = Q / V = Q / VA – VB
∴ C – Q 4 π∈a ra rh / Q (rb – ra)
C = 4π∈0 ra rb / (rb – ra)
The capacity of cylindrical air capacitor is given by
C = 2π∈0L / log (rb / ra)
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