Electrostatics - Spherical Capacitor

Spherical Capacitor

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 

V_A = Q / 4π∈₀r – Q / 4π ∈₀ r 

V_A = Q / 4π ∈₀ [1 / r – 1 / r] = Q / 4π∈₀(r_b – r_a) / r_a r_b

As outer sphere B is earthed therefore, its potential V_B – 0

Potential difference between the two spheres 

A and B = V_A – V_B = Q / 4π ∈₀ (r_b – r_a) / r_a r_b

As C = Q / V = Q / V_A – V_B

∴ C – Q 4 π∈a r_a r_h / Q (r_b – r_a)

C = 4π∈₀ r_a r_b / (r_b – r_a)

The capacity of cylindrical air capacitor is given by 

C = 2π∈₀L / log (r_b / r_a)

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