(a)    When the outer sphere is earthed: Consider a of two concentric spheres A and B of radii a and b metres respectively separated by a dielectric of relative metres respectively separated by a dielectric of relative permittivity ∈_r. outer sphere being earthed.

If a + ve charge of Q coulombs is given to the inner sphere, a charge of –Q coulombs will be induced on the inner surface of the outer sphere B and a charge of +Q coulombs induced on the outer surface of sphere B will got o earth.

Concentric Spherical Capacitor

The electric intensity at any point P in the dielectric, r metres from the centre, due to the charge on the inner sphere is  Q   newtons. The electric intensity at point P due to charge on the outer sphere is zero
                   4π∈_o∈_r
since point P is within the sphere.

P.D. between spheres A and B = Work done carrying a + ve charge of one coulomb from sphere A to B



Capacitance of the condenser, C = Q  =  Q  =  4π∈_o∈_r  ab  Farads
                                                   V     Q(b-a)             b-a
                                                       4π∈_o∈_{r ab}  

Capacitance of spherical condenser with air as medium,

 C = 4π ∈_o ab  =  1  x   ab  Farads.
                b-a   9x10⁹   b-a

(b)    When the inner sphere is earthed: Consider a spherical capacitor consisting of two concentric spheres of radii a and b metres separated by a dielectric of relative permittivity ∈r inner sphere being earthed.

If a positive charge of Q coulombs is given to the outer sphere B, it will distribute itself over both its inner and outer surfaces.

Let the charges of Q₁ and Q₂ coulombs be at the inner and outer surfaces respectively of sphere B where Q = Q₁ +Q₂,

The charge + Q₁ on the inner surface of outer sphere B will induce a charge of –Q₁ coulombs on the outer surface of inner sphere A and + Q₁ coulombs on the inner surface of sphere A, which will go to earth.


Now there are two capacitors connected in parallel.

(i) One capacitor consists outer surface of sphere B and earth having capacitance C₁ =  4π∈_o∈_r b farads and

(ii) Second capacitor consisting of inner surface of outer sphere B and the outer surface of inner sphere A having capacitance

                     C₂ = 4π∈_o∈_r  ab farads
                                         b-a

Total capacitance,  C = C₁ + 2 = 4π∈_o∈_r b+ 4π∈_o∈_r  ab = 4π∈_o∈_r  b2 farads.
                                                                              b-a                b-a

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