Consider an assembly of three conducting concentric spherical shell of radii a, b and c as shown in figure Find the capacitance of the assembly between the points Aand B.
The spherical shells form two spherical capacitors: one made by A and B and the other made by B and C.
The capacitance of the spherical capacitor made by the shells of radii r1 and r2 is given by
`C = (4pi∈_0)/[[1/r_1 - 1/r^2]] = (4pi∈_0r_1r_2)/(r_2 - r_1)`
The capacitance of the capacitor made by A and B is given by
`C_(AB) = (4pi∈_0ab)/(b-a)`
The capacitance of the capacitor made by B and C is given by
`C_(BC) = (4pi∈_0bc)/(c-b)`
As the capacitors are in series, the net capacitance is given by
`1/C = 1/C_(AB) + 1/C_(BC)`
⇒ `C = (C_(AB)C_(BC))/(C_(AB)+C_(BC)) = (((4pi∈_0)^2 ab^2c)/((b-a)(c-b)))/((4pi∈_0ab)/((b-a))+(4pi∈_0bc)/((c-b))`
⇒ `C = ((4pi∈_0ab^2c)/((b-a)(c-b)))/(((ab(c-b)+bc(b-a))/((b-a)(c-b))))`
⇒ `C = (4pi∈_0ab^2c)/[[ab(c-b)+bc(b-a)]]`
⇒ `C = (4pi∈_0ab^2c)/(b^2(c-a)) = (4pi∈_0ac)/((c-a))`