Three concentric spherical metallic shells A, B and C of radii a, b and c (a lt b ltc) have surface charge densities `sigma`, `-sigma` and `sigma` respectively.
`(i) Find the potential of the three shells A, B and C.
(ii) If the shells A and C are at the same potential, obtain the relation between the radii a, b and c.

Explanation

`V _ A = k[ (q _ 1 )/(a) + (q_2)/(b) + (q_3)/(c )] `
` = k 4pi a sigma - k 4pi b sigma + k4 pi c sigma `
` = 4 pi a sigma ( a - b + c ) `
` = (sigma )/(epsi _0) (a - b + c ) ` ` V_B = k [ (q_1)/(a) + (q_2)/(b) + (q_3)/(c ) ] = k[ (4pi a ^2 sigma ) /(b) - 4pi k b sigma + 4pi k c sigma ] `
` = (sigma )/( epsi _ 0) ((a ^2)/(b) - b ^ 2 + c ^2 ) `
` V_C= (sigma )/(epsi _0 c ) (a ^ 2 - b ^ 2 + c ^ 2 ) `
When ` V_A= V _ C `
` (sigma )/(epsi _0) (a - b + c ) = (sigma ) /(epsi _0C) ( a ^ 2 - b ^ 2 + c ^2 ) `
` ac - bc + c ^2 = a ^ 2 - b ^ 2 + c ^ 2 `
` c ( a - b ) = (a - b ) ( a + b ) `
` c = a + b `