Three concentric metallic shells A , B and C of radii a, b and c `(a lt b lt c)` have surface densities `+ sigma ,- sigma` and `+sigma` respectively as shown in Fig.

Obtain the expressions for the potential of three shells A , B and C .

Explanation

We know that potential at a point either on the surface or inside a charged spherical shell having surface density `sigma` is `V = (sigma R)/(in_(0))` , where R is radius of shell . The potential at a point outside the charged shell at a distance (where `r gt R`) is V = `(sigma R^(2))/(in_(0) r)`
In present question `sigma_(A) = + sigma , sigma_(B) = - sigma` and `sigma_(c) = + sigma` , moreover `a lt b lt c`
`therefore` Electric potential at shell A ,
`V_(A) = (sigma A * a)/(in_(0)) + (sigma_(B) * b)/(in_(0)) + (sigma_(C) * c)/(in_(0)) = (sigma a)/(in_(0)) - (sigma b)/(in_(0)) + (sigma_(c) )/(in_(0)) = (sigma)/(in_0) [a -b + c] " " ..... (i)`
Electric potential at shell B ,
`V_(B) = (sigma_(A) * a^(2))/(in_(0) b) + (sigma _(B) * b)/(in_(0)) + (sigma_(c) * c)/(in_(0)) = (sigma * a^(2))/(in_(0) b) - (sigma *b)/(in_(0)) + (sigma * c)/(in_(0)) = (sigma)/(in_(0)) [ (a^(2))/(b) - b + c] " " ..... (ii) `
and electric potential at shell C .
`V_(C) = (sigma_(A) * a^(2))/(in_(0) c) + (sigma_(B) * b^(2))/(in_(0) c) + (sigma_(c) * c)/(in_0)) = (sigma a^(2))/(in_(0) c) - (sigma b^(2))/(in_(0) c) + (sigma_(c))/(in_(0)) = (sigma)/(in_(0)) [ (a^(2) - b^(2))/(c) + c] " " ... (iii)`