Three concentric metallic shells A, B and C of radii a, b and `c(a lt b lt c)` have surface charge densities `+sigma, -sigma and +sigma` respectively as shown. Obtain the expressions for the potential of three shells A, B and C. If shells A and C are at the same potential, obtain the relation between a, b and c.

Explanation

`V_(A)=(kQ_(A))/(a)+(kQ_(B))/(b)+(kQ_(C))/(c )`
`V_(A)=(1)/(4pi epsilon_(0))[(4pi a^(2)sigma)/(a)+(4pi b^(2)(-sigma))/(b)+(4pi c^(2)sigma)/(c )]`
`=(sigma)/(epsilon_(0))[a-b+c]`
`V_(B)=(kQ_(A))/(b)+(kQ_(B))/(b)+(kQ_(C))/(c )`
`V_(B)=(sigma)/(epsilon_(0)) [(a^(2)-b^(2))/(b)+c]`
`V_(C )=(kQ_(A))/(c )+(kQ_(B))/(c )+(kQ_(C ))/(c )=(sigma)/(epsilon_(0))[ (a^(2)-b^(2)+c^(2))/(c )]`
`because V_(A)=V_(C )`
`a-b+c=(a^(2)-b^(2))/(c )+c`
`c=a+b`