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Three concentric spherical metallic shel...

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.

Text Solution

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`(a) V_(A) = K(sigma4pia^(2))/(a)+K((-sigma)4pib^(2))/(b)+K(sigma4pic^(2))/(c)`
`= Ksigma4pi(a-b+c)`
`= (sigma)/(epsilon_(0))(a-b+c)`


`V_(B)= K(4pia^(2)sigma)/(b)+K(4pib^(2)sigma)/(b)+K(4pic^(2)sigma)/(c)`
`=(sigma)/(epsilon_(0))[(a^(2))/(b)-b-+c]`
`V_(c)=(k4pia^(2)sigma-K4pib^(2)sigma+K4pic^(2)sigma)/(c)`
`=(sigma)/(epsilon_(0))((a^(2)-b^(2)+c^(2))/(c))`
(b) `V_(A)=V_(C)`
`implies a-b+c=(a^(2)-b^(2)+c^(2))/(c)`
`implies a-b+c=(a^(2)-b^(2))/(c)+c`
`implies a-b=((a+b)(a-b))/(c)`
`implies c=a+b`
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