There are three concentric metallic spherical shells of radius a, b and c respectively as shown in the figure. The innermost sphere is given a charge q and the outermost sphere is given a charge 4q while the middle sphere is earthed. Find the charges appearing on the inner and outer surfaces of middle sphere.

Explanation

There is no electric field inside the innermost sphere and hence its total charge q must be distributed uniformly on its outer surface. Metal surfaces facing each other always carry equal and opposite cahrges, hence charge on the inner surface of middle sphere must be -q, which becomes one of the answers to this question that is charge on inner surface of middle sphere. We need to find charge on its outer surface also. The middle sphere is an earthed object so its electric potential must be zero. Hence, it will acquire charge on its outer surface accordingly. Let us assume that charge on outer surface of the middle sphere be `q_(1)` then again due to electrical constraint of metals. charge on inner surface of the outermost, sphere must be `-q_(1)`. The outermost sphere is isolated and net charge given is 4q. Hence, charge on its outer surface must be `4q+q_(1)` so that net charge on its remains 4q.
In order to calculate the magnitude of `q_(1)` we need to write electric potential of the middle sphere due to entire charge distributions and then equate it to zero, because an object connected to earth eventually acquires zero potential.
`V=(q)/(4pi epsilon_(0)b)+(q_(1)-q)/(4pi epsilon_(0)b)+(4q)/(4pi epsilon_(0)c)=0`
`rArr (q)/(4pi epsilon_(0)b)=(q_(1))/(4pi epsilon_(0)b)-(q)/(4pi epsilon_(0)b)+(4q)/(4pi epsilon_(0)c)=0 rArr q_(1)=-(4qb)/(c )`
Finally we can say that charge on inner surface of middle sphere is -q and on outer surface it is -4qb/c.