The capacitance of an isolated conducting sphere of radius R becomes k times its original value when it is enclosed by a concentric conducting spherical shell which is connected to earth. The gap width (distance between the inner sphere and the outer shell) is:

Step-by-step solution

The capacitance of the isolated conducting sphere is C = 4 ₀ R . Let the gap width be d . The radius of the outer concentric spherical shell is R + d . When the outer shell is earthed, the capacitance of the spherical capacitor formed is: C' = 4 ₀ R(R+d) (R+d) - R = 4 ₀ R(R+d) d Given that C' = kC , we have: 4 ₀ R(R+d) d = k(4 ₀ R) R+d d = k R + d = kd R = d(k - 1) d = R k-1 Answer: R k-1