One way to double the capacitance of a parallel-plate capacitor is to reduce the plate separation distance by a factor of 2. Doubling the capacitance of a spherical capacitor is a bit more complex, though.

(a) If you had a spherical capacitor consisting of two concentric spherical shells of radii \(R_{\text {inner }}\) and \(R_{\text {outer }}>R_{\text {inner }}\), you could double the capacitance by changing the radius of the outer shell from \(R_{\text {outer }}\) to some value \(d\). What is \(d\) in terms of \(R_{\text {inner }}\) and \(R_{\text {outer }}\) ?

(b) You could also double the capacitance by changing the radius of the inner shell from \(R_{\text {inner }}\) to some value \(d\). What is \(d\) in terms of \(R_{\text {inner }}\) and \(R_{\text {outer }}\) ?

(c) Repeat parts \(a\) and \(b\) for a capacitor consisting of two long concentric cylindrical shells of radii \(R_{\text {inner }}\) and \(R_{\text {outer }}>R_{\text {irner. }}\)