# One way to double the capacitance of a parallel-plate capacitor is to reduce the plate separation distance

## Question:

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 $$R_{\text {outer }}>R_{\text {irner. }}$$

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