Calculating the Capacitance — Question 7

Question 7

Uniform geometric enlargement scales capacitance by one power of length

Every length of a vacuum capacitor is multiplied by s>0 while its shape is preserved. Consider both parallel-plate and spherical examples.

1. Show the resulting capacitance scaling in both cases.

2. At fixed voltage find the charge and characteristic field scalings, then repeat at fixed charge.

Question 7 - Solution

Uniform geometric enlargement scales capacitance by one power of length

(a) Method and calculation

$$C_{\mathrm{plate}}\prime=\epsilon_0\frac{s^2A}{sd}=sC.$$$$C_{\mathrm{sphere}}\prime=4\pi\epsilon_0\frac{(sa)(sb)}{sb-sa}=sC.$$

(b) Interpretation and further calculation

$$V\text{ fixed}:\quad Q\prime=sQ,\quad E\prime=E/s.$$$$Q\text{ fixed}:\quad V\prime=V/s,\quad E\prime=E/s^2.$$

Physical check

The field scaling follows a voltage difference divided by a corresponding length, not capacitance alone.