Calculating the Capacitance — Question 10

Question 10

A coaxial geometry has an optimal inner radius for a specified field ceiling

A long vacuum coaxial capacitor has fixed outer radius b and variable inner radius a. Its voltage is V>0. A prescribed maximum allowable field is E*. Ignore end effects.

1. Express the greatest field in terms of V,a,b and obtain the limiting voltage.

2. Choose a to maximize that voltage for fixed b and E*, and find the corresponding capacitance per length.

Question 10 - Solution

A coaxial geometry has an optimal inner radius for a specified field ceiling

(a) Method and calculation

$$E(r)=\frac{V}{r\ln(b/a)},\quad E_{\max}=E(a).$$$$V_{\mathrm{lim}}(a)=E_*a\ln(b/a).$$

(b) Interpretation and further calculation

$$\frac{dV_{\mathrm{lim}}}{da}=E_*[\ln(b/a)-1]=0\Rightarrow a=b/e.$$$$\frac{d^2V_{\mathrm{lim}}}{da^2}=-E_*/a<0,\quad V_{\max}=E_*b/e,\quad C/L=2\pi\epsilon_0.$$

Physical check

The largest allowable voltage does not occur at the largest capacitance; making the inner radius too large squeezes the gap.