Calculating the Capacitance — Question 5

Question 5

A slowly tapered gap requires averaging inverse separation

Two plates have width w and length L. Their local vacuum gap is $d(x)=d_0(1+\eta x/L)$ for $0\le x\le L$, with $\eta>0$. Assume a small slope so each strip behaves locally as parallel plates at the same voltage.

1. Integrate the strip charges to find total capacitance.

2. For $\eta=1$, compare with using the arithmetic mean gap $\bar d=3d_0/2$.

Question 5 - Solution

A slowly tapered gap requires averaging inverse separation

(a) Method and calculation

$$dQ=\frac{\epsilon_0Vw\,dx}{d(x)},\quad C=\epsilon_0w\int_0^L\frac{dx}{d_0(1+\eta x/L)}.$$$$C=\frac{\epsilon_0wL}{d_0}\frac{\ln(1+\eta)}\eta.$$

(b) Interpretation and further calculation

$$\eta=1:\ C=\frac{\epsilon_0wL}{d_0}\ln2,\quad C_{\mathrm{mean}}=\frac{2\epsilon_0wL}{3d_0}.$$$$C/C_{\mathrm{mean}}=\tfrac32\ln2=1.03972.$$

Physical check

Narrower regions contribute disproportionately; the local strip approximation requires gaps small relative to the plate dimensions.