Calculating the Capacitance — Question 6

Question 6

A sliding overlap provides a linear displacement sensor

Two rectangular vacuum-capacitor plates of width w have overlap length L-x, with $0<x<L$, at fixed gap d. Neglect fringing and use $C=\epsilon_0A/d$.

1. Find C(x) and its displacement sensitivity.

2. At fixed voltage V, find the signed change in positive-plate charge for a small increase in x and infer x from a measured capacitance Cm.

Question 6 - Solution

A sliding overlap provides a linear displacement sensor

(a) Method and calculation

$$A(x)=w(L-x),\quad C(x)=\frac{\epsilon_0w}{d}(L-x).$$$$dC/dx=-\epsilon_0w/d.$$

(b) Interpretation and further calculation

$$\delta Q=V\,\delta C=-\frac{\epsilon_0wV}{d}\delta x.$$$$x=L-\frac{dC_m}{\epsilon_0w},\quad0<C_m<\frac{\epsilon_0wL}{d}.$$

Physical check

The admissible measured range checks whether the ideal overlap model is consistent with the reading.