Capacitance — Question 8

Question 8

A nonlinear charge law has distinct secant and differential capacitances

A device obeys $q(v)=C_0v+\beta v^3$, with $C_0,\beta>0$. Define secant capacitance $C_s=q/v$ for nonzero v and differential capacitance $C_d=dq/dv$.

1. Find Cs and Cd at v=V.

2. Find the first-order voltage response to a small added charge at V and compare it with using Cs.

Question 8 - Solution

A nonlinear charge law has distinct secant and differential capacitances

(a) Method and calculation

$$C_s(V)=C_0+\beta V^2.$$$$C_d(V)=C_0+3\beta V^2.$$

(b) Interpretation and further calculation

$$\delta q=C_d(V)\delta v+O(\delta v^2)\Rightarrow\delta v\simeq\frac{\delta q}{C_0+3\beta V^2}.$$$$\frac{\delta v_{\mathrm{using\ }}C_s}{\delta v_{\mathrm{correct}}}=\frac{C_0+3\beta V^2}{C_0+\beta V^2}>1\quad(V\ne0).$$

Physical check

The tangent slope controls small changes; the origin-to-point slope describes the full charge-to-voltage ratio.