Calculating the Capacitance — Question 9

Question 9

Two gap settings separate plate capacitance from stray capacitance

A measured sensor obeys $C_{\mathrm{meas}}(d)=K/d+C_s$, with K and Cs constant. Measurements give 9.00 pF at gap d0 and 6.00 pF at gap 2d0.

1. Infer K/d0 and Cs.

2. Predict the reading at 3d0 and find the fraction of the original reading attributable to the intended plate gap.

Question 9 - Solution

Two gap settings separate plate capacitance from stray capacitance

(a) Method and calculation

$$K/d_0+C_s=9.00,\quad K/(2d_0)+C_s=6.00\quad\text{in pF}.$$$$K/d_0=6.00\,\mathrm{pF},\quad C_s=3.00\,\mathrm{pF}.$$

(b) Interpretation and further calculation

$$C_{\mathrm{meas}}(3d_0)=6.00/3+3.00=5.00\,\mathrm{pF}.$$$$\frac{C_{\mathrm{plate}}(d_0)}{C_{\mathrm{meas}}(d_0)}=6/9=2/3.$$

Physical check

The offset stays finite as the intended gap capacitance tends toward zero within the supplied model.