Capacitance of a Stacked Mica Capacitor

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Electrostatics Intermediate Capacitor

Source: High school physics (Chinese)

Problem

A mica capacitor is built from $10$ aluminum foil sheets and $9$ mica sheets stacked alternately in parallel layers. The odd-numbered foils are connected together to form one electrode, and the even-numbered foils are connected together to form the other electrode. Each foil and each mica sheet has area $2.5\ \mathrm{cm}^2$; the relative permittivity of mica is $\varepsilon_r = 7.0$; each layer (mica sheet) has thickness $d = 0.15\ \mathrm{mm}$.

Find the capacitance of this capacitor.
Problem image
$C \approx 9.3 \times 10^{-10}$ F $\approx 930$ pF.

Each mica sheet sits between two adjacent aluminum foils --- one on the odd electrode and one on the even electrode --- and so forms one elementary parallel-plate capacitor between the two electrodes. With $9$ mica layers, there are $9$ such elementary capacitors, all in parallel.

Capacitance of one layer:

$$C_1 = \dfrac{\varepsilon_r \varepsilon_0 A}{d}.$$

With $\varepsilon_0 = 8.85 \times 10^{-12}$ F/m, $A = 2.5 \times 10^{-4}\ \mathrm{m}^2$, $d = 1.5 \times 10^{-4}$ m:

$$C_1 = \dfrac{7.0 \times 8.85 \times 10^{-12} \times 2.5 \times 10^{-4}}{1.5 \times 10^{-4}} \approx 1.03 \times 10^{-10}\ \mathrm{F}.$$

Total capacitance:

$$C = 9 C_1 \approx 9.3 \times 10^{-10}\ \mathrm{F} = 930\ \mathrm{pF}.$$