Capacitance of a Dielectric-Filled Plate Capacitor

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Electrostatics Intermediate Capacitor

Source: High school physics (Chinese)

Problem

A parallel-plate capacitor is made of two opposing circular plates of radius $3.0$ cm, separated by $2.0$ mm. The space between the plates is filled with a dielectric of relative permittivity $\varepsilon_r = 6.0$.

What is the capacitance of this capacitor?
$C \approx 7.5 \times 10^{-11}$ F $= 75$ pF.

For a parallel-plate capacitor with a dielectric:

$$C = \dfrac{\varepsilon_r \varepsilon_0 A}{d},$$

where $A = \pi r^2$ is the plate area, $d$ the separation, and $\varepsilon_0 = 8.85 \times 10^{-12}$ F/m.

Plate area: $A = \pi (0.030)^2 = 9\pi \times 10^{-4}\ \mathrm{m^2} \approx 2.83 \times 10^{-3}\ \mathrm{m^2}.$

Capacitance:

$$C = \dfrac{6.0 \times 8.85 \times 10^{-12} \times 2.83 \times 10^{-3}}{2.0 \times 10^{-3}} \approx 7.5 \times 10^{-11}\ \mathrm{F}.$$