Capacitors are devices that accumulate voltage in separated electric charges, but their mechanism and mathematics can describe thermal insulation and the discharge of lightning from cloud to ground.

Consider a spherical capacitor which consists of two concentric spherical shells of radii \(38.0\text{ mm}\) and \(40.0\text{ mm},\) respectively. If we want to make a parallel-plate capacitor with the same separation, what must be the area of each plate for the same capacitance?

The value of electrostatic constant is \(\displaystyle k=\frac{1}{4\pi\varepsilon_0}=8.99 \times 10^9 \text{ N}\cdot\text{m}^2\text{/C}^2\) and the value of the permittivity constant is \(\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.\)

If the radius of the smaller ball is \(5~\mbox{cm}\) and of the bigger sphere \( 15~\mbox{cm}\), which capacitance **in pF** do you expect to obtain?

**Details and assumptions**

- The ball and the sphere are centered around the same point

If you want to make a parallel-plate capacitor of capacitance \(0.9\text{ pF}\) with two metal plates with area \(1.1\text{ cm}^2,\) what must be the separation between the two plates?

The value of the permittivity constant is \(\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.\)

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