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## Capacitors and Transformers

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.

# Capacitors

The two metal plates in the above figure have net charges of $$+140.0\text{ pC}$$ and $$-140.0\text{ pC},$$ respectively. If the potential difference between them is $$10.0\text{ V},$$ what is the capacitance of the system consisting of the two metal plates?

The capacitor in the above figure has a capacitance of $$40.0 \,\mu\text{F}$$ and is initially uncharged. The battery provides a potential difference of $$90.0\text{ V}.$$ After switch S is closed, how much charge will pass through it?

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 $$\varepsilon_0=8.85 \times 10^{-12} \text{ C}^2\text{/N}\cdot\text{m}^2.$$

One can make a homemade capacitor using an aluminum foil and a plastic or a wooden stick (it's important that the stick is an insulator). Make a small ball out of the aluminum foil and wrap it around the stick. Then make a wider foil sphere around the ball so that they don't touch, and your capacitor is ready!

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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