Capacitor

Treat the Earth as an isolated spherical conductor of radius $6400$ km, and let the measured electric field at the surface be $100$ V/m.

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 $\varepsilon_r = 6.0$.

A capacitor charged with charge $Q$ has a potential difference $U$ across its plates. When the charge is increased by $4.0 \times 10^{-8}$ C, the potential difference increases by $20$ V.

An air-filled parallel-plate capacitor has plate separation $d = 1.0$ mm. Both plates are squares of equal area.

The breakdown field strength of dry air is $3 \times 10^{3}$ V/m. A parallel-plate capacitor has plates of area $40 \times 50\ \mathrm{cm}^2$.

Design a capacitor of capacitance $0.1\ \mu\mathrm{F}$ that operates safely at $1000$ V. The dielectric has relative permittivity $\varepsilon_r = 3$ and breakdown field strength $E_{\text{breakdown}} = 10$ kV/mm. Use a safety factor of $4$ on the breakdown field.

Four capacitors, each of capacitance $C$, are connected in two different configurations between terminals $A$ and $B$, as shown in Figure. In configuration (a), four capacitors are arranged in a series chain enclosed in a loop, with $A$ tied to one end of the loop and $B$ tied to the midpoint of the chain (so two capacitors lie between $A$ and $B$ on each side of the loop). In configuration (b), the same four-capacitor loop is used, but $B$ is tied between the third and fourth capacitor (so one branch has three capacitors in series and the other branch has one capacitor).

In the circuit of Figure, $C_1 = 0.25\ \mu\mathrm{F}$, $C_2 = 0.15\ \mu\mathrm{F}$, $C_3 = 0.20\ \mu\mathrm{F}$. $C_2$ and $C_3$ are connected in parallel with each other, and that parallel combination is in series with $C_1$ between terminals $A$ and $B$. The voltage across $C_1$ is $50$ V.

In the circuit of Figure, $C_1 = 20\ \mu\mathrm{F}$ and $C_2 = 5\ \mu\mathrm{F}$. Switch $S$ first connects $C_1$ to a battery of voltage $U = 1000$ V, fully charging $C_1$ (with $C_2$ disconnected). The switch is then thrown to the other side, disconnecting the battery and connecting $C_1$ in parallel with $C_2$.

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 $d = 0.15\ \mathrm{mm}$.