Capacitance

A capacitor of capacitance $C_1$ is connected in series with a second capacitor of capacitance $C_2$.

Show that the combined capacitance $C$ of the two capacitors is given by

Three identical capacitors, each of capacitance $C$, are connected in a network as shown in Fig. 5.1.

The variation of the charge $Q$ with the potential difference (p.d.) $V$ between the terminals X and Y is shown in Fig. 5.2.

Show that $C$ is equal to $44\ \mu\text{F}$.

Determine the time constant $\tau$ of the circuit. Give a unit with your answer.

$\tau$ = ______ unit ______

Determine the time taken for the discharge current to reduce to $15\%$ of the initial discharge current.

time = ______ $\text{s}$

State the relationship between $V_C$ and $V_R$.

the capacitance $C$, in $\mu\text{F}$

$C$ = ______ $\mu\text{F}$

the resistance $R$, in $\text{k}\Omega$

$R$ = ______ $\text{k}\Omega$

the time constant $\tau$ of the circuit.

$\tau$ = ______ $\text{s}$

Use Fig. 7.2, Fig. 7.3 and your answer in (a) to explain why the variation of $Q$ with $t$ is exponential in nature.

Two parallel plate capacitors $C_1$ and $C_2$ are connected to a supply that has a potential difference (p.d.) $V_S$. The capacitors may be connected in series or in parallel.

The supply provides charge $Q_S$ and the plates of the two capacitors acquire charges $Q_1$ and $Q_2$ respectively. The p.d.s across the plates of the capacitors are $V_1$ and $V_2$ respectively.

Complete Table 6.1 to indicate how $Q_S$, $Q_1$ and $Q_2$ relate to each other, and how $V_S$, $V_1$ and $V_2$ relate to each other, for series and parallel connections of the capacitors to the supply.

Table 6.1

Calculate the p.d. across the capacitor.

p.d. = ______ $\text{V}$

Calculate the charge on the capacitor.

charge = ______ $\text{C}$

The capacitor is now connected in parallel with a capacitor of capacitance $180\ \mu\text{F}$ that is initially uncharged.

Determine the total energy, in mJ, now stored in the two capacitors.

energy = ______ $\text{mJ}$

Two parallel plate capacitors $C_1$ and $C_2$ are connected to a supply that has a potential difference (p.d.) $V_S$. The capacitors may be connected in series or in parallel.

The supply provides charge $Q_S$ and the plates of the two capacitors acquire charges $Q_1$ and $Q_2$ respectively. The p.d.s across the plates of the capacitors are $V_1$ and $V_2$ respectively.

Complete Table 6.1 to indicate how $Q_S$, $Q_1$ and $Q_2$ relate to each other, and how $V_S$, $V_1$ and $V_2$ relate to each other, for series and parallel connections of the capacitors to the supply.

Table 6.1

Calculate the p.d. across the capacitor.

p.d. = ______ $\text{V}$

Calculate the charge on the capacitor.

charge = ______ $\text{C}$

The capacitor is now connected in parallel with a capacitor of capacitance $180\ \mu\text{F}$ that is initially uncharged.

Determine the total energy, in mJ, now stored in the two capacitors.

energy = ______ $\text{mJ}$

Explain the shape of the line in Fig. 6.2.

resistance $R$

$R$ = ______ $\Omega$

the time constant $\tau$ of the circuit in Fig. 6.1.

$\tau$ = ______ $\text{s}$

Use your answers in (b) to determine capacitance $C$.

$C$ = ______ $\text{F}$

Two capacitors X and Y are connected in series to a power supply of voltage $V$, as shown in Fig. 6.1.

The capacitance of X is $C_X$ and the capacitance of Y is $C_Y$.

Derive an expression, in terms of $C_X$ and $C_Y$, for the combined capacitance $C_T$ of the capacitors in this circuit.

Explain your reasoning.

Show that the supply voltage $V$ is $5.0\ \text{V}$.

Calculate the total energy, in $\text{mJ}$, stored in the capacitors when $C_Q$ has its maximum value.

total energy = ______ $\text{mJ}$

On Fig. 6.2, sketch the variation of the total energy $E$ stored in the capacitors with $C_Q$, as $C_Q$ varies from $0$ to $400\ \mu\text{F}$.

Explain the shape of the line in Fig. 6.2.

resistance $R$

$R$ = ______ $\Omega$

the time constant $\tau$ of the circuit in Fig. 6.1.

$\tau$ = ______ $\text{s}$

Use your answers in (b) to determine capacitance $C$.

$C$ = ______ $\text{F}$

Define the capacitance of a parallel-plate capacitor.

On Fig. 7.1, sketch the variation of charge with p.d. for capacitor X as the p.d. increases from 0 to $V$.

Determine an expression, in terms of $Q$ and $V$, for the work $W$ done on capacitor X during the charging process. Explain your reasoning.

$W$ = ______

Complete Table 7.1 to show expressions, in terms of $Q$ and $V$, for the final p.d.s across, and the final charges on, the two capacitors.
Use the space below for any working that you need.

Table 7.1

State whether the total energy stored in the two capacitors is less than, the same as, or greater than the energy initially stored in capacitor X.

Show that the capacitance of the capacitor is $1.5\ \mu\text{F}$.

Determine the resistance of R.

resistance = ______ $\Omega$

Calculate the energy stored in the capacitor at time $t = 0$.

energy = ______ $\text{J}$

A second identical resistor is now connected in parallel with R.

The switch is initially in position S. When the capacitor is fully charged, the switch is moved to position T so that the capacitor discharges. At time $t$ after the switch is moved the charge on the capacitor is $Q$.

On Fig. 5.2, sketch a line to show the variation of $\ln(Q/\mu\text{C})$ with $t$ between time $t = 0$ and time $t = 5.0\ \text{s}$.

the initial charge $Q_A$ on the plates of capacitor A

$Q_A$ = ______

the initial energy $E_A$ stored in capacitor A.

$E_A$ = ______

State and explain what happens to the charge that was initially on the plates of capacitor A.

Show that the final potential difference (p.d.) $V_B$ across capacitor B is given by

Explain your reasoning.

Determine an expression, in terms of $V$ and $C$, for the decrease $\Delta E$ in the total energy that is stored in the capacitors as a result of the change of the position of the switch.

$\Delta E$ = ______