Capacitance as charge per unit potential difference, the energy stored on a capacitor, and the exponential charge and discharge of a capacitor through a resistor with the time constant.

What this dot point is asking

Edexcel wants you to define capacitance as charge per unit potential difference, calculate the energy stored on a charged capacitor, and analyse the exponential charging and discharging of a capacitor through a resistor using the time constant $\tau = RC$.

The answer

Capacitance

A larger capacitance stores more charge at a given voltage. Practical capacitors are measured in microfarads or smaller because a farad is enormous. Capacitors in parallel add ($C = C_1 + C_2 + \ldots$) since they share the same voltage and the charges add; capacitors in series combine reciprocally ($\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + \ldots$) since they carry the same charge and the voltages add.

Energy stored

The factor of one half appears because the potential difference grows from zero to its final value as charge is added; the average voltage during charging is half the final voltage. On a $V$ against $Q$ graph the stored energy is the triangular area $\frac{1}{2}QV$. When a capacitor charges through a resistor, exactly half the energy delivered by the supply is dissipated as heat in the resistor, regardless of the resistance, with the other half stored on the capacitor.

Charging and discharging

When a capacitor discharges through a resistor, the current is proportional to the charge remaining, which gives exponential decay:

The time constant $\tau = RC$ has units of seconds and sets the timescale: after one time constant the discharging quantity has fallen to $\frac{1}{e} \approx 0.37$ of its start value, and after $5\tau$ the capacitor is essentially fully discharged or charged. A larger $R$ or $C$ slows the process. Taking natural logs of the discharge equation gives $\ln V = \ln V_0 - \frac{t}{RC}$, so a graph of $\ln V$ against $t$ is a straight line of gradient $-\frac{1}{RC}$, the standard way to find $\tau$ experimentally.

Examples in context

A camera flash stores energy on a large capacitor and dumps it rapidly through the flash tube, the short $RC$ giving a brief, intense pulse. Smoothing capacitors in a power supply charge during voltage peaks and discharge into the load between them, reducing ripple. Defibrillators store hundreds of joules on a capacitor and release it in milliseconds. The long $RC$ time of a backup capacitor keeps a clock running through a brief power cut.

Try this

Q1. Define capacitance. [1 mark]

• Cue. The charge stored per unit potential difference, $C = \frac{Q}{V}$, in farads.

Q2. A $50$ microfarad capacitor is charged to $20$ V. Find the energy stored. [2 marks]

• Cue. $W = \frac{1}{2}CV^2 = \frac{1}{2} \times 50 \times 10^{-6} \times 20^2 = 1.0 \times 10^{-2}$ J.

Q3. A capacitor discharges through a resistor with time constant $\tau = 4.0$ s. Find the fraction of charge remaining after $4.0$ s. [2 marks]

• Cue. $\frac{Q}{Q_0} = e^{-t/\tau} = e^{-1} = 0.37$, so about $37\%$ remains.