Capacitance and the energy stored, charging and discharging through a resistor with the time constant, and the behaviour of capacitors in d.c. and a.c. circuits.

What this key area is asking

The SQA wants you to use capacitance $C = \frac{Q}{V}$ and the energy stored $E = \tfrac{1}{2}QV = \tfrac{1}{2}CV^2$, analyse the charging and discharging of a capacitor through a resistor including the time constant $\tau = RC$, and describe the behaviour of capacitors in d.c. and a.c. circuits.

Capacitance and energy stored

The energy stored is not simply $QV$, because the voltage grows from zero to its final value as the capacitor charges. Averaging, the energy is:

The factor of one half is the most commonly forgotten detail. A graph of charge against voltage is a straight line, and the energy stored is the area under it, a triangle of area $\tfrac{1}{2}QV$.

Charging and discharging: the time constant

A larger resistance or capacitance gives a longer time constant and slower charging. On discharge, the current is largest at the start (when the voltage is highest) and decays away; on charge, the current starts large and falls to zero as the capacitor fills. The exponential shape, and the fact that the capacitor is effectively fully charged after about five time constants, are standard exam content.

Capacitors in d.c. and a.c. circuits

Because a capacitor opposes a change in voltage, it lets through rapidly changing (high-frequency) signals while blocking steady (d.c.) ones. This is why capacitors are used to block d.c. while passing a.c. signals, and in filters that select frequencies. The opposition a capacitor offers to a.c. decreases as frequency rises, the opposite of an inductor.

Examples in context

Camera flashes charge a capacitor slowly and release the energy $\tfrac{1}{2}CV^2$ in a brief, bright burst. Smoothing capacitors in power supplies charge during the peaks of rectified a.c. and discharge between them, flattening the output. Timing circuits use the $RC$ time constant to set delays in flashing indicators and timers. Coupling capacitors in audio amplifiers block d.c. bias while passing the a.c. signal between stages.

Try this

Q1. Write the relationship for the energy stored on a capacitor in terms of $C$ and $V$. [1 mark]

• Cue. $E = \tfrac{1}{2}CV^2$.

Q2. Write the relationship for the time constant of an RC circuit. [1 mark]

• Cue. $\tau = RC$.

Q3. State what a fully charged capacitor does to a steady direct current. [1 mark]

• Cue. It blocks it (no steady current flows once fully charged).