Capacitance

19.1 Capacitors and capacitance

Capacitance is the charge stored per unit potential:

$C = \frac{Q}{V}$

The unit is the farad, where 1 F = 1 C/V.

• For a parallel plate capacitor, it is the charge on one plate per unit p.d. between the plates.
• For an isolated spherical conductor, it is the charge on it per unit potential at its surface. Combining $C = Q/V$ with $V = Q/(4\pi\varepsilon_0 r)$ gives $C = 4\pi\varepsilon_0 r$.

A capacitor's plates hold equal and opposite charges. The net charge is zero, and $Q$ means the charge on one plate.

Capacitors in series and parallel

In parallel, the p.d. is the same across each, and the charges add: $Q = Q_1 + Q_2$. Dividing by $V$ gives:

$C = C_1 + C_2 + \dots$

In series, each capacitor holds the same charge, and the p.d.s add: $V = V_1 + V_2$. Dividing by $Q$ gives:

$\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + \dots$

19.2 Energy stored in a capacitor

Charging a capacitor means moving charge against an increasing p.d., so the work done is the area under the potential–charge graph, which is a triangle.

$W = \tfrac{1}{2}QV = \tfrac{1}{2}CV^2 = \frac{Q^2}{2C}$

It is not $QV$, because the p.d. builds up from zero. The cell does transfer $QV$, and half of that goes to heat in the circuit's resistance, whatever the size of the resistance.

19.3 Discharging a capacitor

When a capacitor discharges through a resistor, the current is large at first, because the p.d. is large. As the charge falls, the p.d. falls and the current falls. So the rate of loss of charge is proportional to the charge left, which gives an exponential decay.

$x = x_0 e^{-t/RC}$

where $x$ can be the charge, the p.d. or the current.

The time constant

The time constant is:

$\tau = RC$

It is the time for the charge to fall to $1/e$, or about 37%, of its starting value. After $5RC$ the capacitor is effectively discharged. A larger $R$ makes the current smaller, and a larger $C$ means there is more charge to remove, so both make discharge slower.

Finding the time constant from a graph

• From an exponential curve, read the time at which the value falls to 37%.
• Taking natural logs gives $\ln x = \ln x_0 - t/RC$. So a graph of $\ln V$ against $t$ is a straight line with gradient $-1/RC$.