Capacitors: capacitance, energy and exponential charge/discharge

Core ideas

What a capacitor does

A capacitor is two conducting plates separated by an insulator. When it is connected to a supply, electrons are pushed onto one plate and pulled off the other, so the plates carry $+Q$ and $-Q$. The net charge is zero, and no charge crosses the gap. Current flows in the external circuit until the capacitor’s pd equals the supply emf. Capacitance is the charge stored per unit pd:

Real values are usually μF, nF or pF.

Parallel plates and dielectrics

$C$ is proportional to the plate overlap area and inversely proportional to the separation. The relative permittivity $\varepsilon_r$ is the factor by which the dielectric increases the capacitance: $\varepsilon_r = C_{\text{dielectric}}/C_{\text{vacuum}}$.

Dielectric action (polar-molecule model): the dielectric contains polar molecules. In the field between the plates they rotate to align with the field, with their positive ends towards the negative plate. Their own field opposes the applied field, so the resultant field is smaller. For the same charge the pd is lower ($V = Ed$), so $C = Q/V$ is larger. Equivalently, more charge can be stored at the same pd.

Energy stored

Adding a small charge $\Delta Q$ at pd $V$ takes work $V\Delta Q$. The pd rises linearly with charge, so the total work is the area under the $V$–$Q$ graph, a triangle:

Choose the form that uses the quantity held constant. When charging from a battery, the battery transfers $QV$, the capacitor stores $\tfrac{1}{2}QV$, and the other half is dissipated in the resistance. This is true whatever the value of $R$.

Series and parallel

These are the reverse of the resistor rules. In series, the smaller capacitor takes the larger pd.

Discharge through a resistor

The current is $I = V/R = Q/RC$, so charge drains at a rate proportional to what is left: $\dfrac{\Delta Q}{\Delta t} = -\dfrac{Q}{RC}$. This gives exponential decay:

• Time constant $\tau = RC$ (in seconds when $R$ is in Ω and $C$ in F) is the time for $Q$, $V$ or to fall to of its starting value.

Charging through a resistor

From uncharged, with a supply of emf $\varepsilon$:

The current decays during charging as well as during discharge. The pd across the resistor falls as the pd across the capacitor rises, and at every instant $V_R + V_C = \varepsilon$.

Getting $RC$ from data

Take natural logs: $\ln V = \ln V_0 - \dfrac{t}{RC}$. Plot on the -axis against . The graph is a straight line with and intercept . In the practical, choose of tens of seconds (such as 470 μF with 100 kΩ, giving 47 s) for stopwatch timing, or use a data logger or oscilloscope for small .

Iterative model (OCR A and B): $\Delta Q = -\dfrac{Q}{RC}\Delta t$, applied step by step. Each step uses the current at the start of the interval, which is the largest current in it. The model therefore slightly over-estimates the charge lost, and it improves as $\Delta t$ gets smaller.

Worked examples

Exam technique

• Graph toolkit. The gradient of a $Q$–$t$ graph is the current. The area under an $I$–$t$ graph is the charge. The area under a $V$–$Q$ graph is the energy. The gradient of a – graph is .

Common mistakes

Quick recap

• $C = Q/V$, and $C = A\varepsilon_0\varepsilon_r/d$ for parallel plates. Aligned polar molecules oppose the field and raise by a factor .