7.4.3 Energy stored by a capacitor

Energy Stored in a Capacitor

1. Graphical Interpretation:

• For a capacitor, charge ($Q$) is directly proportional to potential difference ($V$). This relationship produces a straight line through the origin when plotted on a graph, as shown in the diagram.
• Since the area under this $Q$ -$V$ graph forms a right-angled triangle, we can calculate the area (and hence the energy stored) as:

$$E = \frac{1}{2} QV$$

2. Alternate Forms of the Energy Equation:

• By using the formula for capacitance $C = \frac{Q}{V}$, we can rearrange and substitute into the energy formula to find two other versions that may be useful depending on the given quantities.
• These variations are:

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

Where:

• $Q$ is the charge stored on the plates (in coulombs, $C$),
• $V$ is the potential difference across the plates (in volts, $V$),
• $C$ is the capacitance of the capacitor (in farads, $F$).
• $E$ is the energy stored (in joules, $J$),

Explanation of Each Form

• $E = \frac{1}{2} QV$: This form is most direct if both the charge and potential difference are known.

• $E = \frac{1}{2} CV^2$: Useful when you know the capacitance and potential difference across the capacitor.

• $E = \frac{Q^2}{2C}$: Convenient if you have the charge and capacitance but not the potential difference.