Edexcel A Level (IAL) Physics-4.21 Energy Stored by a Capacitor- Study Notes- New Syllabus

Energy Stored in a Capacitor

A capacitor stores energy when it is charged. This energy is stored in the electric field between the capacitor plates.

Energy Stored in a Capacitor

The energy \( W \) stored in a capacitor is given by:

\( W = \dfrac{1}{2}QV \)

• \( W \) = energy stored (J)
• \( Q \) = charge stored (C)
• \( V \) = potential difference across the capacitor (V)

Derivation Using the \( V \)–\( Q \) Graph

For a capacitor:

\( V = \dfrac{Q}{C} \)

This shows that potential difference is directly proportional to charge. A graph of potential difference \( V \) against charge \( Q \) is therefore a straight line through the origin.

Work done in charging:

When a small charge \( \mathrm{d}Q \) is added at potential difference \( V \):

\( \mathrm{d}W = V\,\mathrm{d}Q \)

Total energy stored equals the area under the \( V \)–\( Q \) graph.

The graph is triangular, so:

\( W = \dfrac{1}{2}QV \)

Deriving \( W = \dfrac{1}{2}CV^2 \)

From the capacitance equation:

\( Q = CV \)

Substitute into \( W = \dfrac{1}{2}QV \):

\( W = \dfrac{1}{2}(CV)V \)

\( W = \dfrac{1}{2}CV^2 \)

Deriving \( W = \dfrac{1}{2}\dfrac{Q^2}{C} \)

Rearrange the capacitance equation:

\( V = \dfrac{Q}{C} \)

Substitute into \( W = \dfrac{1}{2}QV \):

\( W = \dfrac{1}{2}Q \left(\dfrac{Q}{C}\right) \)

\( W = \dfrac{1}{2}\dfrac{Q^2}{C} \)

Choosing the Correct Equation

• Use \( W = \dfrac{1}{2}QV \) when both charge and voltage are known.
• Use \( W = \dfrac{1}{2}CV^2 \) when capacitance and voltage are known.
• Use \( W = \dfrac{1}{2}\dfrac{Q^2}{C} \) when charge and capacitance are known.