Energy stored in a capacitor

Also called Capacitor energy, U_C

The energy stored in a capacitor equals the work an external agent had to do to separate its charge, and both AP equation sheets give it in one form only: one half the charge times the potential difference. The two familiar alternatives have to be derived.

Both CEDs state the physical meaning first: the electric potential energy stored in a capacitor is equal to the work done by an external force to separate that amount of charge on the capacitor (10.6.A.4 in AP Physics 2, 10.3.A.4 in AP Physics C: E&M). Then the equation:

U_C = \tfrac{1}{2} Q \Delta V

That is the only version either sheet prints. Rendering the Table of Information appendix of both booklets confirms it: the AP Physics 2 Electricity column and the AP Physics C: E&M column each show $U_C = \tfrac{1}{2}Q\Delta V$ and nothing else for capacitor energy. So the two forms every textbook lists are yours to produce:

Substitute	Result	On a sheet
$Q = C\Delta V$	$U_C = \tfrac{1}{2} C (\Delta V)^2$	no
$\Delta V = Q/C$	$U_C = \dfrac{Q^2}{2C}$	no

All three are the same energy. Choosing between them is a question of which quantity a problem holds fixed. A capacitor still connected to a battery has $\Delta V$ fixed, so $\tfrac{1}{2}C(\Delta V)^2$ answers questions about changing the plate separation. A capacitor charged and then disconnected has $Q$ fixed, so $Q^2/2C$ answers the same questions with the opposite sign of change.

Where the one half comes from. The potential difference climbs from zero as charge accumulates, so the last coulomb costs far more work than the first. The energy is the average, not the final voltage times the total charge.

See capacitance for the ratio itself, and the capacitor entry for the device.

Back to the physics glossary