What is the energy stored in a capacitor with capacitance C and voltage V?

• A. 1/2 C V^2
• B. C V
• C. 1/2 V^2 / C
• D. C^2 V

Answer: (A)

When a capacitor is charged, the energy stored is tied to how much charge sits on the plates and the voltage across them. You do work to move charges onto the plates as the voltage builds, so that work ends up stored in the electric field between the plates.

Consider a small amount of charge dQ added at the current voltage V. The incremental energy is dE = V dQ. For a capacitor, V = Q/C, so dE = (Q/C) dQ. Integrating from zero to the final charge Q gives E = ∫0^Q (Q'/C) dQ' = Q^2/(2C). Since Q = C V, this becomes E = (1/2) C V^2.

This same result can be written as E = (1/2) QV, which is equivalent because Q = CV. The energy is measured in joules, and it increases with the square of the voltage and linearly with the capacitance.