Which expression is an equivalent form for the energy stored in a capacitor with capacitance C charged to voltage V?

• A. U = QV
• B. U = 2 C V^2
• C. U = Q^2/(2C)
• D. U = V^2 / C

Answer: (C)

When a capacitor charges, energy is the work done to move charges against the electric field between the plates. As the charge q builds from 0 to Q, the voltage across the plates is V = q/C. The small amount of work to add an incremental charge dq is dW = V dq = (q/C) dq. Integrating from 0 to Q gives the total energy stored: U = ∫0^Q (q/C) dq = Q^2/(2C). Since Q = C V, this is also U = (1/2) C V^2.

That’s why U = Q^2/(2C) is the correct form. It’s equivalent to U = (1/2) C V^2. If you plug in the relations between Q, C, and V, the other expressions don’t match the actual energy: QV would be twice as large, 2C V^2 would be four times as large, and V^2/C does not reduce to the correct dependence on Q and C.