Which expression correctly gives the energy stored in a capacitor in terms of its capacitance C and voltage V?

Which expression correctly gives the energy stored in a capacitor in terms of its capacitance C and voltage V?

• A. U = (1/2) C V^2.
• B. U = QV.
• C. U = (1/2) Q^2 V.
• D. U = 2 C V^2.

Answer: (A)

When a capacitor is charged, the work done to move charge into it builds up the stored energy. The differential work is dW = V dq, because each small amount of charge dq is moved against the instantaneous voltage V. For a capacitor, the voltage is related to the charge by V = Q/C, so dW = (q/C) dq. Integrating from zero to the final charge Q gives U = ∫0^Q (q/C) dq = Q^2/(2C). Since Q = CV, this becomes U = (1/2) C V^2. This form is equivalent to U = (1/2) QV, since Q = CV. The other expressions aren’t correct in general: U = QV would only be valid if the voltage stayed constant while charging, which isn’t the case for a capacitor, and U = (1/2) Q^2 V has the wrong dependence on Q and V (it doesn’t simplify to the correct energy with the right units).

When a capacitor is charged, the work done to move charge into it builds up the stored energy. The differential work is dW = V dq, because each small amount of charge dq is moved against the instantaneous voltage V. For a capacitor, the voltage is related to the charge by V = Q/C, so dW = (q/C) dq. Integrating from zero to the final charge Q gives U = ∫0^Q (q/C) dq = Q^2/(2C). Since Q = CV, this becomes U = (1/2) C V^2.

This form is equivalent to U = (1/2) QV, since Q = CV. The other expressions aren’t correct in general: U = QV would only be valid if the voltage stayed constant while charging, which isn’t the case for a capacitor, and U = (1/2) Q^2 V has the wrong dependence on Q and V (it doesn’t simplify to the correct energy with the right units).