Which expression is also a valid form for the energy stored in a capacitor?

Which expression is also a valid form for the energy stored in a capacitor?

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

Answer: (A)

Energy stored in a capacitor can be written in several equivalent forms because charge and voltage are linked by Q = C V. If you start from the work done to build up the charge, U = ∫0^Q V dq, and use V = q/C, you get U = ∫0^Q (q/C) dq = Q^2/(2C) = (1/2) Q^2 / C. That same energy can also be written as (1/2) Q V, since V = Q/C gives (1/2) Q V = (1/2) Q (Q/C) = Q^2/(2C). Another common form is (1/2) C V^2, which follows from U = ∫0^V C V dV = (1/2) C V^2; using Q = C V makes it consistent with the other expressions. Therefore (1/2) Q V is a valid form for the energy stored. Note that expressions like QV or C V^2 without the 1/2 do not equal the stored energy (they would be too large by a factor of 2).

Energy stored in a capacitor can be written in several equivalent forms because charge and voltage are linked by Q = C V. If you start from the work done to build up the charge, U = ∫0^Q V dq, and use V = q/C, you get U = ∫0^Q (q/C) dq = Q^2/(2C) = (1/2) Q^2 / C. That same energy can also be written as (1/2) Q V, since V = Q/C gives (1/2) Q V = (1/2) Q (Q/C) = Q^2/(2C). Another common form is (1/2) C V^2, which follows from U = ∫0^V C V dV = (1/2) C V^2; using Q = C V makes it consistent with the other expressions. Therefore (1/2) Q V is a valid form for the energy stored.

Note that expressions like QV or C V^2 without the 1/2 do not equal the stored energy (they would be too large by a factor of 2).