What is the energy required to assemble a capacitor with final voltage V and capacitance C?

What is the energy required to assemble a capacitor with final voltage V and capacitance C?

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

Answer: (B)

The main idea is that the energy required to charge a capacitor equals the work done by the charging source, which is the integral of voltage times incremental charge as the capacitor builds up from 0 to its final state. With V = Q/C, this work becomes W = ∫0^Q (Q'/C) dQ' = Q^2/(2C). Since Q = C V, this is also W = (1/2) C V^2. So expressing the energy as Q^2/(2C) is just another correct form of the same quantity, useful when you know the charge and the capacitance. The form QV would imply the full final voltage applied at all times, which isn’t the case during charging, and would overestimate the work by a factor of 2. The form C V^2 is also off by a factor of 2 for the same reason.

The main idea is that the energy required to charge a capacitor equals the work done by the charging source, which is the integral of voltage times incremental charge as the capacitor builds up from 0 to its final state. With V = Q/C, this work becomes W = ∫0^Q (Q'/C) dQ' = Q^2/(2C). Since Q = C V, this is also W = (1/2) C V^2. So expressing the energy as Q^2/(2C) is just another correct form of the same quantity, useful when you know the charge and the capacitance. The form QV would imply the full final voltage applied at all times, which isn’t the case during charging, and would overestimate the work by a factor of 2. The form C V^2 is also off by a factor of 2 for the same reason.