In a capacitor, which expression is a correct alternative representation of the energy stored in terms of charge Q and capacitance C?

In a capacitor, which expression is a correct alternative representation of the energy stored in terms of charge Q and capacitance C?

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

Answer: (C)

Charging a capacitor requires work because you must move charge onto the plates against the increasing voltage as the charge grows. The energy stored is the work done in charging, which can be written as W = ∫ from 0 to Q of V dq. Since the voltage on a capacitor is V = q/C, substitute to get W = ∫0^Q (q/C) dq = (1/2) Q^2 / C = Q^2/(2C). This expression uses only the quantities Q and C, matching the requirement. It also aligns with the familiar forms W = (1/2) C V^2 and W = (1/2) QV, which are all equivalent when you relate the variables by V = Q/C. The other forms either omit the necessary 1/2 factor in the Q and V form or involve voltage, which isn’t part of the requested Q–C expression.

Charging a capacitor requires work because you must move charge onto the plates against the increasing voltage as the charge grows. The energy stored is the work done in charging, which can be written as W = ∫ from 0 to Q of V dq. Since the voltage on a capacitor is V = q/C, substitute to get W = ∫0^Q (q/C) dq = (1/2) Q^2 / C = Q^2/(2C). This expression uses only the quantities Q and C, matching the requirement. It also aligns with the familiar forms W = (1/2) C V^2 and W = (1/2) QV, which are all equivalent when you relate the variables by V = Q/C. The other forms either omit the necessary 1/2 factor in the Q and V form or involve voltage, which isn’t part of the requested Q–C expression.