Question

The energy stored in a capacitor of capacitance C having a charge Q under a potential V is

• A. \(\frac{1}{2}Q^2V\)
• B. \(\frac{1}{2}C^2V\)
• C. \(\frac{1}{2}\frac{Q^2}{V}\)
• D. \(\frac{1}{2}QV\)
• E. \(\frac{1}{2}CV\)

Hint:

The energy stored in a capacitor can be expressed in three equivalent forms: 1. \(U = \frac{1}{2}QV\) 2. \(U = \frac{1}{2}CV^2\) 3. \(U = \frac{Q^2}{2C}\) Always choose the form that uses the variables given or required by the question!

Answer 1

The Correct Option is D

Concept: The energy stored in a capacitor represents the work done by an external agent to move charge from one plate to another against the developing electric field.
• Basic Definition: Capacitance is defined as \(C = Q/V\).
• Work Done: The incremental work to move a small charge \(dq\) at potential \(v\) is \(dW = v \cdot dq\).
• Integration: Integrating from zero charge to total charge \(Q\) yields the stored potential energy.

Step 1: Derive the primary energy formula.
Substituting \(v = q/C\) into the work equation: \[ W = \int_0^Q \frac{q}{C} dq = \frac{1}{C} \left[ \frac{q^2}{2} \right]_0^Q = \frac{Q^2}{2C} \] This is the most fundamental form of stored energy (\(U\)).

Step 2: Convert to the form provided in the options.
We know from the definition of capacitance that \(C = Q/V\). Substituting this into our derived formula: \[ U = \frac{Q^2}{2(Q/V)} = \frac{Q^2 \cdot V}{2Q} \] \[ U = \frac{1}{2}QV \] Comparing this result with the given choices, Option (D) matches exactly.