If a capacitor with fixed voltage V has its capacitance increased by a factor κ, the stored energy changes to what?

If a capacitor with fixed voltage V has its capacitance increased by a factor κ, the stored energy changes to what?

• A. U' = κ^2 U
• B. U' = (1/κ) U
• C. U' = κ U
• D. U' = U

Answer: (C)

Holding the voltage across a capacitor fixed, the stored energy scales with the capacitance because U = 1/2 C V^2. If the capacitance increases by a factor κ, the energy becomes U' = 1/2 (κ C) V^2 = κ U. So the stored energy increases linearly with κ. This matches the idea that with the same voltage, a larger C means more charge stored (Q = C V), hence more energy.

Holding the voltage across a capacitor fixed, the stored energy scales with the capacitance because U = 1/2 C V^2. If the capacitance increases by a factor κ, the energy becomes U' = 1/2 (κ C) V^2 = κ U. So the stored energy increases linearly with κ. This matches the idea that with the same voltage, a larger C means more charge stored (Q = C V), hence more energy.