Exponential decay equation for capacitor voltage in a source-free RC circuit.

• A. Vc(t) = Vo e^{-t/RC}
• B. Vc(t) = Vo e^{t/RC}
• C. Vc(t) = Vo e^{-t/r}
• D. Vc(t) = Vo e^{-r/t}

Answer: (C)

In a source-free RC circuit, the capacitor voltage decays exponentially because the resistor draws current proportional to that voltage, while the capacitor’s current is C dVc/dt. With discharge, the current through the resistor equals the capacitor current but in the opposite sense: i = Vc/R and i = C dVc/dt, so C dVc/dt = - Vc/R. This gives dVc/dt = - (1/RC) Vc. Solving this differential equation yields Vc(t) = Vo e^{-t/(RC)}; the exponent is negative and the time constant is RC, which makes the voltage decay over time toward zero. The exponent must be dimensionless, so the denominator is RC (not just R or another improper combination). Expressions lacking RC in the denominator or with a positive exponent would describe growth or incorrect units, not a discharging capacitor.