An RC circuit is discharging through a resistor R with initial capacitor voltage V0. What is Vc(t) as a function of time t?

An RC circuit is discharging through a resistor R with initial capacitor voltage V0. What is Vc(t) as a function of time t?

• A. V0 e^(-tR/C)
• B. V0 e^(-t/RC)
• C. V0 (1 - e^(-t/RC))
• D. V0 e^(-t/(2RC))

Answer: (B)

Exponential decay with a time constant RC. In an RC circuit discharging through a resistor, the current leaving the capacitor equals Vc/R, and the capacitor current is i = C dVc/dt. Since this discharge reduces the voltage, dVc/dt = - (1/RC) Vc. Solving with Vc(0) = V0 gives Vc(t) = V0 e^{-t/(RC)}. This matches the initial voltage and shows the voltage fading to zero over time, with the characteristic time RC. The other forms either mix up the exponent (not dimensionless), predict the wrong long-time limit, or use the wrong factor in the exponent.

Exponential decay with a time constant RC. In an RC circuit discharging through a resistor, the current leaving the capacitor equals Vc/R, and the capacitor current is i = C dVc/dt. Since this discharge reduces the voltage, dVc/dt = - (1/RC) Vc. Solving with Vc(0) = V0 gives Vc(t) = V0 e^{-t/(RC)}. This matches the initial voltage and shows the voltage fading to zero over time, with the characteristic time RC. The other forms either mix up the exponent (not dimensionless), predict the wrong long-time limit, or use the wrong factor in the exponent.