In an RC circuit, which expression defines the time constant τ?

• A. τ = RC
• B. τ = R/C
• C. τ = C/R
• D. τ = R^2 C

Answer: (A)

The time constant is the characteristic time scale of an RC circuit. It is the product of the resistance and the capacitance: τ = RC. This arises from the equations governing charging and discharging: during charging, V(t) = Vs[1 − e^(−t/RC)], and during discharging, V(t) = V0 e^(−t/RC). In both cases, the exponential decay is controlled by the factor t/RC, so the constant that sets the pace is RC.

Practically, after a time equal to τ, the exponential factor is 1/e, so the capacitor has charged to about 63% of its final value in charging, or decayed to about 37% of its initial value in discharging. The units also line up correctly: ohms times farads give seconds. Other expressions would not have the correct units or behavior for the circuit’s time-dependent response.