RL time constant
τ = L/R. The characteristic time for current in an RL circuit to rise to 1−1/e ≈ 63% of its steady-state value, or decay to 1/e ≈ 37% of its initial value.
Definition
The RL time constant τ = L/R is the inductive analogue of the RC time constant. When a battery V₀ is switched on across an RL series circuit, the current rises as I(t) = (V₀/R)(1 − e^(−t/τ)), climbing toward the DC steady-state value V₀/R. When the battery is removed and the inductor is shorted through the resistor, the current decays as I(t) = I₀e^(−t/τ). Both timescales are set by L/R alone — the larger the inductance or smaller the resistance, the slower the circuit reacts to changes.
The governing equation comes from Kirchhoff's voltage law: V₀ = IR + L dI/dt. The dI/dt term is the back-EMF from Lenz's law, resisting the change in current. Solving gives the exponential. Physically, the inductor is storing or releasing magnetic-field energy U = ½LI² as the current ramps, and the resistor dissipates power I²R — the time constant balances how fast L can absorb or release energy against how fast R can dissipate it.
Practical consequences: interrupting an inductive circuit — a relay coil, a solenoid, a motor winding — forces the current to change on a timescale shorter than L/R, which means dI/dt becomes enormous, and the back-EMF rises without bound. The result is a spark across the opening switch contacts, which can destroy semiconductors, weld relay contacts, and emit EMI. Every inductive load in practical electronics needs a flyback diode (reverse-biased across the coil) to provide a safe path for the collapsing current, clamping the back-EMF to a diode-drop above the supply and letting the current decay with τ set by L over the sum of the diode resistance and the circuit resistance.