Result

RL circuit current

I (t) = \frac{ε}{R} (1 - e^{- t / τ}), τ = \frac{L}{R}

Statement

Current rising toward its final value ε/R after a switch closes, with time constant τ = L/R.

Symbols and units

$I (t)$	current at time t	A
$ε$	source emf	V
$R$	resistance	Ω
$τ$	time constant (L/R)	s
$L$	inductance	H

When it applies

A series RL circuit switched on at t = 0, with a constant emf source.

Don't use when

Assumes a series RL circuit switched on at t = 0 with a constant EMF source — a time-varying source requires solving the differential equation directly.

Derivation

The loop rule with an inductor's back-emf gives a first-order differential equation identical in form to RC charging, with current playing the role charge played there:

I (t) = \frac{ε}{R} (1 - e^{- t / τ})

Watch out for

Mathematically identical to RC charging with τ = L/R replacing τ = RC — recognizing the shared form saves re-deriving it from scratch.

Connects to

Charging