RL Circuit Current Growth

Equivalent forms

V_L(t) = V_0\, e^{-t/\tau}

I(t) = I_0\, e^{-t/\tau} \;\; \text{(decay)}

τ = L/R mirrors τ = RC with the roles flipped: inductors resist current change as capacitors resist voltage change — the two transients are duals of each other.

Symbol	Name	SI unit	Dimension	Range
$I$	Currentoutput Circuit current at time t	A	I	0 – 24
$V₀$	Source voltage EMF of the battery driving the circuit	V	M·L²·T⁻³·I⁻¹	1 – 24
$R$	Resistance Series resistance	Ω	M·L²·T⁻³·I⁻²	1 – 20
$L$	Inductance Self-inductance of the coil	H	M·L²·T⁻²·I⁻²	0.1 – 10
$t$	Time Time since the switch closed	s	T	0 – 10

Unit systems

Symbol	SI	CGS	Imperial
$I$	A	statA	A
$V₀$	V	statV	V
$R$	Ω	s/cm	Ω
$L$	H	cm	H
$t$	s	s	s

Where it holds

Valid for ideal linear R and L with a step EMF. Breaks down for saturating iron-core inductors, significant winding capacitance at high frequency, or temperature-dependent resistance.

Dimensional analysis

$[\tau ] = [L]/[R] = (V\cdot s$ /A)/(V/A $) = s \checkmark$

Discovery

Joseph Henry · 1832

Henry noticed vivid sparks when breaking circuits wound with long coils — self-inductance announcing itself. The quantitative exponential transient was worked out by Helmholtz and later systematized by Heaviside's operational calculus in the 1880s.

Try this

Why does yanking a cord from a running vacuum cleaner throw a spark — what is the inductor refusing to give up?

A 3 H inductor in series with a 6 Ω resistor is switched onto a 12 V battery. How fast does the current rise, and what is it after one time constant?

Research status: stable

Real-world applications

Flyback diodes protecting transistors that switch relays and motors
Automotive ignition coils generating spark-plug kilovolts
Soft-start circuits limiting inrush current
Switch-mode power supply energy transfer timing

Common misconceptions

Current cannot jump discontinuously through an inductor — but the voltage across it can
Opening an RL circuit forces dI/dt to spike, producing a large voltage — the spark in switches and the reason flyback diodes exist
$\tau = L/R$ , not LR: more resistance makes the transient faster, not slower (opposite of RC)

Experimental verification

Scope the voltage across R in a series RL circuit driven by a square wave: the exponential rise reaches 63.2% of its final value at

t = L/R

, matching component values to within tolerance. A standard first-year electronics lab.

Derivation

Kirchhoff's loop:

V_{0} = IR + L dI/dt

.

This first-order linear ODE with

I(0) = 0

has solution

I(t) = (V_{0}/R)(1 - e^(-Rt/L))

.

The inductor voltage

V_L = L dI/dt = V_{0} e^(-t/\tau )

decays as the current grows.

Limiting cases

t \to 0

⟶

I \to 0

,

V_L \to V_{0}

At switch-on the inductor's back-EMF fully opposes the battery — it behaves like an open circuit.

t \to \infty

⟶

I \to V_{0}/R

Once current is steady there is no dI/dt; the inductor becomes a plain wire and Ohm's law rules.

t = \tau

⟶

I \approx 0.632\,\mathrm{V}_{0}/R

After one time constant the current has reached

(1 - 1/e) \approx 63.2

% of its final value.

What if…

What if R doubles while L stays fixed?

The time constant halves (faster settling) but the final current also halves — you reach a smaller current sooner.

What if the switch is opened after steady state?

The current must decay through whatever path exists. With no path, dI/dt is enormous and $V = L dI/dt$ arcs across the switch contacts — hence flyback diodes.

What if

L \to 0

?

The transient vanishes: current jumps instantly to $V_{0}/R$ . The circuit degenerates to plain Ohm's law.

1

Current after one time constant

Given ·

V₀:: 12
R:: 6
L:: 3

Find ·

\tau and I(\tau )

Steps

$\tau = L/R = 3/6 = 0.5\,\mathrm{s}$
Final current $I\infty = V_{0}/R = 12/6 = 2$ A
$I(\tau ) = 2(1 - e^{-1}) = 2 \times 0.632 = 1.26$ A

Result ·

\tau = 0.5\,\mathrm{s}

,

I(\tau ) \approx 1.26

A

2

Time to reach 1.5 A

Given ·

V₀:: 12
R:: 6
L:: 3
I target:: 1.5

Find · t

Steps

$1.5 = 2(1 - e^(-t/0.5))$
$e^(-2t) = 1 - 0.75 = 0.25$
$-2t = \ln (0.25) = -1.386$ , so $t = 0.693\,\mathrm{s} \approx \tau \cdot \ln 4$

Result ·

t \approx 0.69\,\mathrm{s}

RC Circuit Charging→Energy Stored in an Inductor→Self-Inductance of a Solenoid→LC Oscillation Frequency→