RC Time Constant

Equivalent forms

V(t) = V_0\left(1 - e^{-t/\tau}\right)

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

Two component values multiply into a single number with units of seconds that governs the entire transient.

Symbol	Name	SI unit	Dimension	Range
$\tau$	Time constantoutput Characteristic charge/discharge time	s	T	0.000001 – 100
$R$	Resistance Series resistance	Ω	M·L²·T⁻³·I⁻²	1 – 10000000
$C$	Capacitance Capacitance being charged	F	M⁻¹·L⁻²·T⁴·I²	1e-12 – 0.001

Unit systems

Symbol	SI
$\tau$	s
$R$	Ω
$C$	F

Where it holds

Exact for a single ideal R and C in series with a step source. Multiple time constants, dielectric absorption, and ESR/ESL of real capacitors cause deviations from the clean single exponential.

Dimensional analysis

$[R][C] = \Omega \cdot F = (V$ /A $)\cdot (C/V) = C$ / $A = s \checkmark$

Discovery

Hermann von Helmholtz / Lord Kelvin · 1853

Kelvin's analysis of signaling on the first transatlantic telegraph cable revealed the RC charging law that limited its speed — the founding insight of the time constant in real systems.

Try this

A blinking turn signal, a camera flash recharging, a defibrillator's whine — all keep time with the same exponential clock. What sets its pace?

A 10 μF capacitor charges through a 100 kΩ resistor. How long until it reaches 63% of the supply voltage, and how long to be 'fully' charged?

Research status: stable

Real-world applications

555-timer and oscillator periods
Camera-flash and defibrillator charging
Debounce and RC filters
Sample-and-hold settling

Common misconceptions

The capacitor is 'fully charged' after $one \tau$ — it's only at 63%; $5\tau$ reaches >99%
Larger R charges faster — larger R slows charging by limiting current

Experimental verification

Scope the capacitor voltage during a step input; the time to reach 63.2% of final reads

\tau

directly, matching

R\times C

across decades of values.

Derivation

KVL on a charging RC:

V_0 = IR + Q/C

with

I = dQ/dt

.

Substituting gives

R dQ/dt + Q/C = V_0

, a first-order linear ODE whose solution is

Q(t) = CV_0(1 - e^{-t/RC})

.

The exponential's time constant

is \tau = RC

;

V(t) = V_0(1 - e^{-t/\tau })

.

Limiting cases

R \to 0

⟶

\tau \to 0

No resistance means an unlimited charging current — the capacitor charges instantly (and dangerously).

C \to \infty

⟶

\tau \to \infty

A huge reservoir never finishes filling through a fixed resistor on human timescales.

What if…

What if you halve the resistor?

$\tau$ halves — the capacitor charges and discharges twice as fast.

What if you use this RC to set a timer?

Adjusting R or C tunes the period; that is exactly how 555 timers and RC oscillators keep time.

1

100 kΩ charging 10 μF

Given ·

R:: 100000
C:: 0.00001

Find ·

\tau and

time

to \sim

full charge

Steps

$\tau = RC = (10^{5})(10^{-5}) = 1\,\mathrm{s}$
After $1\tau the$ voltage reaches 63.2% of V_0
After $5\tau \approx 5\,\mathrm{s}$ it is >99% charged — effectively full

Result ·

\tau = 1\,\mathrm{s}

; 63% in 1 s, essentially full

in \sim 5\,\mathrm{s}

.

Capacitance→rc dischargingenergy in capacitorlow pass filter