Charging a Capacitor

Close the switch and current rushes into the capacitor, but the fuller it gets, the less the supply can push. The voltage rises fast, then creeps toward the supply. An integrating factor solves it exactly.

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How it works

Linear first-order equations

In one sentence

A first-order linear equation $y' + p(t)y = q(t)$ is solved by multiplying by the integrating factor $\mu = e^{\int p\,dt}$ , which turns the left side into $(\mu y)'$ ; here $p = 1/RC$ and $q = V_s/RC$ .

General formula

{\htmlData{sym=0}{R}}{\htmlData{sym=1}{C}}\,\frac{d{\htmlData{sym=2}{V}}}{dt} + {\htmlData{sym=2}{V}} = {\htmlData{sym=3}{V_s}} \quad\Rightarrow\quad {\htmlData{sym=2}{V}}(t) = {\htmlData{sym=3}{V_s}} + \left({\htmlData{sym=4}{V_0}} - {\htmlData{sym=3}{V_s}}\right)e^{-t/{\htmlData{sym=0}{R}}{\htmlData{sym=1}{C}}}

Current flows in proportion to the gap $V_s - V$ , and current is what charges the capacitor, so the gap closes at a rate proportional to itself: the gap decays exponentially.

What the symbols mean

$R$
Resistance
How strongly the resistor limits current, in ohms.
$C$
Capacitance
How much charge the capacitor stores per volt, in farads.
$V$
Capacitor voltage
The unknown function: the voltage across the capacitor at time t.
$V_s$
Supply voltage
The battery or supply the capacitor charges toward.
$V_0$
Starting voltage
The capacitor's voltage when the switch closes.

Kirchhoff's voltage law: the supply $V_s$ equals the drop across the resistor, $RC\,V'$ , plus the capacitor's voltage $V$ .
It is linear in $V$ . Multiply by the integrating factor $e^{t/RC}$ and the left side becomes $\frac{d}{dt}\left(e^{t/RC}V\right)$ .
Integrate and solve: $V(t) = V_s + (V_0 - V_s)e^{-t/RC}$ . The gap to $V_s$ shrinks to 37% every $RC$ seconds.
After about $5RC$ the capacitor is more than 99% charged.

Learn the math

Linear EquationsDiff Eq · Lesson 4, First Order Differential Equations Capacitance and Stored EnergyE&M · Lesson 11, Electric Potential RC CircuitsE&M · Lesson 15, Current and DC Circuits Inductance and RL CircuitsE&M · Lesson 21, Electromagnetic Induction The Displacement CurrentE&M · Lesson 26, Maxwell's Equations and Light From Signals to Bits: Sampling and QuantizationDiscrete · Lesson 36, The Math Inside Computers

Learn the math

Linear EquationsDiff Eq · Lesson 4, First Order Differential Equations

Capacitance and Stored EnergyE&M · Lesson 11, Electric Potential

RC CircuitsE&M · Lesson 15, Current and DC Circuits

Inductance and RL CircuitsE&M · Lesson 21, Electromagnetic Induction

The Displacement CurrentE&M · Lesson 26, Maxwell's Equations and Light

From Signals to Bits: Sampling and QuantizationDiscrete · Lesson 36, The Math Inside Computers