Energy Released by a Discharging Capacitor

A charged capacitor dumps its energy as heat in a resistor, with power that fades exponentially. Adding up that power over time is an integral that needs one substitution, and the answer is exactly the energy that was stored: ½CV².

How it works

The substitution rule

In one sentence

If $u = g(x)$ is differentiable and $f$ is continuous, then $\int f(g(x))g'(x)\,dx = \int f(u)\,du$ , and for definite integrals $\int_a^b f(g(x))g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du$ .

General formula

{\htmlData{sym=0}{E}} = \int_0^{\infty} \frac{{\htmlData{sym=1}{V_0}}^2}{\htmlData{sym=2}{{R}}}e^{-2t/{\htmlData{sym=2}{R}}{\htmlData{sym=3}{C}}}\,dt \;\xrightarrow{\;{\htmlData{sym=4}{u}} = -2t/{\htmlData{sym=2}{R}}{\htmlData{sym=3}{C}}\;}\; \frac{{\htmlData{sym=1}{V_0}}^2}{\htmlData{sym=2}{{R}}}\cdot\frac{{\htmlData{sym=2}{R}}{\htmlData{sym=3}{C}}}{2} = \frac12{\htmlData{sym=3}{C}}{\htmlData{sym=1}{V_0}}^2

Substitution renames the inside of a chained function as one variable. The $dx = \frac{du}{g'(x)}$ factor accounts for how fast the new variable runs.

What the symbols mean

$E$
Energy released
Total heat produced in the resistor.
$V_0$
Starting voltage
The capacitor's voltage before it discharges.
$R$
Resistance
Where the energy turns into heat.
$C$
Capacitance
How much charge the capacitor holds per volt.
$u$
Substitution
u = −2t/RC, which turns the integrand into a plain exponential.

While discharging, $V = V_0e^{-t/RC}$ , so the resistor's power is $V^2/R = \frac{V_0^2}{R}e^{-2t/RC}$ .
Substitute $u = -2t/RC$ , so $dt = -\frac{RC}{2}du$ : the integrand becomes a plain exponential.
Integrate $e^u$ and change the limits ( $t = 0 \to u = 0$ , $t \to \infty \to u \to -\infty$ ).
The result, $\frac12CV_0^2$ , is the energy the capacitor stored: none is lost.

Learn the math

The Substitution RuleCalc I · Lesson 42, Integrals More SubstitutionCalc I · Lesson 43, Integrals Substitution with Definite IntegralsCalc I · Lesson 47, Integrals Capacitance and Stored EnergyE&M · Lesson 11, Electric Potential RC CircuitsE&M · Lesson 15, Current and DC Circuits