Electric Charge in Electric Circuit/L, R, C in Series

Theorem

Consider the electric circuit $K$ consisting of:

a resistance $R$

an inductance $L$

a capacitance $C$

in series with a source of electromotive force $E$ which is a function of time $t$.

The electric charge $Q$ in $K$ is given by the linear second order ODE:

$L \dfrac {\d^2 Q} {\d t^2} + R \dfrac {\d Q} {\d t} + \dfrac 1 C Q = E$

Proof

Let:

$E_L$ be the drop in electromotive force across $L$

$E_R$ be the drop in electromotive force across $R$

$E_C$ be the drop in electromotive force across $C$.

From Kirchhoff's Voltage Law:

$E - E_L - E_R - E_C = 0$

From Ohm's Law:

$E_R = R I$

From Drop in EMF caused by Inductance is proportional to Rate of Change of Current:

$E_L = L \dfrac {\d I} {\d t}$

From Drop in EMF caused by Capacitance is proportional to Accumulated Charge:

$E_C = \dfrac 1 C Q$

where $Q$ is the electric charge $Q$ that has accumulated on $C$.

Thus:

$E - L \dfrac {\d I} {\d t} - R I - \dfrac 1 C Q = 0$

which can be rewritten:

$L \dfrac {\d I} {\d t} + R I + \dfrac 1 C Q = E$

By definition of electric current:

$I = \dfrac {\d Q} {\d t}$

and so:

$L \dfrac {\d^2 Q} {\d t^2} + R \dfrac {\d Q} {\d t} + \dfrac 1 C Q = E$

$\blacksquare$

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.13$: Simple Electric Circuits