Current in Electric Circuit/L, R in Series

Theorem
Consider the electrical circuit $K$ consisting of:
 * a resistance $R$
 * an inductance $L$

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


 * [[File:CircuitRLseries.png]]

The electric current $I$ in $K$ is given by the linear first order ODE:
 * $L \dfrac {\mathrm d I} {\mathrm d t} + R I = E$

Proof
Let:
 * $E_L$ be the drop in electromotive force across $L$
 * $E_R$ be the drop in electromotive force across $R$

From Kirchhoff's Voltage Law:
 * $E - E_L - E_R = 0$

From Ohm's Law:
 * $E_R = R I$

From Drop in EMF caused by Inductance is proportional to Rate of Change of Current:
 * $E_L = L \dfrac {\mathrm d I} {\mathrm d t}$

Thus:
 * $E - L \dfrac {\mathrm d I} {\mathrm d t} - R I = 0$

which can be rewritten:
 * $L \dfrac {\mathrm d I} {\mathrm d t} + R I = E$