Current in Electric Circuit/L, R in Series/Constant EMF at t = 0

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]]

Let the electric current flowing in $K$ at time $t = 0$ be $I_0$.

Let a constant EMF $E_0$ be imposed upon $K$ at time $t = 0$.

The electric current $I$ in $K$ is given by the equation:
 * $I = \dfrac {E_0} R + \paren {I_0 - \dfrac {E_0} R} e^{-R t / L}$

Proof
From Electric Current in Electric Circuit: L, R in Series:
 * $L \dfrac {\d I} {\d t} + R I = E_0$

defines the behaviour of $I$.

This can be written as:
 * $(1): \quad \dfrac {\d I} {\d t} = \dfrac R L \paren {\dfrac {E_0} R - I}$

$(1)$ is in the form:
 * $\dfrac {\d y} {\d x} = k \paren {y_a - y}$

where:
 * $k \in \R: k > 0$
 * $y = y_0$ at $x = 0$

This is an example of the Decay Equation, where:
 * $k = \dfrac R L$
 * $y_a = \dfrac {E_0} R$
 * $y_0 = I_0$.

whose particular solution is:
 * $y = y_a + \paren {y_0 - y_a} e^{-k x}$

Hence the particular solution to $(1)$ is:
 * $I = \dfrac {E_0} R + \paren {I_0 - \dfrac {E_0} R} e^{-R t / L}$