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

Theorem

Consider the electric 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 $t$.

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:

$I = \dfrac {E_0} R + \paren {I_0 - \dfrac {E_0} R} e^{-R t / L}$

Corollary 1

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 $t = 0$.

After a sufficiently long time, the electric current $I$ in $K$ is given by the equation:

$E_0 = R I$

Corollary 2

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

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:

$I = \dfrac {E_0} R \paren {1 - e^{-R t / L} }$

Corollary 3

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

Let EMF imposed upon $K$ be zero.

The electric current $I$ in $K$ is given by the equation:

$I = I_0 e^{-R t / L}$

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:

$I = \dfrac {E_0} R + \paren {I_0 - \dfrac {E_0} R} e^{-R t / L}$

$\blacksquare$

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.13$: Simple Electric Circuits: Example $1$