Current in Electric Circuit/L, R in Series/Constant EMF at t = 0/Corollary 2

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 $0$.

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:

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

Proof

From Electric Current in Electric Circuit: L, R in Series: Constant EMF at $t = 0$:

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

Setting $I_0 = 0$:

\(\ds I\)

\(=\)

\(\ds \dfrac {E_0} R + \paren {0 - \dfrac {E_0} R} e^{-R t / L}\)

\(\ds \)

\(=\)

\(\ds \dfrac {E_0} R \paren {1 - e^{-R t / L} }\)

Hence the result.

$\blacksquare$

Sources

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