Current in Electric Circuit/L, R in Series/Constant EMF at t = 0/Corollary 1

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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 $t = 0$.

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

$E_0 = R I$


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

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

We have that:

$\ds \lim_{t \mathop \to \infty} e^{-R t / L} \to 0$

and so:

$\ds \lim_{t \mathop \to \infty} I \to \dfrac {E_0} R$

Hence the result.