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

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 $t$.


 * [[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 $t = 0$.

After a sufficiently long time, the electric current $I$ in $K$ is given by the equation:
 * $E_0 = R I$

Proof
From Electric Current in Electric Circuit: L, R in Series: Constant EMF at $t = 0$:
 * $I = \dfrac {E_0} R + \paren {I_0 - \dfrac {E_0} R} e^{-R t / L}$

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.