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

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## 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 $t = 0$.

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.

$\blacksquare$

## Sources

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