Current in Electric Circuit/L, R in Series/Constant EMF at t = 0/Corollary 1
Jump to navigation
Jump to search
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 $1$