Current in Electric Circuit/L, R in Series/Condition for Ohm's Law

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 $t$.

Ohm's Law is satisfied by $K$ whenever the current $I$ is at a maximum or a minimum.

Proof

From Electric Current in Electric Circuit: L, R in Series:

$L \dfrac {\d I} {\d t} + R I = E$

defines the behaviour of $I$.

Let $I$ be at a maximum or a minimum.

Then from Derivative at Maximum or Minimum:

$\dfrac {\d I} {\d t} = 0$

and so:

\(\ds E\)

\(=\)

\(\ds 0 + R I\)

\(\ds \)

\(=\)

\(\ds R I\)

which is Ohm's Law.

$\blacksquare$

Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.13$: Simple Electric Circuits: Problem $2 \ \text{(a)}$