Current in Electric Circuit/L, R in Series/Condition for Ohm's Law
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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 $t$.
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 $2 \ \text{(a)}$