Current in Electric Circuit/L, R in Series/Maximum Current implies Decreasing EMF

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 current $I$ be at a maximum.

Then the EMF $E$ is decreasing.

Proof

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

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

defines the behaviour of $I$.

Taking the derivative:

$L \dfrac {\d^2 I} {\d t^2} + R \dfrac {\d I} {\d t} = \dfrac {\d E} {\d t}$

From Second Derivative of Real Function at Minimum:

$L \dfrac {\d^2 I} {\d t^2} \le 0$

while from Derivative at Maximum or Minimum:

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

and so:

$\dfrac {\d E} {\d t} \le 0$

The result follows from Decreasing Function has Negative Derivative.

$\blacksquare$

Sources

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