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

Theorem
Consider the electrical 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 $t$.


 * [[File:CircuitRLseries.png]]

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.