Current in Electric Circuit/L, R in Series/Minimum Current implies Increasing EMF
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 current $I$ be at a minimum.
Then the EMF $E$ is increasing.
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:
- $L \dfrac {\d^2 I} {\d t^2} \ge 0$
while from Derivative at Maximum or Minimum:
- $R \dfrac {\d I} {\d t} = 0$
and so:
- $\dfrac {\d E} {\d t} \ge 0$
The result follows from Increasing Function has Positive Derivative.
$\blacksquare$
Sources
- 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $\S 2.13$: Simple Electric Circuits: Problem $2 \ \text{(b)}$