Question

When an a.c. source is connected across an ideal inductor, show on a graph the nature of variation

of the voltage and the current over one complete cycle.​

Answer 1

Given:

An ideal inductor has been connected to an alternating current source in a circuit.

To find:

Graph representing the nature of variation of voltage and current over one complete cycle.

Calculation:

As per the potential difference source in the circuit the voltage can be represented as:

$\therefore \: V= V_{0} \: \sin( \omega t)$

Now, we know that:

$\therefore \: V = - L \times \dfrac{di}{dt}$

$\implies \: di = - \dfrac{1}{L} \times V \: dt$

$\displaystyle \implies \: \int di = - \dfrac{1}{L} \times \int V \: dt$

$\displaystyle \implies \: \int di = - \dfrac{1}{L} \times V_{0} \int \sin( \omega t) \: dt$

$\displaystyle \implies \: \int di = - \dfrac{V_{0}}{L} \int \sin( \omega t) \: dt$

$\displaystyle \implies \: i = - \dfrac{V_{0}}{ \omega L} \times \{ - \cos( \omega t) \}$

$\displaystyle \implies \: i = \dfrac{V_{0}}{ \omega L} \times \cos( \omega t)$

$\displaystyle \implies \: i = \dfrac{V_{0}}{ \omega L} \times \sin(\omega t-\frac{\pi}{2})$

So, current lags from voltage by phase π/2 (refer to graph attached).