Question

When an AC signal of frequency 1kHz is applied across a coil of resistance 100Ω, then the applied voltage leads the current by ${45^0}$. The inductance of the coil is:

A. $16mH$
B. $12mH$
C. $8mH$
D. $4mH$

Answer 1

Hint: In case of DC circuit if there is resistor we can measure using an ammeter but if there are inductors in the AC circuit we can’t measure the obstruction for the flow of current hence there comes a term inductive reactance and it depends on source angular frequency too.

Formula used:
$\eqalign{
  & {X_L} = \omega L \cr
  & \tan \phi = \dfrac{{{X_L}}}{R} \cr} $

Complete step by step answer:
In case alternating currents there will be angular velocity of a wave and we can find frequency of an alternating wave from that. That frequency determines the capacitive reactance and inductive reactance and total impedance of the AC circuit. If a capacitor is present in the AC circuit then we include the capacitive reactance and if not then only resistance and inductor reactance determines the impedance.
Normally AC voltages will be in the form $V = {V_0}\sin (\omega t)$ where ${V_0}$ is the peak voltage while $\omega $ is the angular frequency.
In AC circuits there will be some phase difference between the current and the potential in the circuit. That phase difference is given as ${45^0}$.
So in the given circuit we have the inductor and the resistor, hence we have the inductive reactance (${X_L}$) and the resistance(R) which are related to the phase difference as
$\tan \phi = \dfrac{{{X_L}}}{R}$
$\eqalign{
  & \Rightarrow \tan \phi = \dfrac{{\omega L}}{R} \cr
  & \Rightarrow \tan {45^0} = \dfrac{{2\pi fL}}{R} \cr
  & \Rightarrow 2\pi fL = R \cr
  & \Rightarrow L = \dfrac{R}{{2\pi f}} \cr
  & \Rightarrow L = \dfrac{{100}}{{2\pi \times {{10}^3}}} \cr
  & \therefore L = 16mH \cr} $
Hence option A will be the answer.

Note:
In order to remember whether current leads voltage or voltage leads current one should remember the terms IPL and CCL. IPL means in the inductive circuit potential leads the current and CCL means in capacitive circuit current leads the potential. While current and voltage across the resistor will be in phase always. Resonance occurs when inductive reactance equals the capacitive reactance.