Question

In an AC circuit, a resistance of $R$ ohm is connected in series with an inductance $L$. If the phase angle between voltage and current be ${45^ \circ }$, the value of inductive reactance will be
(A) $\dfrac{R}{4}$
(B) $\dfrac{R}{2}$
(C) $R$
(D) cannot be found with the given data.

Answer 1

The given problem can be solved using the formula formulated for the inductive reactance in an A.C. circuit from the phase angle between voltage and current and the resistance and the inductance of the resistor and the inductor in an A.C. circuit.

Formula used:
The formula to find the value of the inductive reactance is given as;
$\tan \,\theta = \dfrac{{\omega L}}{R}$
Where, $\theta$ denotes the phase angle between voltage and current in the A.C. circuit, $L$ denotes the inductance of the inductor in the A.C. circuit, $R$ denotes the resistance of the resistor in the A.C. circuit.

Complete step by step answer:
The data given in the problem is;
The resistance of the resistor in the A.C. circuit is,$R$.
The inductance of the inductor in the A.C. circuit is, $L$.
phase angle between voltage and current is, $\theta = {45^ \circ }$.
The formula to find the value of the inductive reactance is given as;
$\tan \,\theta = \dfrac{{\omega L}}{R}\,\,..........\left( 1 \right)$
Since $\omega L = {X_L}$
Where, ${X_L}$ denotes the inductive reactance in an A.C. circuit.
Substitute the values of $\omega L$ and the value of $\theta$ in the equation (1);
$\Rightarrow \tan \left( {{{45}^ \circ }} \right) = \dfrac{{{X_L}}}{R}$
Since, $\tan \left( {{{45}^ \circ }} \right) = 1$
$\Rightarrow \dfrac{{{X_L}}}{R} = 1$
Since we only need inductive reactance in an A.C. circuit;
$\Rightarrow {X_L} = R$

Therefore, the value of inductive reactance will be ${X_L} = R$. Hence the option (C) $R$ is the correct answer.

Note:
Inductive reactance is the name assigned to the opposition to changing current flow. The impedance of inductive reactance is measured in ohms, just like the resistance of the resistor. In inductors, voltage leads current by ninety degrees.