Question

An inductive coil has a resistance of $100\Omega $ When an AC signal of frequency $1000Hz$ is applied to the coil, the voltage leads the current by ${45^ \circ }$. The inductance of the coil is
a. $\dfrac{1}{{10\pi }}$
b. $\dfrac{1}{{20\pi }}$
c. $\dfrac{1}{{40\pi }}$
d. $\dfrac{1}{{60\pi }}$

Answer 1

Hint: The property of a component that opposes the change in flow of current through it. If the current in a coil of a wire increases, then they will produce an emf which restricts the current. Here the frequency, resistance of inductance of the coil is given. Using the above data determine the inductance of the coil.

Formula used:
$L = \dfrac{{{X_L}}}{{2\pi f}}$
$L$ is the inductance, $f$ is the frequency.

Complete step by step solution:
If the current in a coil of a wire increases, then they will produce an emf which restricts the current. Then by applying Kirchhoff's voltage law we see the effect of this emf on the circuit equation. Emf opposes the current change and is an example of Lenz's law. The inductance of a coil follows Faraday's law.

Inductor is produced by a self-induced emf due to their changing magnetic field. The effect of emf induced in the same circuit in which the current is changing, this effect is called Self-induction. The property of the coil due to which it opposes the change of current in the other coil is called mutual inductance between two coils. When the current in the other coil or neighboring coil changes, a changing flux emf is induced in the coil. It is called mutual induced emf. The mutual inductance depends on cross sectional area, closeness of two coils and number of turns in the secondary coil.

Permeability of the medium surrounding the coils is directly proportional to mutual inductance. The magnetic field in one of the coils tends to link with each other when the two coils are brought close together. This property of coil changes the current and voltage in the secondary coil.
$L = \dfrac{{{X_L}}}{{2\pi f}}$
$L = \dfrac{{100}}{{2 \times \pi \times 1000}} = \dfrac{1}{{20\pi }}$ Henry.

Hence, the correct answer is option (B).

Note: A galvanometer connected to the coil measures the induced emf. The flux linking with other coils changes when the current flowing through the primary coil is changed to the value of variable resistor $R$. Inductor is produced by a self-induced emf due to their changing magnetic field.