Question

The inductive reactance of a coil is $1000\Omega$ . If its self-inductance and frequency both are increased two times, then inductive reactance will be
(A) $1000\Omega$
(B) $2000\Omega$
(C) $4000\Omega$
(D) $16000\Omega$

Answer 1

Hint : The inductive reactance is proportional to the frequency of the alternating current. The inductive reactance is also proportional to the self inductance.

Formula used: In this solution we will be using the following formula;
$\Rightarrow {X_L} = 2\pi fL$ where ${X_L}$ is the inductive reactance, $f$ is the frequency of the alternating current flowing through a coil, and $L$ is the inductance (self inductance in the case where it is caused the ac current through the coil).

Complete step by step answer
In general, when an ac current is flowing through a wire or coil, it generates a changing magnetic field around the coil or wire. Now this magnetic field, due to the faraday’s law of electromagnetic induction induces its own current in the coil which is in the opposite direction as the input alternation current thus, creating an additional resistance in the wire. This resistance is called inductive reactance.
The inductive reactance is given by
$\Rightarrow {X_L} = 2\pi fL$ where ${X_L}$ is the inductive reactance, $f$ is the frequency of the alternating current flowing through a coil, and $L$ is the inductance (self inductance in the case where it is caused the ac current through the coil)
Hence, if we double the frequency and double the self inductance, we have
$\Rightarrow {X_L} = 2\pi (2f)(2L) = 4\left( {2\pi fl} \right)$
Hence, the inductive reactance is 4 times the original value.
For our question, the initial inductive reactance is $1000\Omega$ , then after both the frequency and inductance have been doubled, we will have
$\Rightarrow {X_L} = 4000\Omega$
Hence, the correct option is C.

Note
For clarity, inductance is dependent on the strength of the magnetic field generated due to the alternating current, hence the stronger the magnetic field the higher the inductance. Hence, inductance is increased by increasing the number of turns on a wire which in turn increases the inductive reactance (which is a form of resistance to current). This is why transmission and distribution lines are made to travel around the city in a straight path as possible. Coils in the lines are completely avoided even at joints.