Question

If the cross section area and length of a long solenoid are increased $ 3 $ times, then its self-inductance will be changed by how many times?
(A) $ 1 $
(B) $ 2 $
(C) $ 3 $
(D) $ 4 $

Answer 1

Hint
To solve this problem, we need to first find out how the area of cross section and length of the solenoid are related to self-inductance. Later on, by making the changes in these quantities, we can see by how much the self-inductance changes.
Formula used: Self-inductance of a solenoid, $ L = {\mu _0}{n^2}Al $ , where $ {\mu _0} $ is the permeability of free space, $ n = \dfrac{N}{l} $ is the number turns on the solenoid per unit length, $ N $ is the total number of turns on the solenoid, $ A $ is the cross sectional area and $ l $ is the length of the solenoid.

Complete step by step answer
It is given that the area of cross section and the length of the solenoid are increased three times. But we know that the self-inductance is given by:
 $\Rightarrow L = {\mu _0}{n^2}Al $
But as we know that, $ n = \dfrac{N}{l} $ , where $ N $ is the total number of turns on the solenoid and $ l $ is the length of the solenoid. Therefore, the self-inductance can be again written as:
 $\Rightarrow L = \dfrac{{{\mu _0}{N^2}Al}}{{{l^2}}} = \dfrac{{{\mu _0}{N^2}A}}{l} $
So, when we increase the area and length three times, the new inductance will be
 $\Rightarrow {L_{new}} = \dfrac{{{\mu _0}{N^2}3A}}{{3l}} $ ’
and on cancelling $ 3 $ on the numerator and denominator, we get:
 $\Rightarrow {L_{new}} = \dfrac{{{\mu _0}{N^2}A}}{l} = L $
Hence, we observe that by increasing the length and the area of the cross section by three times, there is no change in the self-inductance.
Therefore, the correct answer is option (A).

Note
Self inductance is the phenomenon of induction of an emf due to the current flowing through a solenoid. This is due to the magnetic field produced due to the current passing through the coil. The magnetic field also produces the magnetic flux in the coil and the rate of change of this flux is the emf as per Faraday's law.