Question

A coil of wire of a certain radius has 100 turns and a self-inductance of 15mH. The self inductance of a second similar coil of 500 turns will be
a)75mH
b)375mH
c)15mH
d)none of these

Answer 1

In the above question let us consider the coil be a solenoid. The self inductance of the coil of 100 turns is given to us as 15mH. Hence using the equation of the self inductance we will determine the other parameters of the coil. It is also given to us that the second coil is similar to the first coil means its geometry and condition in which the coil exists is similar to the first. Hence further we will again use the equation of the inductance of the coil to determine the self-inductance of the second coil with 500 turns.
Formula used:
$L={{\mu }_{o}}{{n}^{2}}Al$

Complete answer:
Let us consider a coil of ‘n’ turns similar to a solenoid. Let the length of the coil be ‘l’ and its area of cross section be ‘A’. Then the self inductance (L)of the coil is given by,
$L={{\mu }_{o}}{{n}^{2}}Al$
It is given to us that the first coil has a self inductance of 15mH and 100 turns. Hence from the above equation we get,
$\begin{aligned} & L={{\mu }_{o}}{{n}^{2}}Al \\\ & \Rightarrow 15mH={{\mu }_{o}}{{(100)}^{2}}Al \\\ & \Rightarrow {{\mu }_{o}}Al=15\times {{10}^{-7}} \\\ \end{aligned}$
Further it is given to us that the two coils are similar. Hence the self inductance of the second coil with 500 turns we get as,
$\begin{aligned} & L={{\mu }_{o}}{{n}^{2}}Al \\\ & \Rightarrow L={{\mu }_{o}}{{(500)}^{2}}Al\text{, }\because {{\mu }_{o}}Al=15\times {{10}^{-7}} \\\ & \Rightarrow L={{(500)}^{2}}15\times {{10}^{-7}} \\\ & \Rightarrow L=250000\times 15\times {{10}^{-7}}=625\times {{10}^{-3}}=625mH \\\ \end{aligned}$

Hence the correct answer of the above question is option d.

Note:
It is to be noted that we have considered the coil to be a solenoid. We can actually use any expression for self inductance of a coil as all the expressions are proportional to square of the number of turns of the coil. If we see the above expression for self inductance we can conclude that it does not depend on the amount of current flowing through it, but is related to the geometry of the coil.