Question

A circular coil having ‘N’ turns and diameter ‘d’ carries a current ‘I’. It is unwound and then rewound to make another coil of diameter ‘2d’ current ‘I’ remaining the same. Calculate the ratio of magnetic moments of the new coil and the original coil.
$\begin{align}
  & (A)2:1 \\
 & (B)4:3 \\
 & (C)3:4 \\
 & (D)3:1 \\
\end{align}$

Answer 1

Hint: Since, the same coil is unwound and wound together back again, the total length of the coil initially and afterwards remains the same. We will use this to calculate the new number of turns of the coil. Also, the magnetic moment of a coil is given by the product of, its current, total number of turns in the coil and the cross-sectional area of the coil.

Complete answer:
It has been given in the question, the initial and final parameters of the solenoid as:
Initial number of turns in the coil: N
Initial diameter of the coil: d
Initial current through each coil: I
Now, the coil has been unwound and rewound again, therefore its length will remain the same. Also, the diameter of the new coil is doubled and current is given to be the same as before. The number of turns can be calculated as follows:
Let the number of turns in the new coil be given by ${{N}_{0}}$ .
$\Rightarrow N(\pi d)={{N}_{0}}(\pi .2d)$
$\Rightarrow {{N}_{0}}=\dfrac{N}{2}$
Now, let the magnetic moment of the coil initially be denoted by $M$ and afterwards be denoted by ${{M}_{0}}$ .
So, the magnetic moment (say $M$) of a solenoid will be given by the formula:
$\Rightarrow M=INA$
Thus, the ratio of magnetic moments will be equal to:
$\Rightarrow \dfrac{{{M}_{0}}}{M}=\dfrac{I{{N}_{0}}{{A}_{0}}}{INA}$
Putting the values of all the respective terms in right-hand side of the equation, we get:
$\begin{align}
  & \Rightarrow \dfrac{{{M}_{0}}}{M}=\dfrac{I\dfrac{N}{2}[\pi {{(2d)}^{2}}]}{IN[\pi {{(d)}^{2}}]} \\
 & \Rightarrow \dfrac{{{M}_{0}}}{M}=\dfrac{4\pi {{d}^{2}}IN}{2\pi {{d}^{2}}IN} \\
 & \therefore \dfrac{{{M}_{0}}}{M}=\dfrac{2}{1} \\
\end{align}$
Hence, the ratio of magnetic moment of the new coil to the old coil comes out to be 2:1 .

Hence, option (A) is the correct option.

Note:
We have worked under the assumption that the coil is sufficiently long and ideal. Only under these assumptions most of our formulas can work properly. Also, as these are questions of ratio and proportionality, one should be very careful in writing the ratios, as a:b could be mistakenly written as b:a and this would make our whole solution incorrect.