Question

A circular coil of $N$ turns and radius $R$ carries a current $I$. It is unwound and rewound to make another coil of radius $\dfrac{R}{2}$, current $I$ remaining the same. Calculate the ratio of the magnetic moments of the new coil and original coil.

Answer 1

Hint:Here first we have to find the initial magnetic moment of the coil, then the length of the new coil. With this length we can find the magnetic moment of the new coil.Magnetic moment is the magnetic force and position of a magnet or other object that creates a magnetic field.

Complete step by step answer:
Most elementary particles act as magnetic dipoles. The electron itself, for example, behaves like a magnetic dipole and has a Spin Magnetic Dipole moment. This magnetic moment is implicit in the fact that the electron does not have a region A (it is a point object) or spin around it, but is central to the essence of the life of the electron.
Area of the original coil $A = \pi {R^2}$
Magnetic moment of the original coil
$
\mu = N \times I \times A \\
\Rightarrow\mu = N \times I \times \pi {R^2} \\
$
Length of the original coil $L = N \times 2\pi \times R$
Radius of the new coil $R' = \dfrac{R}{2}$
Let the number of turns in the new coil be $N'$ .
According to the question the coil is rewound to make the new coil.
So, the length of the two soils must be equal.
$L' = L$
$
\Rightarrow N' \times 2\pi \times \dfrac{R}{2} = N \times 2\pi \times R \\
\Rightarrow N' = 2N \\
$
So, the area of the new coil is
$\Rightarrow A' = \dfrac{{\pi {R^2}}}{4}$
Therefore, magnetic moment of new coil is:
$
\mu ' = N' \times I \times A' \\
\Rightarrow\mu ' = 2 \times N \times I \times \dfrac{{\pi \times {R^2}}}{4} \\
\Rightarrow\mu ' = \dfrac{{N \times I \times \pi \times {R^2}}}{2} \\
$
$\therefore\dfrac{{\mu '}}{\mu } = \dfrac{1}{2}$

Hence, the ratio of the magnetic moments of the new coil and original coil is $\dfrac{1}{2}$.

Note:Here we have to be careful while calculating the ratio. If we change the values then the ratio would be wrong.Moreover,magnetic moment refers to the moment of a magnetic dipole, a part of a magnetic moment that can be expressed by a magnetic dipole. The magnetic dipole is a magnetic pole of the north and a magnetic pole of the south separated by a slight distance.