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 $\frac{r}{2}$ current $I$ remaining the same. calculate the ratio of the magnetic moments of the new coil and the original coil.

Answer 1

Step 1: Given data

$\text{Original turns}=n$

$\text{Original radius}=r$

$\text{Original current}=I$

$\text{New radius}=\frac{r}{2}$

$\text{New current}=I$

$\text{New Turns}=2n$

Step 2: To find

The ratio of the magnetic moments of the new coil and the original coil.

Step 3: Calculate the ratio

$\frac{\mu \text{'}}{\mu }=\frac{n\text{'}IA\text{'}}{nIA}$

Here,

$\mu$ and $\mu \text{'}$ is the magnetic moment.

$n$ is the circular coil turns.

$n\text{'}$ is the new circular coil turns.

$I$ is the current.

$A$ is the area of the coil.

$A\text{'}$ is the area of the new coil.

$$\begin{gathered} \frac{\mu \text{'}}{\mu }=\frac{2n\times {\displaystyle \frac{\pi }{4}}\times {\left({\displaystyle \frac{r}{2}}\right)}^2}{n\times I\times {\displaystyle \frac{\pi }{4}}\times r^2} \\ \frac{\mu \text{'}}{\mu }=\frac{1}{2} \end{gathered}$$

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