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.

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Solution

## Step 1: Given data$\text{Originalturns}=n$$\text{Originalradius}=r$$\text{Originalcurrent}=I$$\text{Newradius}=\frac{r}{2}$$\text{Newcurrent}=I$$\text{NewTurns}=2n$Step 2: To findThe ratio of the magnetic moments of the new coil and the original coil.Step 3: Calculate the ratio$\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.$\frac{\mu \text{'}}{\mu }=\frac{2n×\frac{\pi }{4}×{\left(\frac{r}{2}\right)}^{2}}{n×I×\frac{\pi }{4}×{r}^{2}}\phantom{\rule{0ex}{0ex}}\frac{\mu \text{'}}{\mu }=\frac{1}{2}$Hence, the ratio of the magnetic moments of the new coil and the original coil is $\frac{1}{2}$.

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