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(D) The magnetic moment $M$ of a circular coil is given by the formula $M = NIA$, where $N$ is the number of turns, $I$ is the current, and $A$ is the

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$1: \sqrt{3}$

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(D) The magnetic moment $M$ of a circular coil is given by the formula $M = NIA$, where $N$ is the number of turns, $I$ is the current, and $A$ is the area of the coil.Since the coil is circular, the area $A = \pi r^2$, where $r$ is the radius of the coil.Thus, $M = NI(\pi r^2)$.Given for coil $A$: $N_A = 300$, $M_A = M_1$, $r_A = r_1$.Given for coil $B$: $N_B = 200$, $M_B = M_2$, $r_B = r_2$.We are given $M_A / M_B = 1/2$ and $I_A = I_B = I$.Taking the ratio of the magnetic moments:$\frac{M_A}{M_B} = \frac{N_A I_A \pi r_A^2}{N_B I_B \pi r_B^2} = \frac{N_A}{N_B} \cdot \left(\frac{r_A}{r_B}\right)^2$.Substituting the known values:$\frac{1}{2} = \frac{300}{200} \cdot \left(\frac{r_A}{r_B}\right)^2$.$\frac{1}{2} = \frac{3}{2} \cdot \left(\frac{r_A}{r_B}\right)^2$.$\left(\frac{r_A}{r_B}\right)^2 = \frac{1}{2} \cdot \frac{2}{3} = \frac{1}{3}$.Therefore, $\frac{r_A}{r_B} = \frac{1}{\sqrt{3}}$.

The number of turns of two circular coils $A$ and $B$ are $300$ and $200$ respectively. The magnetic moments of the two coils $A$ and $B$ are in the ratio $1:2$. If the two coils carry equal currents, then the ratio of radii of coils $A$ and $B$ is

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