A 2 amp current is flowing through two different small circular copper coils having radii ratio 1 : 2. The ratio of their respective magnetic moments will be

For a coil, the magnetic moment $\mu$ is given by:

$\mu =NIA$

where $N$ is the number of turns, $I$ the current, and $A$ the area of the coil.

Both coils are single‑turn (the problem states “small circular copper coils” without mentioning multiple turns), so $N=\; 1$ for each. The current in each coil is the same, $2A$, so the factor $I$ cancels when taking a ratio.

The area of a circular coil is $A=\; \pi \; r²$. Let the radii be $\mathrm{r₁}$ and $\mathrm{r₂}$ with the given ratio:

$\mathrm{r₁}:\mathrm{r₂}=\; 1\; :\; 2$

Thus the area ratio is:

$\mathrm{A₁}:\mathrm{A₂}=\; \pi \; r₁²\; :\; \pi \; r₂²\; =\; r₁²\; :\; r₂²\; =\; 1²\; :\; 2²\; =\; 1\; :\; 4$

Since the magnetic moment is proportional to the area (with equal current), the ratio of magnetic moments is the same as the area ratio:

$\mathrm{\mu ₁}:\mathrm{\mu ₂}=\; 1\; :\; 4$

Therefore, the correct ratio of the magnetic moments of the two coils is 1 : 4.