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

The correct option is 1:4.

To understand why, let us break down the magnetic moment of a current-carrying loop step-by-step:

Step 1: Understand the formula for magnetic moment
The magnetic moment ($M$) of a circular coil carrying current is directly proportional to the current flowing through it and the area enclosed by the loop. Mathematically, it is given by:

$M=N·I·A$

where:
- $N$ is the number of turns in the coil (assumed to be 1 for each coil, or equal for both),
- $I$ is the electric current flowing through the coil,
- $A$ is the area of the circular coil.

Step 2: Relate area to the radius
Since the coils are circular, the area $A$ of a coil with radius $r$ is:

$A=\pi ·r^2$

Substituting this into the magnetic moment equation gives:

$M=N·I·\pi ·r^2$

Step 3: Analyze the given values and ratios
Both coils carry the same current of $I=2\text{A}$. Since the current $I$, number of turns $N$, and $\pi$ are constant for both coils, the magnetic moment $M$ is directly proportional to the square of the radius:

$M\propto r^2$

Therefore, the ratio of their magnetic moments is:

$\frac{M_1}{M_2}={\left(\frac{r_1}{r_2}\right)}^2$

Step 4: Calculate the final ratio
We are given the ratio of their radii as:

$\frac{r_1}{r_2}=\frac{1}{2}$

Substituting this value into the ratio of the magnetic moments:

$\frac{M_1}{M_2}={\left(\frac{1}{2}\right)}^2=\frac{1}{4}$

Thus, the ratio of their respective magnetic moments is 1:4.