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.

Step-by-Step Explanation:

The magnetic moment ($M$) of a current-carrying circular coil is defined as the product of the current flowing through it ($I$) and the area enclosed by the loop ($A$):

$M=IA$

For a circular coil of radius $r$, the area is given by:

$A=\pi r^2$

Substituting the area formula into the magnetic moment equation gives:

$M=I\pi r^2$

Since the same current of 2 A flows through both coils, the current $I$ remains constant. Therefore, the magnetic moment of the coils is directly proportional to the square of their radii:

$M\propto r^2$

Thus, the ratio of their magnetic moments ($M_1:M_2$) can be written as:

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

Given that the ratio of the radii of the two coils is $r_1:r_2=1:2$, we substitute this value into the equation:

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

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