The ratio of the radii of two circular coils is 1:2. The ratio of currents in the respective coils such that t

Question

The ratio of the radii of two circular coils is 1:2. The ratio of currents in the respective coils such that the same magnetic moment is produced at the centre of each coil is:

Accepted Answer

1.  **Identify Formula:** The magnetic dipole moment ($M$) of a current-carrying circular coil is given by $M = N I A$, where $N$ is the number of turns, $I$ is the current, and $A$ is the area of the coil.
2.  **Analyze Area:** The area of a circle is $A = \pi r^2$. Thus, $M = N I (π r^2)$.
3.  **Set Condition:** The problem states that the magnetic moments are the same ($M_1 = M_2$) and implies single-turn coils or equal turns ($N_1 = N_2$) as specific numbers aren't given.
    $$I_1 \pi r_1^2 = I_2 \pi r_2^2$$
4.  **Calculate Ratio:** Rearrange to find the ratio of currents ($I_1 / I_2$):
    $$\frac{I_1}{I_2} = \frac{r_2^2}{r_1^2} = \left( \frac{r_2}{r_1} \right)^2$$
5.  **Substitute Values:** Given the ratio of radii $r_1 : r_2 = 1 : 2$, implies $r_2 / r_1 = 2 / 1 = 2$.
    $$\frac{I_1}{I_2} = (2)^2 = 4$$
6.  **Conclusion:** The ratio of the currents is 4:1.

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