A uniform conducting wire of length $12a$ and resistance $R$ is wound up as a current-carrying coil in the sha

Question

A uniform conducting wire of length $12a$ and resistance $R$ is wound up as a current-carrying coil in the shape of:
(i) an equilateral triangle of side $a$
(ii) a square of side $a$
The magnetic dipole moments of the coil in each case respectively are:

Accepted Answer

1.  **Identify Formula:** The magnetic dipole moment ($M$) of a current-carrying 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 loop.
2.  **Case (i) Equilateral Triangle:**
    *   Total length of wire $L = 12a$.
    *   Perimeter of one turn (equilateral triangle of side $a$) = $3a$.
    *   Number of turns $N_1 = \frac{L}{	ext{Perimeter}} = \frac{12a}{3a} = 4$.
    *   Area of equilateral triangle $A_1 = \frac{\sqrt{3}}{4} a^2$.
    *   Magnetic Moment $M_1 = N_1 I A_1 = 4 	imes I 	imes \frac{\sqrt{3}}{4} a^2 = \sqrt{3} I a^2$.
3.  **Case (ii) Square:**
    *   Total length of wire $L = 12a$.
    *   Perimeter of one turn (square of side $a$) = $4a$.
    *   Number of turns $N_2 = \frac{12a}{4a} = 3$.
    *   Area of square $A_2 = a^2$.
    *   Magnetic Moment $M_2 = N_2 I A_2 = 3 	imes I 	imes a^2 = 3 I a^2$.
4.  **Conclusion:** The magnetic moments are $\sqrt{3} I a^2$ and $3 I a^2$ respectively.

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