A wire of length $L$ meters carrying a current of $I$ ampere is bent in the form of a circle. What is its magn

Question

A wire of length $L$ meters carrying a current of $I$ ampere is bent in the form of a circle. What is its magnetic moment?

Accepted Answer

1.  **Geometry Constraint:** When a wire of length $L$ is bent into a circular loop of radius $R$, the circumference of the loop equals the length of the wire.
    $$L = 2\pi R \Rightarrow R = \frac{L}{2\pi}$$
2.  **Area Calculation:** The area $A$ of the circular loop is given by:
    $$A = \pi R^2 = \pi \left( \frac{L}{2\pi} ight)^2 = \pi \frac{L^2}{4\pi^2} = \frac{L^2}{4\pi}$$
3.  **Magnetic Moment Formula:** The magnetic moment $M$ of a current-carrying loop is defined as the product of the current $I$ and the area $A$ (for a single turn, $N=1$).
    $$M = I 	imes A$$
4.  **Substitution:** Substituting the expression for area:
    $$M = I 	imes \frac{L^2}{4\pi} = \frac{IL^2}{4\pi}$$
    The unit is Ampere-meter squared (A-m$^2$).

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