Question

The relation between magnetic moment (M) of a current carrying circular coil and length (L) of the wire used is

• A. $M \propto L$
• B. $M \propto \frac{1}{L}$
• C. $M \propto L^2$
• D. $M \propto \frac{1}{L^2}$

Hint:

For a fixed length of wire, a single large circular turn provides the maximum possible magnetic moment.

Answer 1

The Correct Option is C

Step 1: Magnetic Moment Formula
$M = n I A = (1) \cdot I \cdot (\pi r^2)$ for a single turn.
Step 2: Relation with Length
Length of wire $L = 2\pi r \Rightarrow r = \frac{L}{2\pi}$.
Step 3: Substitution
$M = I \cdot \pi \cdot (\frac{L}{2\pi})^2$
$M = I \cdot \pi \cdot \frac{L^2}{4\pi^2} = \frac{I L^2}{4\pi}$
Step 4: Proportionality
Since $I$ is constant, $M \propto L^2$.
Final Answer:(C)