Step 1: Understanding the Question:
A wire of length $L$ is bent into a circular coil of $N$ turns. We need to find how the magnetic moment $M$ depends on $L$.
Step 2: Key Formula or Approach:
1. Magnetic moment $M = N I A$, where $A = \pi r^2$.
2. Length of wire $L = N(2\pi r)$.
Step 3: Detailed Explanation:
From the length equation, radius $r = \frac{L}{2\pi N}$.
Substitute $r$ into the area formula:
\[ A = \pi r^2 = \pi \left(\frac{L}{2\pi N}\right)^2 = \frac{L^2}{4\pi N^2} \]
Now, substitute $A$ into the magnetic moment formula:
\[ M = N I A = N I \left(\frac{L^2}{4\pi N^2}\right) \]
\[ M = \frac{I L^2}{4\pi N} \]
For a fixed number of turns $N$ and current $I$:
\[ M \propto L^2 \]
Step 4: Final Answer:
The relation is $M \propto L^2$.