The magnetic moment of a current I carrying circular coil of radius r and number of turns N varies as

Options

A. `1/(r^(2))`
B. `1/r`
C. r
D. `r^(2)`

Explanation

To find how the magnetic moment \( m \) of a current-carrying circular coil varies with the radius \( r \), number of turns \( N \), and current \( I \), we can follow these steps:

### Step 1: Understand the Formula for Magnetic Moment
The magnetic moment \( m \) of a circular coil is given by the formula:
\[ m = N \cdot I \cdot A \]
where:
- \( N \) is the number of turns,
- \( I \) is the current flowing through the coil,
- \( A \) is the area of the coil.

### Step 2: Calculate the Area of the Coil
The area \( A \) of a circular coil with radius \( r \) is calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]

### Step 3: Substitute the Area into the Magnetic Moment Formula
Substituting the expression for area \( A \) into the magnetic moment formula:
\[ m = N \cdot I \cdot (\pi r^2) \]

### Step 4: Simplify the Expression
This can be simplified to:
\[ m = N \cdot I \cdot \pi \cdot r^2 \]
From this expression, we can see that the magnetic moment \( m \) is directly proportional to \( r^2 \):
\[ m \propto r^2 \]

### Step 5: Conclusion
Thus, we conclude that the magnetic moment of a current-carrying circular coil varies as the square of the radius \( r \). Therefore, the answer to the question is:
\[ m \propto r^2 \]