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

### Step-by-Step Solution:

1. **Understanding Magnetic Moment**:
The magnetic moment \( M \) of a coil is defined as:
\[ M = N \cdot I \cdot A \]
where:
- \( N \) = number of turns in the coil,
- \( I \) = current flowing through the coil,
- \( A \) = area of the coil.

2. **Calculating the Area of the Coil**:
The area \( A \) of a circular coil is given by the formula:
\[ A = \pi r^2 \]
where \( r \) is the radius of the coil.

3. **Substituting the Area into the Magnetic Moment Formula**:
Now, substituting the area \( A \) into the magnetic moment formula:
\[ M = N \cdot I \cdot (\pi r^2) \]

4. **Identifying the Relationship**:
From the equation \( M = N \cdot I \cdot \pi r^2 \), we can see that:
- \( N \) and \( I \) are constants for a given coil.
- The term \( \pi \) is also a constant.

Therefore, we can express the magnetic moment as:
\[ M \propto r^2 \]
This indicates that the magnetic moment is directly proportional to the square of the radius of the coil.

5. **Conclusion**:
Thus, the magnetic moment \( M \) of a current-carrying circular coil varies as \( r^2 \).

### Final Answer:
The magnetic moment of a current \( I \) carrying circular coil of radius \( r \) and number of turns \( N \) varies as \( r^2 \).