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 solve the problem of how the magnetic moment (M) of a current-carrying circular coil varies with the current (I), 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 current-carrying coil is defined as the product of the current (I) flowing through the coil and the area (A) of the coil. Mathematically, it is expressed as:
\[ M = I \cdot A \]

2. **Calculating the Area of the Coil**:
For a circular coil, the area (A) can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
where \( r \) is the radius of the coil.

3. **Considering the Number of Turns**:
If the coil has \( n \) turns, the total magnetic moment will be the product of the magnetic moment of one turn and the number of turns:
\[ M = n \cdot (I \cdot A) = n \cdot (I \cdot \pi r^2) \]

4. **Final Expression for Magnetic Moment**:
Thus, we can express the magnetic moment as:
\[ M = n \cdot I \cdot \pi r^2 \]
This shows that the magnetic moment varies directly with the current (I), the number of turns (n), and the square of the radius (r²).

5. **Identifying the Variation**:
From the equation \( M = n \cdot I \cdot \pi r^2 \), we can conclude that:
- The magnetic moment (M) is directly proportional to the current (I).
- The magnetic moment (M) is directly proportional to the number of turns (n).
- The magnetic moment (M) is directly proportional to the square of the radius (r²).

### Conclusion:
Therefore, the magnetic moment of a current-carrying circular coil of radius \( r \) and number of turns \( n \) varies as:
\[ M \propto n \cdot I \cdot r^2 \]