A circular coil of radius `R` carries an electric current. The magnetic field due to the coil at a point on the axis of the coil located at a distance `r` from the centre of the coil, such that `r gtgt R`, varies as

Options

A. (a)`1//r`
B. (b)`1//r^(3//2)`
C. (c)`1//r^(2)`
D. (d)`1//r^(3)`

Answer

To solve the problem of determining how the magnetic field \( B \) due to a circular coil of radius \( R \) carrying an electric current varies at a point on the axis of the coil located at a distance \( r \) from the center of the coil (where \( r \gg R \)), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Setup**:
We have a circular coil of radius \( R \) carrying a current \( I \). We need to find the magnetic field at a point on the axis of the coil, located at a distance \( r \) from the center of the coil.

2. **Magnetic Field Formula**:
The magnetic field \( B \) at a distance \( r \) on the axis of a circular coil is given by the formula:
\[ B = \frac{\mu_0 I R^2}{2(R^2 + r^2)^{3/2}} \]
where \( \mu_0 \) is the permeability of free space.

3. **Condition \( r \gg R \)**:
Since we are given that \( r \) is much greater than \( R \) (i.e., \( r \gg R \)), we can simplify the expression. In this case, \( R^2 \) becomes negligible compared to \( r^2 \).

4. **Simplifying the Expression**:
Under the condition \( r \gg R \), we can approximate:
\[ R^2 + r^2 \approx r^2 \]
Therefore, the formula for the magnetic field simplifies to:
\[ B \approx \frac{\mu_0 I R^2}{2(r^2)^{3/2}} = \frac{\mu_0 I R^2}{2r^3} \]

5. **Identifying the Variation**:
From the simplified expression, we can see that:
\[ B \propto \frac{1}{r^3} \]
This indicates that the magnetic field \( B \) is inversely proportional to the cube of the distance \( r \).

6. **Conclusion**:
Thus, we conclude that the magnetic field due to the coil at a point on its axis, where \( r \gg R \), varies as:
\[ B \propto \frac{1}{r^3} \]

### Final Answer:
The magnetic field \( B \) varies as \( \frac{1}{r^3} \).