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. `1//r`
B. `1//r^(3//2)`
C. `1//r^(2)`
D. `1//r^(3)`

Answer

To solve the problem, we need to find how the magnetic field \( B \) at a point on the axis of a circular coil varies with the distance \( r \) from the center of the coil, given that \( r \gg R \) (where \( R \) is the radius of the coil).

### Step-by-Step Solution:

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

2. **Consider the Condition \( r \gg R \)**:
Since \( r \) is much greater than \( R \), we can simplify the expression. In this case, \( r^2 + R^2 \) can be approximated as:
\[ r^2 + R^2 \approx r^2 \]
because \( R^2 \) becomes negligible compared to \( r^2 \).

3. **Substitute the Approximation into the Formula**:
Now we substitute this approximation into the magnetic field formula:
\[ B = \frac{\mu_0 I R^2}{(r^2)^{3/2}} \]

4. **Simplify the Expression**:
The term \( (r^2)^{3/2} \) simplifies to \( r^3 \):
\[ B = \frac{\mu_0 I R^2}{r^3} \]

5. **Identify the Variation of Magnetic Field**:
From the simplified expression, we can see that the magnetic field \( B \) varies inversely with the cube of the distance \( r \):
\[ B \propto \frac{1}{r^3} \]

### Final Answer:
Thus, the magnetic field \( B \) due to the circular coil at a point on its axis, located at a distance \( r \) from the center of the coil (where \( r \gg R \)), varies as:
\[ B \propto \frac{1}{r^3} \]