Magnetic field at the centre of a circular coil of radius R due to curent i flowing through it is B. The magnetic field at a point along the axis at distance R from the centre is :

Options

A. `B/2`
B. `B/4`
C. `B/(sqrt(8))`
D. `sqrt(8)B`

Explanation

To find the magnetic field at a point along the axis of a circular coil at a distance \( R \) from its center, we can follow these steps:

### Step 1: Understand the Given Information
We know that the magnetic field at the center of a circular coil of radius \( R \) carrying a current \( i \) is given as \( B \). The formula for the magnetic field at the center of the coil is:
\[ B = \frac{\mu_0 i}{2R} \]
where \( \mu_0 \) is the permeability of free space.

### Step 2: Identify the Point of Interest
We need to find the magnetic field at a point along the axis of the coil at a distance \( R \) from the center. Let's denote this magnetic field as \( B' \).

### Step 3: Use the Standard Formula for Magnetic Field Along the Axis
The standard formula for the magnetic field \( B' \) at a distance \( x \) along the axis of a circular coil is given by:
\[ B' = \frac{\mu_0 i R^2}{2(R^2 + x^2)^{3/2}} \]
In our case, since we are looking for the magnetic field at a distance \( R \) from the center, we set \( x = R \).

### Step 4: Substitute \( x = R \) into the Formula
Substituting \( x = R \) into the formula, we get:
\[ B' = \frac{\mu_0 i R^2}{2(R^2 + R^2)^{3/2}} = \frac{\mu_0 i R^2}{2(2R^2)^{3/2}} \]

### Step 5: Simplify the Expression
Now, simplify the expression:
\[ B' = \frac{\mu_0 i R^2}{2(2^{3/2} R^3)} = \frac{\mu_0 i R^2}{2 \cdot 2\sqrt{2} R^3} = \frac{\mu_0 i}{4\sqrt{2} R} \]

### Step 6: Relate \( B' \) to \( B \)
Since we know \( B = \frac{\mu_0 i}{2R} \), we can express \( B' \) in terms of \( B \):
\[ B' = \frac{B}{2\sqrt{2}} = \frac{B}{\sqrt{8}} \]

### Final Answer
Thus, the magnetic field at a point along the axis at a distance \( R \) from the center of the coil is:
\[ B' = \frac{B}{\sqrt{8}} \]