The distance at which the magnetic field on axis as ompared to the magnetic field at the centre of the coil carrying current I and radius R is `1//8`, would be

Options

A. R
B. `sqrt(2)R`
C. 2R
D. `sqrt(3)R`

Answer

To solve the problem, we need to find the distance \( x_0 \) from the center of a circular coil carrying a current \( I \) and having a radius \( R \), such that the magnetic field at that point on the axis of the coil is \( \frac{1}{8} \) times the magnetic field at the center of the coil.

### Step-by-Step Solution:

1. **Magnetic Field at the Center of the Coil**:
The magnetic field \( B \) at the center of a circular coil carrying current \( I \) and having radius \( R \) is given by the formula:
\[ B_{\text{center}} = \frac{\mu_0 I}{2R} \]

2. **Magnetic Field on the Axis of the Coil**:
The magnetic field \( B \) at a distance \( x_0 \) from the center along the axis of the coil is given by:
\[ B_{P} = \frac{\mu_0 I R^2}{2(R^2 + x_0^2)^{3/2}} \]

3. **Setting Up the Equation**:
According to the problem, we want the magnetic field at point \( P \) to be \( \frac{1}{8} \) times the magnetic field at the center:
\[ B_{P} = \frac{1}{8} B_{\text{center}} \]
Substituting the expressions for \( B_P \) and \( B_{\text{center}} \):
\[ \frac{\mu_0 I R^2}{2(R^2 + x_0^2)^{3/2}} = \frac{1}{8} \cdot \frac{\mu_0 I}{2R} \]

4. **Canceling Common Terms**:
We can cancel \( \mu_0 I \) and \( \frac{1}{2} \) from both sides:
\[ \frac{R^2}{(R^2 + x_0^2)^{3/2}} = \frac{1}{8R} \]

5. **Cross Multiplying**:
Cross-multiplying gives:
\[ 8R \cdot R^2 = (R^2 + x_0^2)^{3/2} \]
Simplifying this:
\[ 8R^3 = (R^2 + x_0^2)^{3/2} \]

6. **Taking the Power of \( \frac{2}{3} \)**:
To eliminate the exponent, we take both sides to the power of \( \frac{2}{3} \):
\[ (8R^3)^{\frac{2}{3}} = R^2 + x_0^2 \]
Simplifying the left side:
\[ 4R^2 = R^2 + x_0^2 \]

7. **Solving for \( x_0^2 \)**:
Rearranging gives:
\[ x_0^2 = 4R^2 - R^2 = 3R^2 \]

8. **Finding \( x_0 \)**:
Taking the square root of both sides:
\[ x_0 = \sqrt{3} R \]

### Final Answer:
The distance \( x_0 \) at which the magnetic field on the axis is \( \frac{1}{8} \) times the magnetic field at the center of the coil is:
\[ x_0 = \sqrt{3} R \]