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 \( x \) from the center of a circular current-carrying coil where the magnetic field is \( \frac{1}{8} \) of the magnetic field at the center of the coil.

### Step-by-Step Solution:

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

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

3. **Setting Up the Equation**:
According to the problem, the magnetic field at point \( P \) (which is at distance \( x \) from the center) is \( \frac{1}{8} \) of the magnetic field at the center:
\[ B = \frac{1}{8} B_{center} \]
Substituting the expressions for \( B \) and \( B_{center} \):
\[ \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} = \frac{1}{8} \left(\frac{\mu_0 I}{2R}\right) \]

4. **Simplifying the Equation**:
Canceling \( \mu_0 I \) from both sides:
\[ \frac{R^2}{2(R^2 + x^2)^{3/2}} = \frac{1}{16R} \]
Multiplying both sides by \( 16R \):
\[ 8R^3 = (R^2 + x^2)^{3/2} \]

5. **Cubing Both Sides**:
Cubing both sides to eliminate the square root:
\[ (8R^3)^2 = R^2 + x^2 \]
This simplifies to:
\[ 64R^6 = R^2 + x^2 \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ x^2 = 64R^6 - R^2 \]

7. **Factoring Out \( R^2 \)**:
\[ x^2 = R^2(64R^4 - 1) \]

8. **Finding \( x \)**:
Taking the square root gives:
\[ x = R\sqrt{64R^4 - 1} \]

9. **Final Result**:
Since we need to express the distance \( x \) in terms of \( R \), we can evaluate \( x \) numerically or leave it in this form, depending on the context of the problem.