A circular current carrying coil has a radius R. The distance from the centre of the coil on the axis where the magnetic induction will be `(1//8)^(th)` of its value at the centre of the coil is,

Options

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

Answer

To solve the problem, we need to find the distance \( x \) from the center of a circular current-carrying coil along its axis, where the magnetic induction is \( \frac{1}{8} \) of its value at the center of the coil.

### Step-by-Step Solution:

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

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. **Set Up the Equation**:
We need to find \( x \) such that:
\[ B = \frac{1}{8} B_{\text{center}} \]
Substituting the expressions for \( B \) and \( B_{\text{center}} \):
\[ \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} = \frac{1}{8} \left(\frac{\mu_0 I}{2R}\right) \]

4. **Simplify the Equation**:
Cancel out common terms:
\[ \frac{R^2}{(R^2 + x^2)^{3/2}} = \frac{1}{8R} \]
Cross-multiplying gives:
\[ 8R \cdot R^2 = (R^2 + x^2)^{3/2} \]
This simplifies to:
\[ 8R^3 = (R^2 + x^2)^{3/2} \]

5. **Cube Both Sides**:
To eliminate the exponent, we cube both sides:
\[ (8R^3)^2 = R^2 + x^2 \]
This results in:
\[ 64R^6 = R^2 + x^2 \]

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

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

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

9. **Final Expression**:
Since we want the distance \( x \) where the magnetic induction is \( \frac{1}{8} \) of its value at the center, we can express it as:
\[ x = R \sqrt{3} \]

### Final Answer:
The distance from the center of the coil on the axis where the magnetic induction will be \( \frac{1}{8} \) of its value at the center of the coil is:
\[ x = R \sqrt{3} \]