Considering magnetic field along the axis of a circular loop of radius `R`, at what distance from the centre of the loop is the field one eighth of its value at the centre ?

Options

A. `4R`
B. `sqrt(3) R`
C. `3R`
D. `6R`

Explanation

To solve the problem of finding the distance from the center of a circular loop where the magnetic field is one-eighth of its value at the center, we can follow these steps:

### Step-by-Step Solution:

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

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

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

4. **Simplifying the Equation**:
Canceling \( \mu_0 I \) from both sides:
\[ \frac{R^2}{(R^2 + x^2)^{3/2}} = \frac{1}{16R} \]
Cross-multiplying gives:
\[ 16R R^2 = (R^2 + x^2)^{3/2} \]
Simplifying this results in:
\[ 16R^3 = (R^2 + x^2)^{3/2} \]

5. **Cubing Both Sides**:
To eliminate the exponent, we cube both sides:
\[ (16R^3)^2 = R^2 + x^2 \]
This simplifies to:
\[ 256R^6 = R^2 + x^2 \]

6. **Rearranging the Equation**:
Rearranging gives:
\[ x^2 = 256R^6 - R^2 \]

7. **Finding \( x \)**:
We can factor out \( R^2 \):
\[ x^2 = R^2(256R^4 - 1) \]
Taking the square root gives:
\[ x = R\sqrt{256R^4 - 1} \]

8. **Final Calculation**:
To find the specific value of \( x \) when the magnetic field is one-eighth at a distance from the center, we can substitute \( R \) into the equation to find the numerical value.

### Final Answer:
The distance \( x \) from the center of the loop where the magnetic field is one-eighth of its value at the center is:
\[ x = R\sqrt{256R^4 - 1} \]