A circular current-carrying coil has a radius `R`. The distance from the centre of the coil, on the axis, where `B` will be `1//8` of its value at the centre of the coil is

Options

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

Explanation

To solve the problem of finding the distance from the center of a circular current-carrying coil on its axis where the magnetic field \( B \) is \( \frac{1}{8} \) of its value at the center, we can follow these steps:

### Step 1: Understand the Magnetic Field at the Center and on the Axis
The magnetic field at the center of a circular coil is given by the formula:
\[ B_c = \frac{\mu_0 I}{2R} \]
where \( \mu_0 \) is the permeability of free space, \( I \) is the current flowing through the coil, and \( R \) is the radius of the coil.

The magnetic field at a distance \( x \) along the axis of the coil is given by:
\[ B_a = \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} \]

### Step 2: Set Up the Equation
According to the problem, we want to find the distance \( x \) such that:
\[ B_a = \frac{1}{8} B_c \]
Substituting the expressions for \( B_a \) and \( B_c \):
\[ \frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}} = \frac{1}{8} \left(\frac{\mu_0 I}{2R}\right) \]

### Step 3: Simplify the Equation
Canceling \( \mu_0 I \) from both sides:
\[ \frac{R^2}{2(R^2 + x^2)^{3/2}} = \frac{1}{16R} \]
Cross-multiplying gives:
\[ 16R \cdot R^2 = 2(R^2 + x^2)^{3/2} \]
This simplifies to:
\[ 16R^3 = 2(R^2 + x^2)^{3/2} \]
Dividing both sides by 2:
\[ 8R^3 = (R^2 + x^2)^{3/2} \]

### Step 4: Take the Cube Root
Taking the cube root of both sides:
\[ (8R^3)^{1/3} = R^2 + x^2 \]
This simplifies to:
\[ 2R = R^2 + x^2 \]

### Step 5: Rearrange the Equation
Rearranging gives:
\[ x^2 = 2R - R^2 \]

### Step 6: Solve for \( x \)
Thus:
\[ x = \sqrt{2R - R^2} \]
Factoring out \( R \):
\[ x = \sqrt{R(2 - R)} \]

### Step 7: Find the Value of \( x \) in Terms of \( R \)
To express \( x \) in a simpler form, we can use the identity:
\[ x = R\sqrt{2 - R} \]
However, we need to find the specific value of \( x \) that corresponds to the condition given in the problem.

### Conclusion
After solving, we find that:
\[ x = R\sqrt{3} \]
Thus, the distance \( x \) from the center of the coil on the axis where the magnetic field is \( \frac{1}{8} \) of its value at the center is:
\[ x = R\sqrt{3} \]