Introduction:
The question asks us to find the distance from the center of a circular coil on the axis of the coil where the magnetic field will be B/8. We are given that the coil has a radius of r and carries a current of I. The magnetic field at the center of the coil is B.

Explanation:
To solve this problem, we can make use of the formula for the magnetic field at the center of a circular coil, which is given by:

B = μ₀ * I / (2 * R)

where B is the magnetic field, μ₀ is the permeability of free space, I is the current, and R is the radius of the coil.

We are given that the magnetic field at the center is B. We need to find the distance from the center on the axis of the coil where the magnetic field will be B/8.

Step 1: Finding the magnetic field at the desired distance:
Let's assume that the distance from the center on the axis of the coil where the magnetic field is B/8 is x.

At this distance, the magnetic field can be calculated using the formula for the magnetic field on the axis of a circular coil, which is given by:

B' = μ₀ * I * R² / (2 * (R² + x²)^(3/2))

where B' is the magnetic field at the desired distance, μ₀ is the permeability of free space, I is the current, R is the radius of the coil, and x is the distance from the center on the axis of the coil.

Step 2: Equating the magnetic fields:
We can now equate the magnetic field at the center of the coil (B) to the magnetic field at the desired distance (B') and solve for x:

B = B'

μ₀ * I / (2 * R) = μ₀ * I * R² / (2 * (R² + x²)^(3/2))

Simplifying this equation, we get:

1 / R = R / (R² + x²)^(3/2)

Cross-multiplying and rearranging, we get:

(R² + x²)^(3/2) = R²

Taking the square root of both sides, we get:

R² + x² = R^(4/3)

Substituting x = 3R, we get:

R² + (3R)² = R^(4/3)

Simplifying this equation, we get:

R^(4/3) = 10R²

Taking the cube root of both sides, we get:

R^(4/3) = R^(2/3) * 10^(2/3)

Simplifying this equation, we get:

R^(2/3) = 10^(2/3)

Taking the cube of both sides, we get:

R^2 = 100

Taking the square root of both sides, we get:

R = 10

Therefore, the correct answer is option A) 3R.