Understanding the Problem
We are given a current-carrying coil of radius R and we need to find the ratio of the magnetic field at the center of the coil to the magnetic field at a distance 3R on its axis.

Key Concepts
1. Magnetic field due to a current-carrying coil: The magnetic field at the center of a current-carrying coil is given by the formula B = μ₀nI, where B is the magnetic field, μ₀ is the permeability of free space (a constant), n is the number of turns per unit length, and I is the current passing through the coil.

Solution
To find the ratio of the magnetic field at the center of the coil to the magnetic field at a distance 3R on its axis, we can use the formula for the magnetic field due to a current-carrying coil.

Step 1:
Let's denote the magnetic field at the center of the coil as B₁ and the magnetic field at a distance 3R on its axis as B₂.

Step 2:
We can write the formula for B₁ and B₂ as follows:
B₁ = μ₀n₁I₁
B₂ = μ₀n₂I₂

Step 3:
Since the coil is the same, the current passing through the coil is the same, i.e., I₁ = I₂ = I.

Step 4:
Also, the number of turns per unit length at the center of the coil (n₁) is equal to the number of turns per unit length at a distance 3R on its axis (n₂).

Step 5:
Hence, we can simplify the ratio of B₁ to B₂ as follows:
B₁/B₂ = (μ₀n₁I)/(μ₀n₂I)
= n₁/n₂

Step 6:
Now, let's substitute the values given in the question. The radius of the coil is R, and the distance on the axis is 3R. Hence, the number of turns per unit length at the center of the coil (n₁) is 1/R, and the number of turns per unit length at a distance 3R on its axis (n₂) is 1/(3R).

Step 7:
Substituting these values in the ratio B₁/B₂, we get:
B₁/B₂ = (1/R)/(1/(3R))
= 3

Final Answer
Therefore, the ratio of the magnetic field at the center of the coil to the magnetic field at a distance 3R on its axis is 3.