Introduction:
The magnetic field produced by a current-carrying coil varies with distance from the center of the coil. The ratio of the magnetic field at the center of the coil to the magnetic field at a distance 'a' from the center of the coil and perpendicular to the axis of the coil is an important parameter that determines the strength of the magnetic field.

Explanation:
The magnetic field at the center of the coil can be calculated using the formula:

B = (μ₀NI)/2R

Where B is the magnetic field, μ₀ is the permeability of free space, N is the number of turns in the coil, I is the current flowing in the coil, and R is the radius of the coil.

The magnetic field at a distance 'a' from the center of the coil and perpendicular to the axis of the coil can be calculated using the formula:

B' = (μ₀NIa²)/(2(a²+R²)^(3/2))

The ratio of the magnetic field at the center of the coil to the magnetic field at a distance 'a' from the center of the coil and perpendicular to the axis of the coil can be calculated as:

B/B' = (2(a²+R²)^(3/2))/a²

Conclusion:
The ratio of the magnetic field at the center of a current-carrying coil of radius a and at a distance 'a' from the center of the coil and perpendicular to the axis of the coil is given by (2(a²+R²)^(3/2))/a². This ratio is an important parameter that determines the strength of the magnetic field and is used in various applications such as in designing transformers, motors, and generators.