**Introduction**

The magnetic field produced by a current-carrying coil is an important concept in electromagnetism. When a current flows through a coil, it generates a magnetic field in its surroundings. The strength of this magnetic field varies at different locations within and outside the coil. In this explanation, we will discuss the ratio of the magnetic field at the center of a current-carrying coil to the magnetic field at a distance 'a' from the center of the coil and perpendicular to its axis.

**Explanation**

To understand the ratio of the magnetic field at the center of the coil to the magnetic field at a distance 'a' from the center, we need to consider the properties of a current-carrying coil. A coil carrying a current I creates a magnetic field that is proportional to the current and the number of turns in the coil.

**Magnetic Field at the Center of the Coil**

The magnetic field at the center of a current-carrying coil depends on the geometry of the coil. When we consider a coil of radius a, the magnetic field at the center can be calculated using Ampere's law. Ampere's law states that the line integral of the magnetic field around a closed loop is equal to the product of the current enclosed by the loop and the permeability of free space.

For a coil, the magnetic field lines are circular and concentric with the coil. At the center of the coil, the magnetic field lines are perpendicular to the plane of the coil. By applying Ampere's law to a circular loop of radius a centered at the center of the coil, we find that the enclosed current is equal to the total current I flowing through the coil.

Therefore, the magnetic field at the center of the coil can be given by the equation:

B_center = μ₀ * (I / 2a)

where μ₀ is the permeability of free space.

**Magnetic Field at a Distance 'a' from the Center**

Now, let's consider a point outside the coil at a distance 'a' from the center and perpendicular to the coil's axis. At this point, the magnetic field lines are not perpendicular to the plane of the coil but are at an angle to it. The magnetic field at this location can also be calculated using Ampere's law.

To do this, we consider a circular loop of radius a centered at the point of interest. The enclosed current within this loop will be less than the total current I, as the loop does not contain the entire coil. Let's denote this enclosed current as I_enclosed.

By applying Ampere's law to this loop, we find that the magnetic field at this distance 'a' can be given by the equation:

B_a = μ₀ * (I_enclosed / 2a)

**Ratio of Magnetic Fields**

To find the ratio of the magnetic field at the center of the coil to the magnetic field at a distance 'a' from the center, we divide the equation for the magnetic field at the center (B_center) by the equation for the magnetic field at a distance 'a' (B_a).

By substituting the equations for B_center and B_a, we get:

Ratio = (B_center / B_a) = (I / I_enclosed)

Since the enclosed current I_enclosed is less than the total current I, the ratio of the magnetic field at the center of the coil to the magnetic field at a distance 'a' from the

Magnetic field at centre is inversely proportional to the radius 'a' and the magnetic field at a distance is proportional to a²/(a²+r²)³½ *[the equation in denominator is raised to 3/2]* where r is the perpendicular distance from centre
now as r=a solving it gives 2√2