The magnitude of magnetic field at the distance from the centre along the axis is given as
Where, is the magnetic field, is number of turns, is the radius, is the distance at which the field is to be calculated, is the current following through the coil and is the permeability of free space.
By using Biot –Savart Law magnetic field due to length is calculated.
The perpendicular components of the field that is will cancel each other the only remaining component will be along that is .
From the , we get
Now, on integrating the above equation over the circle of radius , we get
The magnetic field due to a circular coil having turns along the axis at the distance will be given as,
The magnetic field at the centre of the circular coil is given as,
By substituting in the above equation, we get
As the direction of the current flowing in the two coils is same so the two coils attract each other and the resultant field will be the sum of the field due to the coil and .
Therefore, we can write,
As is very small as compared to the so we can neglect it. Then, the above equation becomes,
On solving further and taking as common from denominator, we get
Thus magnetic field at the mid-point between the coils is uniform over a small distance as compared to and it will be,