Use Biot-Savart law to derive the expression for the magnetic field on the axis of a current carrying circular loop of radius R.
Draw the magnetic field lines due to a circular wire carrying current I.
The Biot-Savart law represents the magnetic field at the center of the circular loop due to the small element “dl” as
Where,
μ is the permeability of air,
r is the position vector.
The direction of dB is the direction of the vector dl × r
I is the current in the wire.
We suppose a circular loop of radius “R” with the center “P”. The current flowing in it is taken as “I”. We suppose the circular loop to be made of small elements having “dl” length.
Here,
|r| = (x2 + R2 )1/2
Since, r is the position vector from the center “P”,
Radius of the loop = R = r
The angle between dl and R is 90° .
Therefore,
⇒ 
⇒ B = ∫dB = 
Also, ∫dl = 2πR i.e. the perimeter of the circular coil.
And If the coil is made up of n circular turns we get,
Magnetic field lines due to a circular wire carrying current I can be given as
