Last time I have discussed and derived the Biot-Savart law. Let us today discuss the APPLICATION OF BIOT-SAVART’S LAW that is

Magnetic Field At The Centre Of The Circular Coil Carrying Current:-

Consider a circular coil of radius r with centre O,

Let I be the current flowing in the circular coil in the direction

Let the circular coil is made of a large number of current elements each of length dl.(Try to make the figure yourself).

According to Biot-Savart’s law ,the magnetic field at the centre of the circular coil due to the current element Idl is given by

dB=μ₀/4π(dl*r/r³)

where r is the position vector of point O from the current element.

As the angle between dl and r is 90⁰ i.e θ=90⁰

dB=μ₀/4π Idl r sin θ/r³

=μ₀/4π Idl sin θ/r²

Or                                   dB=μ₀/4π Idl/r²

Here, the direction of dB is perpendicular to the plane of the current loop and is directed inward. Since the current through all the elements of the circular coil will contribute to the magnetic field in the same direction,therefore ,the total magnetic field at point O due to current in the whole circular coil can be obtained by integrating Equation(1).Thus

B=∫dB=∫μ₀/4π Idl/r²

=μ₀/4π I/r²∫dl

But, ∫ dl =total length of the circular coil=circumference of the current loop=2πr

B=μ₀/4π I/r²2πr

=μ₀I/2r

If the circular coil consists of n turns ,then

B=μ₀nI/2r

This is the Magnetic Field At The Centre Of The Circular Coil Carrying Current that is one of the application of the Biot-Savart’s law.