Find the magnetic field induction at a point on the axis of a circular coil carrying current.

 

 

Consider a circular coil of radius (a ) with centre O. Let current I be flowing in the coil. Suppose P is any point on the axis of the circular coil at s diastance x from its centre O.

Consider two small elements of the coil each of length dl at C and D which are situated at diametrically opposite edges as shown in figure b.

As per the figure b, PC = PD =

CPO = Ø = DPO

According to Biot Savart"s law, the magnetic field induction at P due to current element at C is

           (Because θ = , as a is small)

           ……………………1

Similarly, the magnitude of magnetic field at P due to current element at D is

          ……………………..2

From eq. 1 and 2

dB = dB"

Resolving  and into two rectangular components:

i. dBcosØ acts along PY and dBsinØ acts along PX.

ii. dB"cosØ acts along PY" and dB"sinØ acts along PX.

Since the component of the magnetic field along PY and PY" are equal and opposite, they cancel each other and the components of the magnetic field along PX and PX" (along the axis of the coil) being in the same direction are added up. Therefore the total magnetic field induction at P due to current through the whole circular coil is given by

            …………………eq.3

But  and

Putting both the values in eq.31

If the coil has n turns then

               (Because)