Magnetic Field On The Axis Of Circular Current Loop - Practice Questions & MCQ

Magnetic field on the axis of circular current loop: 

In the figure, it is shown that a circular loop of radius R carrying a current . Application of Biot-Savart law to a current element of length   at angular position  with the axis of the coil. 

 the current in the segment causes the field   which lies in the x-y plane as shown below.

Another symetric   element that is diametrically opposite to previously  element  cause  .

Due to symmetry the components of   perpendicular to the x-axis cancel each other. I.e these components add to zero.

The x-components of the  's combine to give the total field  at point P.

We can use the law of Biot-Savart to find the magnetic field at a point P on the axis of the loop, which is at a distance  from the center.

  and are perpendicular and the direction of field caused by this particular element lies in the x-y plane.

The net magnetic field is   .

Since

the magnitude of the field due to element  is: 

The components of the vector  are 

To obtain the x-component of the total field , we integrate equation (1), including all the 's around the loop.

Everything in this expression except  is constant and can be taken outside the integral.

The integral  of  is just the circumference of the circle, i.e., 
   

So, we finally get 

If x>>R, then    .

At centre ,