Q) A wire of length L  carrying an electric current i  , is  bent in the form of a polygon of  number of sides=n .Then what is the magnetic field induction at the centre of the polygon ? 

 

Solution) The approach to solve this problem is as follows:

 

Use Biot-Savart's law.

 

Here since the magnetic induction due all sides of polygon will be same.

 

so just find magnetic induction at center due to any one side of the polygon = B (let)

 

then magnetic induction at the center due to polygon = nB

 

here length of one side AB = L/n.

 

If center of the polygon be 'C'

 

then angle between between CA and AB = 180/n

 

similarly angle b/w CB and AB = 180/n

 

Now biot savart's law is = dB = o i x dl / r ² 

 

where r = distance of element dl (through which current i is flowing ) from center C.

now integrate this expression between the two limits to find B due to side AB of the polygon.

 

Now total magnetic field will be = nB