Derive an expression for magnetic field at the centre of circular current carrying coil.

Explanation

According to Biot Savart.s law, the magnetic field at the centre of the loop is given by
`d vec (B) = ( mu _(0))/( 4pi) (I d veclxx vec r )/( r ^(3))`
`d B = ( mu _(0))/( 4pi) ( I dl r sin theta)/( r ^(3)) [{:(because theta = 90),( d l bot r ):}]`
` d B = ( mu _(0))/( 4pi) (I dl)/(r ^(2)) " "...(i)`
Integrating both sides

`int d B = ( mu _(0))/( 4pi) ( l )/(r ^(2)) int dl`
`implies B = ( mu _(0))/( 4pi) (l)/(r ^(2)) . 2 pi r = ( mu _(0) l )/( 2r)`
For n turns magnetic field centre of circular coil is `B _(C) = ( mu _(0) n l)/( 2 r)`