Using Biot-Savart's law, derive an expression for magnetic field at the centre of a current carrying circular coil. What will be the direction of this magnetic field?

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)`