Consider a circular coil of radius .a. with centre 0. Let the plane of the coil be perpendicular to the plane of paper.
Let I be the current flowing through the coil as shown in the adjacent figure. Take two current elements at C and D of length dl each.
Let `angle CPO = angle DPO =phi`
According to Biot- Savart.s law, magnetic field d B at point P, due to current element at C is given by
`d B = ( mu _(0) I dl sin 90^(@))/( 4 pi r ^(2))`
(Here `phi = 90^(@)` is the angle between `vec (dl)` and the line CP )
`implies d B = ( mu _(0) I dl . I )/( 4pi ( a ^(2) + x ^(2)))implies d B = ( mu _(0) I dl )/( 4pi ( a ^(2) + x ^(2)))`
According to right hand screw rule the direction of `vec (dB)` is along `vec (PE).` Now accoring to Biot-Savert.s law, magnetic field dB. at point P due to urrent element at D is given by
`dB. = ( mu _(0) I d l sin 90^(@))/( 4pi r ^(2)) = ( mu _(0) I dl . l )/( 4 pi r ^(2)) = ( mu _(0) I dl )/( 4pi ( a ^(2) + x ^(2)))`
According to right hand screw rule, the direction of `vec (dB)` is along `vec (PF).`
Resove `vec (dB)` into two components i.e., dB ` cos phi `along PY and d B sin `phi` along PX.
Resolve `+Z45vec (dB)` into two components i.e., `vec(dB) cos phi` along PY. and `vec (dB.) sin phi` along PX. The components dB ` cos phi` along PY and dB. cos `phi` along PY. are equal and opposite, hence hey cancel each other.
Total magnetic field induction `vec B ` at point P, due to whole circular coil is given by :
`B = int d B sin phi implies B = int ( mu _(0) I dl )/( 4pi ( a ^(2) + x ^(2)) ) sin phi`
`B = ( mu _(0) I )/( 4pi (a ^(2) + x ^(2))) sin phi int dl`
From right angled `Delta COP.`
`sin phi = (a)/( (a ^(2) + x ^(2)) ^( 1//2))and intdl` = circumference of coil `= 2 pi a `
`implies B = ( mu _(0) I )/( 4pi (a ^(2) + x ^(2))). (a)/( (a ^(2) + x ^(2)) ^(1//2)) . 2 pi a `
`implies B = ( mu _(0) I a ^(2))/( 2 (a ^(2) + x ^(2)) ^( 3//2))`
If the coil has n-turns, then
`B = ( m mu _(0) I a ^(2))/( 2 (a ^(2) + x ^(2)) ^( 3//2)) " "[because vec B= B hati]`
`vec B = ( m mu _(0) I a ^(2))/( 2 (a ^(2) + x ^(2)) ^( 3//2))hati `
