State and explain Biot-Sevart's law . Using this law, derive an expression for the magnetic field.

Explanation

This law deals with the magnetic field induction, at a point due to small current element. According to this law, magnetic field induction dB at a point P due to small current element dl is given by the relation
`d B = ( mu _(0))/(4 pi) ( I dl sin theta)/( r ^(2))`
Consider a conductor XY, carrying current I from X to Y. Let AB = dl be the current element. Let .P. be the point at which magnetic field induction is to be calculated.
Let `vec r` be the position vector of point P from the current element `vec dl` and let `theta` be the angle between `vec dl and vec r`
According to Biot-Savart.s law, magnetic field induction dB at point P due to current element depends upon following factors :
(i) `d B prop I`
(ii) `d B prop dl`
(iii) ` dB prop (l)/(r ^(2))`
(iv)` d B prop sin theta `
Combining all these factors, we get
`d B prop ( I dl sin theta )/( r^(2)) implies d B = k ( I dl sin theta)/( r ^(2))`
ln SI units `k = ( mu _(0))/( 4pi )`
where `mu_(0) =` Absolute permeability of free space
also ` mu _(0) = 4 pi xx 10 ^(-7) Wb A ^(-1) m ^(-1)`
`implies d B = ( mu _(0))/( 4pi) . ( I dl sin theta )/(r ^(2)) " "...(i)`
In cgs units `k =l rarr dB = ( I dl sin theta)/( r ^(2)) " "...(ii)`
Direction of `vec(dB):` The direction of `vec (dB)` is given by right hand screw rule. In this case the direction of `vec (dB)` is perpendicular to the plane containing `vec(dl)` and I and is directed inwards.