State Biot-Savart's law.

Explanation

Biot and savart experimentally observed that the magnitude of magnetic field `d vec(B)` at a point P at a distance r from the small elemental length taken on a conductor carrying current veries
(i) directly as the strength of the current I
(ii) directly as the magnitude of the length element `d vec(l)`
(iii) directly as the sine of the angle (say,`theta`) between d `vec(l)` and `hatr`
(iv) inversely as the square of the distance between the point P and length element d`vec(l)`.this is expressed as
`dB prop (I dl)/(r^(2) sin theta`
dB = k `(I dl)/(r^(2)) sin theta`
where k = `(mu_(0))/(4 pi ) ` in SI units and k = 1 in CGS units.
In vector notation,
d`vec(B) = (mu_(0))/(4 pi) (Id vec(l) xx hat(r))/(r^(2)) " "`....(1)
here vector d`vec(B)` is perpendicular to both I d`vec(l)` (pointing the direction of current flow ) and the unit vecotr `hatr` directed from `d vec(l)` toward point P
The equation 1 is used to compute themgnetic field only due to a small elemental length dl of the conductor. the net mangnetic field at P due to the conductor is obtained form principle of superposition by considering the contribution from all current elements `I d vec(l)` . Hence integrating equation (1), we get
`vec(B) = int d vec(B) = (mu_(0) I)/(4 pi ) int (d vec(l) xx hat(r))/(r^(2))`
where the integral is taken over the entier current distribution.
cases
1. If the point P lies on the conductor, then `theta = 0^(@)`. Therefore , `d vec(B)` is zero.
2. If the point lies perpendicular to the conductor , then `theta = 90^(@)`. Therefore, d`vec(B)` is maximum and is given by `d vec(B) = (I dl)/(r^(2))hatn`.
where `hatn` is the unit vector perpendicular to both I d`vec(l)` and `hatr `