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 varies
(i) directly as the strength of the current I
(ii) directly as the magnitude of the length element \(\vec {dl}\)
(iii) directly as the sine of the angle (say,θ) between \(\vec {dl}\) and\(\hat r\).
(iv) inversely as the square of the distance between the point P and length element \(\vec {dl}\).
This is expressed as
Here vector d\(\vec B\) is perpendicular to both I \(\vec{dl}\) (pointing current carrying conductor the direction of current flow) and the unit vector and \(\hat r\) directed from \(\vec{dl}\) toward point P The equation 1 is used to compute the magnetic field only due to a small elemental length \(\vec{dl}\) of the conductor. The net magnetic field at P due to the conductor is obtained from principle of superposition by considering the contribution from all current elements I \(\vec{dl}\) . Hence integrating equation (1), we get
where the integral is taken over the entire current distribution.
Case:
1. If the pont P lies on the ckonductor, then θ = 0°. Therefore, d is zero.
2. If the point lies perpendicular to the conductor, then θ = 90°. Therefore, d\(\vec{B}\) is maximum and is given by d\(\vec{B}\) = \(\frac{Idl}{r^2}\)\(\hat n\).
where \(\hat n\) is the unit vector perpendicular to both l \(\vec{dl}\) and \(\hat r\)
