Jean-Baptiste Biot and Felix Savart jointly gave the Biot-Savart’s law in the year 1820. This is a rule or law which helps to calculate the magnitude of the magnetic field due to a current carrying conductor.
Let ‘dl’ be a current element of a conductor ‘XY’ carrying current ‘I’. Let ‘P’ be a point at a distance ‘r’ from ‘dl’, where the magnetic field is to be determined. Let θ be the angle between ‘dl’ and ‘r’.
Now, according to Biot-Savart’s law, the elementary magnetic field ‘dB’ at the point ‘P’ due to current element ‘dl’ is:
i) directly proportional to magnitude of current passing through it, i.e. dB ∝ I …. (i)
ii) directly proportional to the length of current element, i.e. dB ∝ dl …. (ii)
iii) directly proportional to sine angle between ‘dl’ and ‘r’, i.e. dB ∝ sinθ …. (iii)
iv) inversely proportional to square of the distance between ‘dl’ and ‘P’, i.e. dB ∝ 1/r2 …. (iv)
Now, combining the above given relations (i), (ii), (iii) and (iv), we get,
dB ∝ Idlsinθ/r2
∴ dB = kIdlsinθ/r2, where k is proportionality constant.
In S.I. system,
k = μo/4π, where μo = 4π × 10-7 H/m.
In C.G.S. system, k=1. So,
dB = μo/4π×Idlsinθ/r2 …. (v)
∴ This is the mathematical form of Biot-Savart’s law.
The total magnetic field due to the whole conductor is obtained by integrating the equation (v),
∴ B =∫dB
or, B = ∫μIdlsinθ/4πr2
∴ B = μo/4π∫Idlsinθ/r2