Definition and explanation of Biot- Savart law
Biot and Savart experimentally observed that the magnitude of magnetic field d at a point P (Figure 3.37) 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
iii.directly as the sine of the angle (say,θ) between and r ^ .
iv.inversely as the square of the distance between the point P and length element .
This is expressed as
in SI units and k = 1 in CGS units. In vector notation,
Here vector d is perpendicular to both I (pointing the direction of current flow) and the unit vector r ^ directed from toward point P (Figure 3.38).
The equation (3.37) is used to compute the magnetic field only due to a small elemental length 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 . Hence integrating equation (3.37), we get
where the integral is taken over the entire current distribution.
1. If the point P lies on the conductor, then θ = 0º. Therefore, d is zero.
2. If the point lies perpendicular to the conductor, then θ = 90º. Therefore, d is maximum and is given by where is the unit vector perpendicular to both I and .
Similarities between Coulomb’s law and Biot-Savort’s law
Electric and magnetic fields
· obey inverse square law, so they are long range fields.
· obey the principle of superposition and are linear with respect to source. In magnitude,
Note that the exponent of charge q (source) and exponent of electric field E is unity. Similarly, the exponent of current element Idl (source) and exponent of magnetic field B is unity. In other words, electric field is proportional only to charge (source) and not on higher powers of charge ( q2 , q3 , etc). Similarly, magnetic field is proportional to current element I (source) and not on square or cube or higher powers of current element. The cause and effect have linear relationship.