Definition and explanation of Biot- Savart law

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.

Cases

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.