THE BIOT-SAVART LAW

The Biot Savart Law can be stated as :

The magnetic field at a point which is at a distance from a very short length of a conductor carrying a current I is given by

where θ is the angle between the short length and the line joining it to P.

The direction of is given by the right hand grip rule. is called the current element . The constant of proportionality depends on the medium around the conductor . In vacuum ( or air ) the constant is written as and the equation

becomes

In the vector form the law can be written as

δB P rδl

δB Iδl

μo4π

Note: The law cannot be tested directly because it’s not possible to have a current carrying conductor of length . However it can be used to derive expressions of flux densities of real conductors and these give values that are in agreement with those determined by experiments.

Magnetic Field at the Centre of Plane Circular Coil

According to Biot Savart Law the flux density at P, due to short length δl is given by the equation

The total flux density , B, at P is the sum of the flux densities of all the short lengths ie

Every section of the coil is at distance from P and makes angel with the line joining it to P, and therefore

δl

δB

r 90o

Since is the total length of the coil , ie it’s circumference, , this becomes

That is

If the coil has N turns each carrying current in the same sense , the contribution of each turn adds to that of every other and therefore

∑ δl 2πr

Magnetic field due to an infinitely long conductor

The flux density due to the short length is given by

From the figure , ,

Also , , therefore

Substituting for and gives

sinθ =ar

r =a

sinθ

l = acotθ δl = − acosec2θδθ

r δl

The total flux density, B, at P is the sum of the flux densities of all the short lengths and can be found by integrating over the whole length of the conductor.

Thus ,

The limits of the integration are π and 0 because these are the values of θ at the ends of the conductor. Therefore

i.e.