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Ampere’s Law

The Biot-Savart law in magneto statics is used to calculate the net magnetic field due to current carrying conductor. It can be applied to show that for an infinity long and straight conductor carrying current i, the magnetic field at any arbitrary point P, situated at distance R as shown in Figure 7.27.

B =µ0 i/2πr

whereµ0 is a constant called permeability constant (µo = 4π x 10-7T.m/ A ) and r is the perpendicular distance between the wire and P. The field lines of B are concentric circles around the wire.

 

If we consider any charge distribution and consider only situation of high symmetry,Ampere’s law is preferred over Biot-Savart law to find the magnetic field due to charge distribution.

 

Ampere’s circuital law (or Ampere’s law) can be expressed as

The left hand side of equation (1) represents the dot product B.dl integrated around a closed loop (line integral around a path of radius r) and i be the net current encircled by the loop.

 

Using equation (1), Ampere’s law can be derived

=µ0 i/2πr ʃ dl

=µ0 i/2πr x2πr =µ0 i

 

We can extends this discussion to show that equation (1) holds good for any nature ofthe closed path.

 

 

Differential Form: In a region of distributed current flow, let J be the current density.So the current i = ʃs ].j dS, if dS is an element of the bounded surface area.

µ0ʃS J .ds

Using Stoke’s theorem

Equating the R.H.S. expressions of equations (1) and (2)

Equation (3) is known as differential form of Ampere’s law.