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Ampere's circuital law

According to ampere's circuital law the line integral of magnetic field B around any closed
curve is equal to 0 times the net current i passing through the area enclosed by the closed curve.
A
0
R
P
B
dl
i
B
i.e.
according to Biot-Savart law, magnetic field at P,
 
 B . dl =  0i
where 0 is free space permeability.
Proof : Consider AB as along, straight conductor with current i, as shown
B=
0 i
2 R
...(i)
 B dl  B  dl
...(ii)
and at P line integral
 
 B . dl
Using (i) and
 dl  2 R
=
in (ii)
 
0 i
B
  dl = 2 R  2R 
=  0i
which is Ampere's circuital law.
This is the integral form of Ampere's circuital law.
Conversion to Differential form
As enclosed current I can be stated as
I=
 
 J . ds
...(i)
s


where J is the current density and ds is the small surfaces area of closed path.
From Ampere's circuital law
 
 B.dl =  0 I
= 0
 
 J .ds
...(ii)
s
Stoke's law states that
  
   B  ds =


 
 B . dl
...(iii)
s
from (ii) and (iii)



   B  ds
 
= 0 
 J. ds
s
hence
 

  B  0 J
...(iv)
 
equation (iv) is the differential form of Amperes law. Because   B  0 , magnetic field
is not conservative and its curl has some value.

 
When the points are inside a closed loop for which J  0,   B  0 .