Ampère's circuital law and finite conductor

[vijender]

Can Ampère's circuital law be used to find electric field for a finite (say length l) current carrying this conductor at a finite point away from it?
If yes, then what will be Magnetic field due to a wire extending from (0,-a/2) to (0,a/2) carrying current “I” at a point (b,0) from it, if I consider a loop (circle here) of radius b/2 perpendicular to the x-axis centered at origin?

 

[Born2bwire]

Yes. You can consider a infinitesimal current element and the resulting field. Then you integrate across the strength and length of your current. This has already been derived easily by most textbooks in the form of say the Biot-Savart Law for magnetostatics.

 

[EHT]

No, ampere's law can't be used here. You can always change the length of the wire and get the same answer. This absolutely can't be true
The derivation of Ampere's law requires that \nabla.J=0 everywhere. On the other hand Biot-Savart can be used anytime

 

[Born2bwire]

No, ampere's law can't be used here. You can always change the length of the wire and get the same answer. This absolutely can't be true
The derivation of Ampere's law requires that \nabla.J=0 everywhere. On the other hand Biot-Savart can be used anytime

Biot-Savart is derived from Ampere's Law. Ampere's Law does not make any conditions upon the divergence of the current. When we talk about Ampere's Law, we are talking about the Ampere's Law as it appears in Maxwell's Equations.

Thus, the OP can use Ampere's Law (along with other Maxwell Equations) to derive the Biot-Savart Law for an infinitesimal current element (moving charge with constant velocity which thus requires a non-zero divergence in the current). Then the OP can use this as the basis for his contour integral around his arbitrarily shaped current loop as he has described previously.

 

[EHT]

as in the other thread, the ampere law that I mentioned is the one without displacement current correction in it