Ampere's Law: Understanding Its Complexities

[Yuqing]

I'm a bit confused on the exact workings of Ampere's law.

Firstly, why does the shape of the Amperian loop not matter. Is there a mathematical proof that all Amperian loops are equivalent for the purpose of this law?

Secondly, the law still holds valid in the presence of external magnetic fields (ie a current producing a field but not enclosed in the loop). Clearly the magnitude of the net field will be changed from the superposition of the two fields. How is the integral able to ignore these external magnetic fields.

 

[Bob S]

I'm a bit confused on the exact workings of Ampere's law.

Firstly, why does the shape of the Amperian loop not matter. Is there a mathematical proof that all Amperian loops are equivalent for the purpose of this law?
.

The shape is important to the extent that you want two sides to be orthogonal to the field lines, so that only the field contribution on the other two sides are non-zero if the loop encloses currents.

Secondly, the law still holds valid in the presence of external magnetic fields (ie a current producing a field but not enclosed in the loop). Clearly the magnitude of the net field will be changed from the superposition of the two fields. How is the integral able to ignore these external magnetic fields.

The contribution from currents outside the loop cancel out when Ampere's Law integral is done properly.
Bob S

 

[Yuqing]

Is it possible to show me some mathematical proof?

 

[Bob S]

Is it possible to show me some mathematical proof?

Consider a loop in a constant magnetic field B, with two sides of the loop orthogonal to the field and two sides parallel.

∫B·dl = 0 around the loop is zero because one side of the loop is parallel to the field, and the other antiparallel (the dot vector product is negative).

Bob S