Why do we assume cylindrical symmetry in Ampere's Law?

[maccha]

When you're trying to find the magnetic field inside a current carrying wire using Ampere's Law, how do you know that the magnetic field inside also displays cylindrical symmetry?

 

[kuruman]

You don't. It is an assumption based on lack of additional information. For example, if the cylinder consisted of two half-cylinders joined along the long axis, one half cylinder being copper the other silver, there would not be cylindrical symmetry because the current in each half would be different. You would have to treat them as resistors in parallel, find the current in each, then apply Ampere's Law. Or you could have a resistivity that is not spatially uniform in general. Since none of that is usually mentioned in problems of the sort, you have to assume that it is not the case. If all the information that is not the case were given in a physics problem, then each problem would be pages long and read like a legal document. There is a tacit understanding in physics problems that "what you don't see or can deduce from what you see, is not there."

So in this case, since the cylinder and what it is made of has cylindrical symmetry (by assumption) and since the two end faces of the cylinder are equipotentials, then the B-field inside, just like the B-field outside, should have rotational symmetry about the cylinder axis.