Magnetic field of a wire with non-uniform current distribution

[KDPhysics]

Summary:: Not sure if my solution to a magnetostatics problem is correct

[Mentor Note -- thread moved from the technical forums, so no Homework Template is shown]

I was trying to solve problem 2 from: https://ocw.mit.edu/courses/physics/8-022-physics-ii-electricity-and-magnetism-fall-2006/exams/exam2.pdf

Here is my solution:

 

[kuruman]

Looks fine, except in part 4 you did not specify the field outside.

 

[KDPhysics]

do you mean the curl of the B field?

 

[kuruman]

do you mean the curl of the B field?

Yes, sorry, that's what I meant.

 

[KDPhysics]

Since the current density is zero outside, is the curl also zero?

 

[kuruman]

You have an expression for $\vec B(\vec r)$ outside the wire. Can't you just take its curl?

 

[KDPhysics]

Isn't it faster to just use Ampere's Law in differential form?

 

[KDPhysics]

I tried just evaluating the curl in cylindrical coordinates and found 0, as was expected.

Soo was my initial reasoning using Ampere's law correct? Or was it just a coincidence?

 

[kuruman]

Isn't it faster to just use Ampere's Law in differential form?

It is, but aren't you just curious whether you get the same answer if you did it the other way? Checking the answer by doing it in two different ways shows that it all hangs together and is not coincidence, not to mention reinforcing the belief in yourself that you can do this sort of thing on your own and do it correctly.

 

[KDPhysics]

That is true... Well I guess I now trust Ampere's Law even more than before. Thanks