There is a cross-section wire or radius R with a uniform current density j flowing through it. There is a hole in the wire's cross section (a vacuum) a distance b away from the centre of the cross section. The hole has radius a. What is the magnetic field B in the hole as a function of r?
B(integral of ds) = (mu)(current)
The Attempt at a Solution
If we draw an amperian loop just outside the hole (so r = b + a) we get:
B(1) = (mu)jr/2
and B(2) = (mu)j(a^2)/2r
and B = B(1) - B(2) = (mu)j(r^2 - a^2)/2r
If we do the same just before the hole (so r = b - a) we get:
B = (mu)jr/2 [because the hole does not affect the magnetic field in this case)
But how can we find B in other places inside the hole (in the middle of the hole B = (mu)jb/2 ). My prof says that the magnetic field is uniformly distributed inside the hole is the same (or maybe he said something different but similar), but how can this be? As far as I understand, we can get the B in the hole by visualizing a current in the wire without the hole and then adding a current in the opposite direction (the question says we can assume it's the same density) in order to form the hole. From my understanding, the B of the wire without the hole gets bigger as we get further from the center, right? But if this is true, then wouldn't B be bigger when r = b + a than when r = b, and even bigger than when r = b - a? Can someone please explain this to me? How is B distributed inside the hole? Thx.
Also, if we were to try to find B inside the hole (besides the centre, or r=b+a, or r=b-a)
wouldn't we need to find the area of the hole thats beneath r? So if r was just a little bigger than b, wouldn't the area be just about half of (pi)a^2 ?