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Quick question about magnetic field inside a wire

There is a wire of radius r with a current i flowing through it. There is also a hole of radius a in the wire a distance b from the centre of the wire. The question asks, can you show that the magnetic field inside the hole is uniform? (assume that if you impose a current in the opposite direction where the hole is, that current has the same current density as in the actual conductor.)

My question is: how is the field inside the hole uniform? If the magnetic field gets stronger as r increases (the distance from the centre of the wire to anywhere in the wire), then wouldn't the magnetic field be larger at the outside edge of the hole rather than the inside edge? At both edges of the hole, the imposed opposite current would cause the same magnitude of B, would it not?
 

Answers and Replies

actually I think I got it? There's more total current when you are on the outside of the hole, so it balances out with the increased magnetic field of the wire without the hole?

But I solved B at the middle and outside edge of the hole to be : B = (mu)ib/2(pi)(R^2 - a^2), but when I solve for B on the inside endge of the hole I get
B2(pi)(b-a) = (mu)j(pi)(b-a)^2 where j = i/pi(R^2 - a^2) and eventually get B = (mu)i(b-a)/2pi(R^2 - a^2).

Why aren't they the same?
 
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