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Homework Statement
Consider a two-layered cylindrical wire with inner-layer permeability μ1 and outer-layer permeability μ2. A line current I runs through the center in the z direction. Calculate the bound currents and the magnetic field produced by the bound currents.
Homework Equations
[1] ∫ B⋅dl = Iμ0
[2] ∫ H⋅dl = Ifree
[3] B = μ0 (H+M)
[4] B = μH
[5] Jb = ∇x M
[6] Kb = M x n
The Attempt at a Solution
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Using equation 2 and symmetry, I come up with
H = I/(2πs)
Using equation 4, I found the inner and outer material B fields. These point in the φ direction.
B=Iμ1/(2πs) in the inner material
B=Iμ2/(2πs) in the outer material.
Plugging B and H into equation 3, I found the inner and outer material M fields. These point in the φ direction.
M= I(μ1/μ0 - 1) / (2πs) in the inner material
M= I(μ2/μ0 - 1) / (2πs) in the outer material
Plugging M into equation 5, I calculate that Jb = 0 in both the inner and outer material.
I expected some bound volume current, so this result is strange to me. If there is no bound volume current and I draw an amperian loop within the inner material, the enclosed total current must be equal to I. If that's the case, I should be able to use equation 1 to find that
B = Iμ0/(2πs)
but I already calculated a different inner B field above.
How can I reconcile the different B values in this inner material?