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ItsFootballNo
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Homework Statement
A straight wire, of current I, radius a is centred at (α,β). What are the x and y components of the magnetic field B inside one of the wires?
Homework Equations
∮B.dl = μ I_enc
∫∫J.dS = I
The Attempt at a Solution
Any point (x,y) in the wire has a constant current density J.
Hence:
∫∫J.dS = J pi r^2 = J pi ((x-α)^2 + (y-β)^2)
The wire has total current I and the current density J is uniform, hence:
J = I / (pi a^2)
Therefore:
I_enc = ∫∫J.dS = I ((x-α)^2 + (y-β)^2)/a^2
Therefore:
∮B.dl = μ I ((x-α)^2 + (y-β)^2)/a^2
It is from here that I get stuck, mostly how to evalutate the integral without it becoming one big equation without staying in its components. If I was just looking at magnitude of the magnetic field, I know we could show:
∮B.dl = B (2 pi r)
=> B = μ I r / (2 pi a^2)
But looking at the answers, just the y component comes out as:
B_y = μ I (x-α) / (2 pi a^2) - μ I x / (2 pi [(x+α)^2 + (y+β)^2])
Am I going about this the wrong way or are there any tips on how to get to the next step? Any help is greatly appreciated!