(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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?

2. Relevant equations

∮B.dl = μ I_enc

∫∫J.dS = I

3. 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!

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# Components of a magnetic field in a uniform wire

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