Magnetic field calculation. Method of Images/superimposition.

The question is : A cable of circular cross-section and diameter 2 cm has a long cylindrical hole of
diameter 1 mm drilled in it parallel to the cable axis. The distance between the
axis of the hole and the cable is 5 mm. If the cable has a uniform steady current density of J=10
5 Am-2 flowing in it, calculate the magnetic field:

i) centre of cable
ii) centre of the hole.

So I begin by modelling the hole as 2 wires of diamter 1mm , situated on top of each other with opposite and equal current density, J. Let the postive wire be wire A and the negative wire B.

– First of all, I am not entirely sure my understanding of current density as a vector is correct, I am not enirely sure I interpret the term ”uniform steady curent denisty’ correctly, and this might show through in my questions below.

Here are my thoughts : Current density initally has it’s orgin at he centre of the cable. Drilling this hole, in affect, re-defines the origin, such that prior to the hole, the answer to i would be zero. But now, the origin of the distribution is elsewhere. (I’m unsure though, how to picture a origin when the current density has the same magnitude everywhere, moreso in terms of Ienclosed, so when applying to Amp’s law.)Anyway with my understanding here are my questions:

Questions:

– So for part i, I consider wire A and the remaining body now without the hole, let this be C.Superimposing these two, the system is as it was before, the origin is at the centre, so body A and C contribute 0 to the net field at the centre of the cable.

So field at the centre of the cable= Field at the centre of the cable due to wire B.

However, I am not 100% sure on this interpretation as it doesn’t seem correct as I apply it to part ii*, (the origin comments) :

-The correct interpretation is that wire B contributes nothing – my reasoning would be that the origin of it’s current density is at the centre of the hole*, and so no charge is enclosed. But that, again modelling wire A and body C as equivalent to the system before drilling the hole, with its origin at the centre of the cable as before, we get a contribution as it would be at this location with no hole present: Field at centre of hole = field at the centre of the hole due to the cable
(where when I refer to the ‘cable’ i mean before the hole, and body C for the cable with the hole

– So I think my arguements may be flawed as by * couldn’t we aregue the same about wire A, that this does not contribute at the centre of the whole, and so only body C does.
OH, OR is this correct as this interpretation takes us back to were we were before modelling body C; we are now considering body C only, whose origin of the current desnity is now not known, but once you solve for this and calculate the field contribution due to C at the centre of the hole, you would attain the same answer?

Are my thoughts and interprations of the concepts okay?

Many thanks to anyone who can help shed some light !

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