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Maths is perfect; I am not.


Sep
24
awarded  Autobiographer
Sep
22
accepted Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
Sep
22
comment Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
It doesn't demonstrate the magnetic field parallel to the current is zero. You can't let B perp encapsulate both radial and 'parallel' fields because the 2D divergence theorem used in your full calculation only applies in the plane of the 2D loop. That is, your constructed vector n is parallel to B perp, but is orthogonal to B parallel. Thus, you've concluded the radial B is zero, but the 'parallel' (to current) B drops out in the div theorem and is left ambiguous. I've tried to clarify it here: i.imgur.com/WFnOGrc.png Essentially, the 2D div theorem sheds no information of B para I.
Sep
21
comment Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
I'll write up the full solution if I get time and accept it, otherwise I'm happy to accept any other full solution
Sep
21
revised Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
offering solution
Sep
19
comment Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
Let us continue this discussion in chat.
Sep
19
awarded  Commentator
Sep
19
comment Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
blehhhhhhhhh, forgive me. For this situation, we can show the magnetic field tangent to the cylinder axis is zero, by taking the integral of an amperian loop around empty space (eg: a rectangle with lengths parallel to the axis and widths curved circularly around the wire, since we know B circumferentially already). My mind is a mess
Sep
19
comment Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
in fact, even with the translational symmetry in this scenario, how can we be sure there is no component of the magnetic field parallel to the axis of the wire/cylinder? All our Amperian loop informs us is the magnitude of the "circumferential" magnetic field, but sheds no information on the two directions orthogonal to the circumference tangent (although we eliminate the radial component by the translational symmetry and Gauss' law of magnetism). Having thoroughly confused myself, I offer another picture! i58.tinypic.com/259ixww.png
Sep
19
comment Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
Actually, I've become rather unsatisfied mulling over this. In this cylindrical example, by the translational symmetries, we can apply Gauss' law for magnetism to show there's no radial magnetic field. However, for a contrived system without that translational symmetry (just rotational), it's obvious that if the integral of radial magnetic field is zero, so too is the field (by the symmetry). As to why the integral would be zero, alludes me in every possible way. The orthogonality between the differential loop vector and the proposed field seems to halt everything! :c
Sep
19
comment Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
(oh! I've also realized we can exploit the translational symmetry of the wire to turn our loop into a cylindrical closed* surface which then must definitely have a zero 'net' radial B component)
Sep
19
comment Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
For the weird tree example, the rotational symmetry was still there though, wasn't it? (about the axis of the "tree"). I was just trying to wrap my head around the integral of a B field perpendicular to the loop around the loop being zero, since it's 2D. However, I've only just now realized that invoked integral theorem proves exactly that, accepting the divergence at any point is zero. Do you know exactly which theorem that is? (Is it Stoke's combined with a bit of curl manipulation?) Thanks heaps! :)
Sep
19
comment Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
What about this horrible looking tree looking field? i61.tinypic.com/t8ufpx.png We could make the field have such a geometry that the net magnetic flux through a sphere enclosing it is zero, but we can see that since the field diverges radially upward, we can draw a loop around it with a non-zero net radial component. Doesn't this suggest a non-zero divergence of the magnetic field in/around that loop? (And isn't this a legal / conceivable field?)
Sep
19
comment Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
I think so. So div(B) = 0 says the total "flux" through the loop is absolutely zero? I've just never seen div(B) = 0 applied to a loop, only closed surfaces. I guess I can't contrive a valid 3D analytic B field that results in a 2D loop having a net* non-zero radial component (at least, not immediately). But let me try!
Sep
19
comment Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
Oh, so the zero-divergence of the B field must be respected in not just 3D, but also in 2D (and the zero-divergence says there can be absolutely no 'outward radial' component of B at all, it must be absolutely zero)?
Sep
18
awarded  Student
Sep
18
revised Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
added 592 characters in body
Sep
18
asked Using Ampere's Law without Right-Hand-Rule to derive an expression for the magnetic field around a current
Jan
17
awarded  Supporter
Oct
30
revised Linear motion with variable acceleration
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