Q about shapes of magnetic fields

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Discussion Overview

The discussion revolves around the shapes of magnetic fields generated by different configurations of current-carrying wires, specifically comparing a radial arrangement of wires to a single wire. Participants explore the implications of these configurations on the resulting magnetic fields, including questions about symmetry and uniqueness in magnetic field generation.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that a radial arrangement of wires can generate a magnetic field similar to that of a single wire, questioning the uniqueness of magnetic field shapes.
  • Others argue that it is not possible to have current flowing only inward without generating an electric field, suggesting that the magnetic fields would not be identical.
  • A participant mentions that while a static charge can build at the center for a brief moment, the resulting electric and magnetic fields would not match those of a straight wire.
  • There is a discussion about the symmetry of the magnetic field around a wire, with some noting that the field diminishes with distance and has cylindrical symmetry.
  • One participant expresses confusion about how the fields from the radial wires add up to resemble the field of a single wire, indicating a need for clarification or visual representation.
  • Another participant acknowledges that the magnetic fields from the radial wires could merge, similar to toroidal windings, but expresses skepticism about their equivalence to the field of a single wire.
  • Concerns are raised regarding the assumptions needed to model the fields accurately, particularly about the nature of the current source and the arrangement of the wires.
  • Some participants emphasize that the complete picture of the fields from different configurations is not identical, but question whether similar magnetic fields can arise from different current arrangements in a limited range.

Areas of Agreement / Disagreement

Participants do not reach a consensus on whether the magnetic fields from the radial wires and a single wire can be considered identical. Multiple competing views remain regarding the nature of the fields and the assumptions necessary for their analysis.

Contextual Notes

Participants highlight the need for specific assumptions about the current source and the arrangement of the wires to justify their models. There is also mention of the uniqueness theorems related to electric and magnetic fields, indicating that the discussion is complex and dependent on various factors.

HarryWertM
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Reviewing basics of electrical theory [right hand rule; Faraday's law; etc.]. Come to surprising conclusion:

If a number of wires are arranged like the spokes of an old fashioned wagon wheel and current flows through these wires towards the center, a magnetic field is generated which is the same shape as the field of a single wire perpendicular to the "wheel" & coming out of the center of the "wheel". Are there many [infinitly many?] different geometrical patterns of current which give rise to identical magnetic fields?
 
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What you describe is not possible. You cannot have current just going in.
 
I honestly do not even fully understand the question.
 
What you describe is not possible. You cannot have current just going in.

Sure you can. At least for a few picoseconds. Builds a static charge at the center. Look at a picture of the Los Alamos Z-pinch machine.
 
Sure, but then you have an E-field also and both your E and B fields are time varying even for a steady current. So the fields are not anywhere near the same as those around a straight wire, as you suggested in your OP.
 
The [briefly] steady radials current generate concentric circular B field lines over the radial wires. The increasing central charge E also generates a B field in the form of concentric circles over the "spokes". A suitable time-varying current in a perpendicular central conductor should mimic these B fields, but not the E field.
 
I'm having trouble reconciling your original statement.

The field around a wire has cylindrical symmetry, with the field diminishing with distance from the wire.

(1)The field around the wire emanating from the hub would thus be a series of concentric equipotential cylinders.

(2)The fields around the radial wires would be similar.
How do these add up to (1) ?
 
The magnetic B field lines from the radial wires tend to merge together, just as magnetic field lines in toriodal magnet windings do. The resulting field for the radial wires should approximate the concentric field lines you describe for one wire [which description I totally accept].
 
I realize the fields add up.

Let us say the spokes are in the xy plane.

As I read what you have described the single wire has equipotential lines in the z direction.

The combined equipotential lines from the spokes are in the xy plane or parallel to it.
Proceeding in the z direction yields sets of decreasing equipotential surfaces.

Perhaps I have misunderstood your arrangement, if so a picture might help?
 
  • #10
AFAIK the various uniqueness theorems always treat both the E and B fields, and there is no uniqueness theorem for either field alone.

However, I agree with Studiot, I am not convinced that even just the B field for the spokes would be the same as the single wire. I would have to see the math.
 
  • #11
Good points. The COMPLETE picture of two different fields - B from single wire; B from radials plus charge - is definitely not identical. I guess what I meant was can we have effectively same magnetic field for some small local range from two drastically different current arrangements? This was a surprising result to me, if correct.
 
  • #12
However, I agree with Studiot, I am not convinced that even just the B field for the spokes would be the same as the single wire. I would have to see the math.

It wouldn't, if there are enough spokes, it would be a stack of equipotential disks parallel to the xy plane with decreasing intensity in the z direction. Fewer spokes would give an umbrella like effect

I am assuming the hub forms a perfect source/sink so that we can just allow current to appear in the hub and passoutward along the single infinite wire.

Similarly the current comes in from infinity and disappears down the sink at the hub.

If we do not make these assumptions we are not justified in modelling the field as a series of concentric loops around the wires.

What I am saying is that for a single wire the field is constant along lines in the z direction and decreasing in either the x or y directions.

Whereas for the spokes the combination field is constant in the x and y directions by varies in the z direction.
 

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