# Q about shapes of magnetic fields

## Main Question or Discussion Point

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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Dale
Mentor
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.

Dale
Mentor
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 realise 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?

Dale
Mentor
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.

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.

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.