It isn't supposed to be illustrative of a point charge. There are no point charges. It's supposed to be illustrative of a "vorton" with "intrinsic" spin. I'll have a look at replacing the concentric circles with dipole field lines. As I said I avoided that to keep things simple and to steer clear of the toroidal electron and having to inflate the torus to yield spherical symmetry.

No. It's a thought.

OK, good on both points.

Which lands us right back with the central issue: that E isn't "the" field, it denotes the linear force resulting from the Fμν field interactions.

It's a single field, but it's more than just a vector field. You know that "field strength" diminishes with distance, which starts you off with a scalar. Then the spiral lines have a direction, which steps you up to a vector. But it doesn't stop there. Each line is "curled" in the third dimension rather just curved on the flat, it's more than just a straight line with a direction. You need more information, so it's a tensor field.

More correctly,

**E** is the force a stationary unit test charge feels; it isn't always equal (or proportional) to the force in general! But anyway, all I wanted to illustrate was how the "traditional" electric and magnetic field lines carry actual information about the "traditional" 3-vector fields

**E** and

**B**.

You are correct to say that it is more in keeping with relativity to speak in terms of the tensor

*F*, and that brings us to the very issue under discussion: how do you draw

*F* and get a similar preservation of information? You could just draw both sets of field lines, and that technically works, but it feels a bit like a cop-out.

The moot point is that you don't *create* an electric field by varying a magnetic field, because they're two aspects of the greater whole.

Absolutely. However, do note that it is inaccurate to say that what we mean by electric fields is somehow a linear part of

*F*, while magnetic fields are somehow a rotation part. Like I was saying, you can have rotational parts of the field which people would still call "electric".

A better way of stating it is to say that when

*F* has non-zero space/time components for some observer, that observer will call that an electric field; when

*F* has non-zero space/space components for some observer, that observer will call that a magnetic field. A more Lorentz-invariant way might be this: if

*u*[sup]μ[/sup] are the 4-velocity components of a "traditional" observer (one who does not know about the covariant formulation of electromagnetism), and F[sub]μν[/sub] are the electromagnetic field components, then:

*F*[sub]μν[/sub]*u*[sup]ν[/sup] = the bit of *F* the observer will think of as an electric field;

**F*[sub]μν[/sub]*u*[sup]ν[/sup] = the bit of *F* the observer will think of as a magnetic field.

I've introduced the quantities

**F*[sub]μν[/sub] = (1/2)ε[sub]μνλρ[/sub]

*F*[sup]λρ[/sup], which are the components of the so-called

*dual tensor* to

*F*. They contain exactly the same information as

*F*, but with the components shuffled around.

Please can you try to be cautious with the word create? And please note that those lines don't actually exist. I suppose I can't really complain about creating forces and current, but again my central point is that E denotes the force - which moves the electrons in the electric conduction current. When you pedal a bicycle with a dynamo you can feel a slight resistance when you turn it on. You are "pedalling against the electrons" which you are moving. For an analogy imagine your pedals are connected to a propeller which you dip into the water.

I think it is pretty common to speak loosely and say things like "varying magnetic fields create EMFs", but I agree that it is rather imprecise. That's a good illustration of why mathematics is useful in physics! And I hope I didn't suggest that field lines actually

*exist*; they are definitely just mathematical abstractions which are sometimes useful for visualising some apsects of a field. I hope my meaning was clear, all the same.

Lines of E are lines of force, but lines of B aren't.

Is this what you mean: in covariant language, the Lorentz force is alway given by

*qF*[sub]μν[/sub]

*u*[sup]ν[/sup], but because

*u* = (1,0,0,0) in the rest frame, the force (in that rest frame) on the particle only depends on the time/space components of

*F*, which is what a "traditional" observer ignorant of relativity would call the electric field. In the (3+1) language of that kind of observer, if you look at things in a charged particle's instantaneous rest frame, you discover that the particle only ever "feels" an

*electric* field; in other words, because it is at rest in its own reference frame (i.e.

**v** = 0), the Lorentz force reduces to

**F** =

*q***E**. It's interesting stuff.

And there *are* charges around!

Oh, most definitely. I only meant that I was interested in regions of space where the charge density was non-zero, as opposed to a vacuum somewhere. Did the rest of it make sense?

A charged particle like an electron has an electromagnetic field. It doesn't have an electric field, so "electric charge" is a misnomer. Therefore "magnetic charge" is misguided.

I'm not sure I agree with the "therefore" here, although I don't have a problem with the conclusion that monopoles don't exist, provided that Maxwell's equations are correct of course (as normally written, with div

**B** = 0, or in the covariant formulation

*d*F* = 0).

Think in terms of frame-dragging and thing of the electromagnetic field as a "twist field". If you were moving through it but didn't know it, you would think of it as a "turn field". A magnetic field. But because the frame-dragging is there you know that there can be no region of space that is turning like a roller-bearing disconnected from the surrounding space. Hence magnetic monopoles cannot exist. But do note this.

I'm afraid I didn't grasp your analogy there, but no worries.

I'll think about the magnetic dipole, but life is a constant battle to keep things simple enough for the modern attention span.

In all honesty, my request for the diagrams for electric and magnetic dipoles were more for your benefit; I wanted you to see the ambiguity I mentioned for yourself, rather than expect you to simply take my word for it. Wolfram Alpha (wolframalpha.com) can do streamline plots, incidentally, if you don't have plotting software. You give it a query like this:

Code:

` StreamPlot[ {x/(x^2 + y^2), y/(x^2 + y^2)}, {x, -1, 1}, {y, -1, 1}]`

(that's an inverse square vector field) or like this:

Code:

` StreamPlot[ {x*y/(x^2 + y^2)^(5/2), ((2/3)*y^2 - (1/3)*x^2)/(x^2 + y^2)^(5/2)}, {x, -1, 1}, {y, -1, 1}]`

(that's a dipole vector field).

Maybe I'll do another separate essay which talks about the toroidal dipole and how one "inflates" it to yield a spindle-sphere torus, like this. Topologically it's a torus, hence the electron's magnetic dipole, but geometrically it's a sphere, hence the electron's electric monopole.

That sounds intriguing. Is this a novel model, or something equivalent (in terms of observables) to the standard one?

You're welcome.