Does this look like a possible solution to Maxwell's equations?

[Spinnor]

Using two planes of sheet plastic I sketched electric and magnetic field lines that have zero divergence. Each plane was then slit half way down the axis of symmetry and then slid together. Holding them roughly perpendicular they were photographed. On the two planes do the field lines look like a plausible solution to Maxwell's equations? If so does it look straight forward to fill in the rest of the field lines?

Thanks for any help!

 

[NFuller]

I sketched electric and magnetic field lines that have zero divergence

How do you know these fields have vanishing divergence? What function represents them? It looks like at the axis, which I assume is your time axis, there are points of non-zero divergence.

 

[Dale]

On the two planes do the field lines look like a plausible solution to Maxwell's equations?

No. The B field is the field around a set of parallel wires, but the E field is not. It looks like they are just circles, not like they are intended to be solutions to Maxwell's equations.

 

[Spinnor]

How do you know these fields have vanishing divergence? What function represents them? It looks like at the axis, which I assume is your time axis, there are points of non-zero divergence.

All the field lines form closed loops but I guess that is necessary but not sufficient?

 

[Spinnor]

No. The B field is the field around a set of parallel wires, but the E field is not.

Don't quite understand the above.

It looks like they are just circles

They are ellipses not circles, but not too far from circles.

 

[Dale]

Don't quite understand the above.

What is the source that creates the fields you are showing? Show it mathematically

 

[Spinnor]

The spacing of the ellipses is supposed to be such that the electric and magnetic fields along the central axis is that of a plane electromagnetic wave.

 

[Spinnor]

What is the source that creates the fields you are showing? Show it mathematically

We are told there are no sources in free space. This is supposed to be a vacuum solution.

 

[Dale]

The spacing of the ellipses is supposed to be such that the electric and magnetic fields along the central axis is that of a plane electromagnetic wave.

It is not a plane wave. Please show your math.

We are told there are no sources in free space. This is supposed to be a vacuum solution.

It is not a vacuum solution. The B field solution is the field around a set of parallel wires, not a vacuum solution. What makes you believe that it is a vacuum solution, please show your math.

 

[Spinnor]

It is not a plane wave. Please show your math.

It is not a vacuum solution. The B field solution is the field around a set of parallel wires, not a vacuum solution. What makes you believe that it is a vacuum solution, please show your math.

There are no wires anywhere.

 

[Dale]

There are no wires anywhere.

For the fourth time, please show your math. It is looking to me like you are just drawing random figures and hoping that they represent a plane wave.

You didn't draw the wires, but that is where you get a B-field like the one you have drawn.

 

[Spinnor]

So the equation from Jackson that my sketch is partly based on is from the 1975 edition, problem 7.20, page 333,

Note the fields in the direction of propagation.

There are 4 terms in the square brackets above. If we eliminate the 2nd and the 4th terms I think we convert this solution to a linearly polarized plane wave of finite extent?

We know that in vacuum that the divergence of the electric and magnetic fields is zero so field lines must form loops?

What I sketched does not follow Jackson's requirement of a wave of many wavelengths broad but I think that if we superimpose the fields of many such sketches you would recover the fields given by Jackson above?

Edit: A long time ago I used Mathematica to graph the fields of Jackson's problem above, I don't remember if was the circularly polarized version or my linear version of his problem but if I remember correctly the fields came out looking similar to my sketch which was made even longer ago.

Thanks for any help!

 

[Spinnor]

So the equation from Jackson that my sketch is partly based on is from the 1975 edition, problem 7.20, page 333,

This sentence contradicts with what I wrote later. The sketch was made during my final year of undergrad schooling for some for some extra credit project I did. I still have it. The graphing came latter with one of the early versions of Mathematica. If you take my sketch and stretch it perpendicular to the axis of propagation I think you get the fields Jackson's problem wants you to show.

 

[NFuller]

It sounds like you have answered your own question. If your drawing does in fact match a particular case of the expression shown in Jackson, then it is a solution to Maxwell's equations.

 

[Spinnor]

It sounds like you have answered your own question. If your drawing does in fact match a particular case of the expression shown in Jackson, then it is a solution to Maxwell's equations.

It only matches if my sketch is stretched perpendicular to the axis of propagation. Jackson's problem was to show the expression works provided the beam of light were many wavelengths broad. My sketch is roughly one wavelength broad. So I guess you are right and the answer to my question is no, but the interesting thing (to me) is that if you superimpose many of my sketches, with the right locations, I believe you would recover Jackson's result?

Thanks!