Are there propagating charged waves admitted in Maxwell’s equations?

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  • #1
Phrak
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Are there propagating charged waves admitted in Maxwell’s equations?

(I know many of you guys posting in Tensor Analysis & Differential Geometry are up to this question. I've seen your posts!)

Put again, are Maxwell’s equations not as robust as is commonly believed, that they should allow for unit velocity (v=c) charged waves?

This would be surprising odd to find true, but my math, and some inference, so far seems to support it.

Fundamentally this is a applied math problem best at home in differential topology.
So I’ve left this problem in differential forms on a pseudo-Riemann Manifold of Lorentz Metric where I found it, and where it seems to be notationally simplest.

Some physics reminders:

d*F = *J, where F = dA
d^2F = 0, completes the set of 4 maxwell equations

F is the 2-form of the electric and magnetic fields.
-J is the 4-current 1-form.
A is the covariant form of the 4-vector potential.

Applying the Laplace-De Rham operator, (d +/- *d*)^2 on F, you obtain the wave equation:

d*d*F = dJ ,

where dJ=0 for the homogeneous solutions of interest,

and where dJ=0 must be over some finite region of space-time rather than a single point, I think. Stop me, if I’m wrong.

Following the same program, the 4-current wave equation I’ve obtained is:

*d*dJ = (*d)^4 A ,

where (*d)^4 A = 0 over a region

I don’t see any other velocities associated with a solution, should it exist, other than the unit velocity, c. But a formal argument would be far better than speculation.

To get propagating waves of charge, this all boils down to asking if the amplitude of J_{0} may be a other than zero over a region, and under the given constraints, I think—and if I haven’t made any errors of course.

But I don’t know how to solve this!

Thanks for any guidance,
-phrak
 
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  • #2
to

Dear phrakto,

Thank you for bringing up this interesting question. As you have correctly stated, Maxwell's equations do allow for propagating waves of charge. These are known as electromagnetic waves, and they are governed by the wave equation you have derived.

It is important to note that in Maxwell's equations, the electric and magnetic fields are not fundamental quantities, but rather they are derived from the electromagnetic potential A. Therefore, it is the potential A that carries the information about the propagation of waves, not the fields themselves.

In order for there to be propagating waves of charge, the 4-current J must be non-zero over a finite region of space-time. This means that there must be a source of charge that is producing the current. In vacuum, where there is no source of charge, the 4-current is zero and there are no propagating waves of charge.

However, in the presence of a source of charge, the 4-current can be non-zero and the wave equation you have derived can have non-trivial solutions. These solutions will correspond to propagating waves of charge with a velocity of c.

It is also worth mentioning that the wave equation you have derived is not the only way to describe propagating waves of charge in Maxwell's equations. The equations can also be written in terms of the electric and magnetic fields, and the wave equation for these fields will also have solutions that correspond to propagating waves with a velocity of c.

I hope this helps to clarify your question. Keep up the good work in your studies of differential topology and Maxwell's equations!


 
  • #3


Yes, there are propagating charged waves admitted in Maxwell’s equations. These are known as electromagnetic waves, which were first predicted by Maxwell's equations and later confirmed by experiments. They are waves of electric and magnetic fields that travel at the speed of light, c, and carry energy and momentum.

Maxwell's equations are indeed robust and allow for the existence of these waves. The wave equation you have obtained is the same one that describes the propagation of electromagnetic waves. The fact that the electric and magnetic fields are related to each other through the speed of light, c, is what allows for the existence of these waves.

It is important to note that the wave equation you have obtained is a mathematical representation of the physical phenomenon of electromagnetic waves. It is not simply a manipulation of equations. The solutions to this equation describe the behavior of electromagnetic waves in space and time.

In terms of the 4-current wave equation, the amplitude of J_{0} can indeed be non-zero over a region. This represents the presence of a varying electric current, which is necessary for the generation of electromagnetic waves. So, in summary, Maxwell's equations do allow for propagating charged waves, and their existence has been confirmed by numerous experiments.
 

1. What are propagating charged waves?

Propagating charged waves are electromagnetic waves that carry both electric and magnetic fields and propagate through a medium or vacuum. These waves are produced by moving charged particles and can travel through space at the speed of light.

2. Are propagating charged waves admitted in Maxwell’s equations?

Yes, propagating charged waves are accounted for in Maxwell’s equations, which describe the behavior of electric and magnetic fields and their interactions. These equations show that changing electric fields produce magnetic fields, and changing magnetic fields produce electric fields, leading to the propagation of electromagnetic waves.

3. What is the significance of propagating charged waves in Maxwell’s equations?

The inclusion of propagating charged waves in Maxwell's equations is significant because it shows the fundamental relationship between electricity and magnetism and how they are connected through the propagation of electromagnetic waves. These waves are the basis for many modern technologies such as radio, television, and wireless communications.

4. Can propagating charged waves exist in a vacuum?

Yes, propagating charged waves can exist in a vacuum because they do not require a medium to propagate. This was a key discovery made by Maxwell, which helped unify the previously separate theories of electricity and magnetism.

5. How do propagating charged waves differ from other types of waves?

Propagating charged waves differ from other types of waves, such as mechanical or sound waves, in that they do not require a medium to travel through. They can also travel at the speed of light, whereas other waves have a finite speed. Additionally, propagating charged waves can have both electric and magnetic components, while other waves typically have only one type of field.

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