# Charged Massless Waves in Maxwells Equations?

1. Jan 8, 2008

### Phrak

Hello. This is a first post for me.

Do Maxwell's equations alone allow for propagating waves in charge/current (\phi,s[/B]J)?

I was rather struck dumb by this question out of the blue. I've never seen it addressed, denyed or confirmed.

Schematically the electric and magnetic fields are first derivatives of the vector and electric potential, and can propagate as waves under first derivative constraints of the currents and charges.

Schematically the charge/current is a second derivative of the potentials. It seems it should have waving solutions under fourth derivative constrains of the potentials.

To make thing even better, there is no other velocity in maxwells equations other than c,
so should Maxwell's equations admit charged fields that propagate at the speed of light?

2. Jan 9, 2008

### olgranpappy

there can be charge oscillations (waves). for example, plasma oscillations in a metal.

3. Jan 9, 2008

### Mentz114

All the charges we encounter are a property of matter, and I think all the charged particles we know have mass. This rules out actual charge propagating at light speed.
The electric field component of the EM wave doesn't have a source as such.

Another problem would be that a charge moving at light-speed would create an infinitely powerful magnetic field.

4. Jan 9, 2008

### Troels

I would say there is a flaw in your reasoning; it seems to me that you are refering to the wave-equation for electromagnetic waves in vacum, which does not accurately descripe an EM-wave in pressence of any charge or dielectric.

I think you will be able to compare the problem to that of an EM-wave in a conducter. Such wave propagetes with exponentially decreasing amplitude, because free carge tends to cancel off the electric field as it goes. This effect is dependent on the conductivity of the material. Now, since a gass of charge has infinite conductivity, an EM wave in such gas would in theory be extinguished instantaneous.

So I'd say the answer is no.

5. Jan 9, 2008

### mda

Yes it does. An antenna is an obvious example.

6. Jan 9, 2008

### mda

First I agree the the OP has possibly missed the point that EM waves do not have to travel at c.

However, it's not as simple as you have implied. Firstly a plasma does not actually have infinite conductivity, so the EM wave is not fully extinguished: it does decay but it effectively tunnels to the other side.

Secondly (non-local) plasmons are longitudinal waves, meaning the charge oscillates in the direction of travel. The particles themselves do not have to travel at the speed of light (even for a transverse wave). Dispersion relations of plasmons show that in fact they can have very interesting properties, and can propagate at c.

7. Jan 9, 2008

### Mentz114

After the current in the antenna has stopped, the waves are still propagating. They now have no source.

8. Jan 9, 2008

### Phrak

Thank you all for your generous responses.

I'm afraid I stated the problem poorly, however. I mean to say: Ignoring all the rest of physics, and utilizing only Maxwell's formalism, are propagating waves of 4-currents existant in the formalism? Homogenious, nontrivial solutions to be more precise.

This would be so much easier to state in the language of differential forms, and maybe Laplace-De Rham operators and stuff. But the math can get pretty exotic. Is the Classical Physics category the right place to post this sort of question?

9. Jan 9, 2008

### olgranpappy

look. it's not so clear what ur asking. the currents are typically given, and the fields solved for. what are you asking for?

then do it.

yes. and you could post in the math forums too, i suppose.

10. Jan 9, 2008

### Phrak

I shall try again in the Tensor Analysis & Differential Geometry section with better presentation.

11. Jan 9, 2008

okay