Questions about Maxwell's equations/radiation

[anonymous299792458]

1. Maxwell's equations do not hold in NON INTERTIAL reference frames, right??

2. Let's say you have a charge which was briefly accelerated. You surround it by a closed surface CLOSE to the charge and integrate the Poynting vector over this surface and with respect to time to get the TOTAL energy radiated (the energy which passed through this surface). Now you surround the charge by a closed surface INFINITELY FAR from the charge and find the total energy which went through this surface. Will the second integration yield LESS TOTAL energy than the first? I'm thinking that some of the energy which passes through the closed surface near the charge does not make it to infinity, rather it is "used" to increase the magnetic field of the charge. Remember, the magnetic field will have to increase since the speed of the charge increased, and this energy has to come from somewhere. Now what if the acceleration was negative (i.e. the speed of our charged decreased), will the radiation at infinity be greater than radiation close to the charge? If not, where does the energy which was in the magnetic field end up? Could someone who really knows this stuff tell me which, if any, errors I've made.

 

[dextercioby]

For the first part,NO,THEY DO NOT...The generally covariant equations are called Maxwell-Einstein equations.

Daniel.

 

[pmb_phy]

1. Maxwell's equations do not hold in NON INTERTIAL reference frames, right??

Maxwell's equations can be expressed in what is called covariant form which basically means that its expressed in terms of tensors. As such they hold in all coordinate systems. However if you express them in coordinate form and the spatial coordinates are Cartesian then you'll get a set of equations which are different than the ones corresponding to the inertial frame.

As far as the second part its an excellent question. I don't know the answer though, however I know someone who probably does. In fact he's an exert on this stuff. Let me ask him and I'll get back to you on this if he's available.

The generally covariant equations are called Maxwell-Einstein equations.

Huh? They're the same equations. Most people (e.g. MTW, Wald, Rindler, Weinberg etc.) simply call them "Maxwell's equations." I've rarely, if ever, seen them referred to as Maxwell-Einstein equations. That's simply a different name for the exact same thing, i.e. Maxwell's equations in tensor form.

Pete

 

[dextercioby]

I wouldn't call them Maxwell equations...They come the variational principle imposed to the HE-Mawell action and it really woudn't be fair...

Daniel.

 

[pmb_phy]

I wouldn't call them Maxwell equations...They come the variational principle imposed to the HE-Mawell action and it really woudn't be fair...

Daniel.

So? Why wouldn't it be "fair"?

Pete

 

[vincentchan]

for the second part of the question, I can answer you...
YES,YOU ARE RIGHT, if you have INTRODUCTION TO ELECTRODYNAMICS, 3rd edit, by Griffiths, flip to page 460, it explains clearly... I'll qoute some of its word without the math involve...

However, not all of this energy flux my comment: for accelerating point charge constitutes radiation; some of it is just field energy carried along by the particle as it moves. The radiated energy is the stuff that, in effect, detaches itself from the charge and propagates off to infinity. (It's like flies breeding on a garbage truck: Some of them hover around the truck as it makes its rounds; others fly away and never come back.) To calculate the total power radiated by the particle ... the rest is mathematics, but that he simply said is use the retarded E field calculate the poynting vector, and integrate over a sphere r \rightarrow \infty

hope tis answer your question

 

[dextercioby]

Because they're different from the ones written by Maxwell round 1863.I don't know whether it was Einstein or Hilbert (or even somebody else) the first to write them (the general covariant tensor equations) down,but it certainly wasn't James Clerk Maxwell...

Daniel.

 

[pmb_phy]

1. Maxwell's equations do not hold in NON INTERTIAL reference frames, right??

2. Let's say you have a charge which was briefly accelerated. You surround it by a closed surface CLOSE to the charge and integrate the Poynting vector over this surface and with respect to time to get the TOTAL energy radiated (the energy which passed through this surface). Now you surround the charge by a closed surface INFINITELY FAR from the charge and find the total energy which went through this surface. Will the second integration yield LESS TOTAL energy than the first? I'm thinking that some of the energy which passes through the closed surface near the charge does not make it to infinity, rather it is "used" to increase the magnetic field of the charge. Remember, the magnetic field will have to increase since the speed of the charge increased, and this energy has to come from somewhere. Now what if the acceleration was negative (i.e. the speed of our charged decreased), will the radiation at infinity be greater than radiation close to the charge? If not, where does the energy which was in the magnetic field end up? Could someone who really knows this stuff tell me which, if any, errors I've made.

Note: If you were to find the total mass-energy of a charged particle in the particle's rest frame (energy of rest mass + mass-energy of electric field) then to find the total mass-energy of the moving particle you simply multiply by gamma m_0 where m_0 is the "rest mass + mass-energy of electric field in zero momentum frame". m_0 is called the "electromagnetic rest mass" of the particle.

Pete