Energy conservation in electric field scattering from a discontinuity

[cljc]

Hi,

I'm a EE PhD student working a little bit out of my area, and have just gotten stumped trying to figure out the transient dymanics of a relativistic electron moving past a discontinuity. My little thought problem came up from wakefield interactions of a relativistic electron in a waveguide. Here's a general reference to the field: slac.stanford.edu/cgi-wrap/getdoc/slac-pub-4547.pdf. I tried to include an inline link, but the forum gave me the error: You are welcome to include a link only after reaching 10 posts. I'm not very welcome, I suppose :).

I'm trying to wrap my head around the energy flow into waveguide modes from a relativistic electron moving past a metallic (either resistive or PEC) discontinuity.

I made a quick little graphic in PDF form to attach as a descriptive aid.
judylab.org/lib/exe/fetch.php?id=people%3Ajere_harrison&cache=
cache&media=people:slide.pdf
(You'll need to put www . before that to get to the file because of how our server is set up. Sorry I couldn't attach it - same issue with the number of posts.)

When I think about this in the intertial frame of the waveguide, a Lorentz contracted electric field comes in contact with the scattering boundary at t_1, producing a propagating electromagnetic field. The electric field is essentially contained within a thin disk perpendicular to the direction of motion for γ >> 1. This scattered electromagnetic field catches up to the relativistic electron at t_2. Now, after the electron has been traveling down the waveguide for a while, the steady state field matching all makes sense to me and I can find the wakefield, energy conservation looks solid, etc. But I cannot figure out how the transient behavior of the fields between t_1 and t_2 fits with energy conservation.

I realize that there will be a little bit of electric field from the electron in the direction parallel to the electron velocity, but it is vanishingly small. The vast majority of the energy in the excited TM electromagnetic modes that interact with the electron comes from the transverse component of the electron's field, which doesn't get excited until the electron is right there. As soon as the fields in the waveguide are excited by surface currents in the metal, there's got to be energy flow. But the electron doesn't see the electromagnetic field from the metal wall until it's passed it by a ways.

And if the waveguide is short enough that the electron exits the other side before the wakefield catches back up to the electron... can you bend the electron away with a steering magnet? How will the wakefield catch back up to balance the energy? It just doesn't make sense to me, so I know I'm missing something here.

Thanks!
Jere

 

[Q-reeus]

Hi,

I'm a EE PhD student working a little bit out of my area, and have just gotten stumped trying to figure out the transient dymanics of a relativistic electron moving past a discontinuity. My little thought problem came up from wakefield interactions of a relativistic electron in a waveguide. Here's a general reference to the field: slac.stanford.edu/cgi-wrap/getdoc/slac-pub-4547.pdf. I tried to include an inline link, but the forum gave me the error: You are welcome to include a link only after reaching 10 posts. I'm not very welcome, I suppose :).

I'm trying to wrap my head around the energy flow into waveguide modes from a relativistic electron moving past a metallic (either resistive or PEC) discontinuity.

I made a quick little graphic in PDF form to attach as a descriptive aid.
judylab.org/lib/exe/fetch.php?id=people%3Ajere_harrison&cache=
cache&media=people:slide.pdf
(You'll need to put www . before that to get to the file because of how our server is set up. Sorry I couldn't attach it - same issue with the number of posts.)

When I think about this in the intertial frame of the waveguide, a Lorentz contracted electric field comes in contact with the scattering boundary at t_1, producing a propagating electromagnetic field. The electric field is essentially contained within a thin disk perpendicular to the direction of motion for γ >> 1. This scattered electromagnetic field catches up to the relativistic electron at t_2. Now, after the electron has been traveling down the waveguide for a while, the steady state field matching all makes sense to me and I can find the wakefield, energy conservation looks solid, etc. But I cannot figure out how the transient behavior of the fields between t_1 and t_2 fits with energy conservation.

I realize that there will be a little bit of electric field from the electron in the direction parallel to the electron velocity, but it is vanishingly small. The vast majority of the energy in the excited TM electromagnetic modes that interact with the electron comes from the transverse component of the electron's field, which doesn't get excited until the electron is right there. As soon as the fields in the waveguide are excited by surface currents in the metal, there's got to be energy flow. But the electron doesn't see the electromagnetic field from the metal wall until it's passed it by a ways.

And if the waveguide is short enough that the electron exits the other side before the wakefield catches back up to the electron... can you bend the electron away with a steering magnet? How will the wakefield catch back up to balance the energy? It just doesn't make sense to me, so I know I'm missing something here.

Thanks!
Jere

This article may help, but I didn't find it particularly illuminating: www.worldscibooks.com/etextbook/5835/5835_chap1.pdf
One assumes the slack is taken up in the cross interactions between the wake and particle fields (which are there 'from the beginning'), as the system includes both Lorentz force charge/field, and field/field contributions to energies and momenta. It's the latter that likely would count most re any subsequent magnetic deflection of charge away from wakefield. If the obstacle is a thin annular diaphragm, the radial acting induced diaphragm currents generate a field opposing that producing the current, so one expects diminution of the net field momenta and energy owing to the source charge. As you have noted, eventually it balances out to include a lag force on the charge itself.

[Just got round to looking at your slides at http://www.judylab.org/lib/exe/fetch.php?id=people%3Ajere_harrison&cache=cache&media=people:slide.pdf
While I realize they are just illustrative, the way they are drawn, it's easy to get the impression the induced wakefield is comparable in energy magnitude to that of the source charge. But this will not be true - not for a single passing charge. Although the region of source-charge/wakefield field interference is initially small, field energy being parametric wrt field strength, the generally -ve 2E.E' term in (E+E')² at a point within such region typically dominates over E'² term, where E>>E' refers respectively to the source charge, and induced wakefield, interacting fields, and similarly for B fields.]

 

[cljc]

That textbook chapter actually helped out a lot! I hadn't thought thoroughly about why the field was compressed into a 'pancake' transverse to the electron motion. I was just applying the Lorentz transformation to the electromagnetic field and not realizing that the compression of the field into that 1/γ disk is a bit more nuanced - the further the distance in the 'pancake' from the electron that you sample the field, the further back in time (or position) it is from.
I believe that takes care of the transient energy conservation issues that I had.

Thanks!

 

[Q-reeus]

That textbook chapter actually helped out a lot! I hadn't thought thoroughly about why the field was compressed into a 'pancake' transverse to the electron motion. I was just applying the Lorentz transformation to the electromagnetic field and not realizing that the compression of the field into that 1/γ disk is a bit more nuanced - the further the distance in the 'pancake' from the electron that you sample the field, the further back in time (or position) it is from.

Fig. 1.1 and accompanying text in that article gives a semi Lienard-Weichert retarded-time perspective - retarded-time contributions indicated but field shown in purely radial acting present-time form. Retarded time usage is of course needed if acceleration occurs. Unlike for accelerated charge motion, the SR transformation has that pancake field 'impact' with obstacle occurs when obstacle and source charge are broadside, regardless of how far laterally apart. Well not exactly as there is always finite angular spread, but true for the field peak. The wakefield then propagates as that from an impiulse excited damped oscillator. Detailed calculations will be difficult, and the easy way is to trust that the Poynting energy theorem holds exactly. But thanks for the thanks!