# Why are electric field lines straight around electrons moving at relativistic speed?

• shoestring
At a distance ct from the charge, why don't the field lines point to the location where the charge was t seconds earlier?The field lines are bent around the charge because it's moving. From the perspective of an observer at a distance ct, the field lines would be bent around the charge because it has been moving since the observer's time is not instantaneous.

#### shoestring

Imagine an electron in uniform motion moving at a speed near the speed of light.

Pictures I've seen of the electric field around such a charge look like this one from wikipedia:

The field lines are straight, but clustering around the plane perpendicular to the direction of motion.

Why are the lines perfectly straight? It looks to me as if that means that the field can predict the future location of the charge. At a distance ct from the charge, why don't the field lines point to the location where the charge was t seconds earlier?

One thing that really stuck out with me was electrons orbiting around an atom will vanish or jump at orbits that the wavelength is a distance where the previous orbit of the electron will not match up at a full wavelength. It is if they know the round trip will not allow them to wave at the same locations every time around so then they vanish and jump to different valence shells. They only like orbitals where the circumfrance is divisable by a full wave length.

So, I would blame it on spacetime dialation... It gets fuzzy around there, but I think contracted worldlines are involved there for the particles themselves. They know exactly where they are and how fast they are moveing they just don't like to share that knowledge with us :P

I could have asked the same question for a much slower electron, but in that case the effect would have been negligible. For a slow electron the field lines would look very straight even if they were slightly bent.

Short Answer: all motion is relative. If the charge was accelerating, then the lines would bend.

The diagram is valid if the charge has always been moving at constant velocity since $t=-\infty$, so that the information about its motion has had infinite time to propagate.

If the charge's velocity changes, the lines won't be straight. There will be kinks that propagate outward.

shoestring said:
...The field lines are straight, but clustering around the plane perpendicular to the direction of motion...Why are the lines perfectly straight? It looks to me as if that means that the field can predict the future location of the charge. At a distance ct from the charge, why don't the field lines point to the location where the charge was t seconds earlier?
Good question. It's the difference between a 'present position' presentation of field lines (as shown in your borrowed illustration), and a retarded position presentation. As you have figured, the charge location from the point where the field originates will be a function of distance from the moving charge. The further out the observer, the further back (to the left) the charge must have been at the time it acted as a source of field wrt that observer. From that retarded field perspective (indispensable when treating accelerated motion for say charge oscillating in an antenna), the lines are not straight, even for constant speed! Remember that in SR a charge moving at say +v in your frame is equivalent to you moving at -ve past a 'stationary' charge, and so it is 'natural' to expect a relativistic length contraction of field lines along the axis of motion. Accelerated motion on the other hand will not give you reciprocal effect - acceleration in your frame past a non-accelerated charge will give a different result than vice versa.

Q-reeus said:
From that retarded field perspective (indispensable when treating accelerated motion for say charge oscillating in an antenna), the lines are not straight, even for constant speed!

Is that the right way to treat it, meaning that the field lines are actually bent for constant speed too? Or does the field "travel with" the charge as long as it is in uniform motion?

For example, I've heard it said that the Earth's magnetic field rotates with the earth, so that as long as I'm stationary relative the surface of the earth, I'm not "travelling through" the Earth's magnetic field, even though the Earth is rotating. Meaning that a stationary human body shouldn't experience a Hall voltage due to the Earth's rotation and magnetic field, despite moving at a speed of order 1000 km/h. Sitting on a plane, on the other hand, moving at a similar speed relative the earth, could result in a Hall voltage. Or perhaps I've got it completely wrong ...?

shoestring said:
Is that the right way to treat it, meaning that the field lines are actually bent for constant speed too? Or does the field "travel with" the charge as long as it is in uniform motion?
Bent or not bent comes down to 'point of view'. The 'straight lines' picture is what you get when all 'stationary' observers (field points) are taking a 'snapshot' of the source charge's field at the same instant, regardless of how distant each are from the source charge. It's as though the field propagated instantly, a viewpoint not possible if acceleration occurs. As discussed previously, that means the charge's retarded position (ie, the point it was when the field emanated) is in general different for those observers. These retarded field lines are in consequence not straight, but for constant straight line motion happen to arrive at just the right moment for each observer to give the 'illusion' of emanating as straight lines from a single point. The full retarded field equations can be studied at eg: http://www.physics.usyd.edu.au/~kuncic/lectures/HEA_L5.pdf [Broken] (eq'n (4), p44 there) Omit the acceleration parts and you will find the electric field equation for a charge in uniform motion - doesn't give straight lines. But remember this is a retarded time usage, mostly intended to deal with accelerated charge motion.
For example, I've heard it said that the Earth's magnetic field rotates with the earth, so that as long as I'm stationary relative the surface of the earth, I'm not "travelling through" the Earth's magnetic field, even though the Earth is rotating. Meaning that a stationary human body shouldn't experience a Hall voltage due to the Earth's rotation and magnetic field, despite moving at a speed of order 1000 km/h. Sitting on a plane, on the other hand, moving at a similar speed relative the earth, could result in a Hall voltage. Or perhaps I've got it completely wrong ...?
I doubt the Earth's magnetic field can be considered locked to the Earth's rotation (believed generated by an internal dynamo in the core that circulates in a complex way). Best thought of as non-rotating, and you are moving through. Very weak effect that basically gets canceled out from gross electrical neutralization in Earth's crust, atmosphere etc. No danger of frying cell-phones etc! Can be fascinating to study what amounts to a Faraday Disk type of effect: http://en.wikipedia.org/wiki/Faraday_paradox

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