electric field of a moving charge

[ilyal]

im taking a physics cource, and we were given the following question:

suppose there is a charge, whose notion is as follows:

from t=0 to t=1 it moves at 0.5c,
then from t=1 to t=2 it moves at -0.5c

the motion is describied with the following graph:
the y axis is location and x axis is time

  /\
 /  \
/    \

now were asked how would the motion look for an observer in point p
whoe at 3 light seconds away from both the begging end the end of the motion

p
  /\
 /  \
/    \
s end
t
a
r
t

and the anser give is
the charge will wont move until t=3,
then the observer would recive the information of the charge start of movesmnet and t=0
and therefore would assume the charge was moving at 0.5c for 3 second and would see the chage
at x = 1.5, then at t = 4 it would recive the information that the charge arrived to point x=1
and would assume the charge was moving at -0.5c for 3 seconds and therefore is located in x = -1.
and at time t=5 it would recive the news that the charge has stopped.

the graph is as follows:
         /
        /

------     --

          \
           \


i dont understand why the charge jumps thru space, i i asked why if we could actully see the charge, if that what we would actuly see, and they said no, thats what you will "see" according to measurments of the electric field at poing p,


i dont understand why the ansers to those questions are diffrent, both questions ask how a em field propogates thru space,
shouldnt they both give the same answer?


thx,
ilya

 

[Born2bwire]

The answer to the two questions are the same. Either way of measuring the charge is being done by the electromagnetic wave propagating from the charge.

I don't understand your statement of the problm though.

Since the charge and distance scales are relativistic, you are dealing with retarded fields. It takes finite amount of time for the electric field to propagate and this causes the lag in information. If one were to make the appropriate transformations via the special rule of relativity and retarded potentials, you would not see any "jumps" in the position. I believe that there is a jump here because the observer is naively assuming Galilean transformations. If one were to measure the retarded field at the instant that it starts, you would get it's starting position I guess by deduction from the field strength. Then the field would be distorted until it stops and changes velocity. When it stops, you can then make a second undistorted field observation and note that the position is different from what you expected it to be due to the lag from the retarded fields. Likewise with the last field measurement. Instead, the Lorentz transformations of the fields will distort the electric field and introduce a distortion to the magnetic field. This distortion is dependent upon the speed of the fields' source and the direction of travel with respect to the observer. In the charge's reference frame, it will be emitting a constant electric field, but this is not so in the observer's reference frame where you will have an electromagnetic field being propagated.

So I don't understand what the position of the observer is in relation to the path of the charges motion, nor do I understand the actual path of the charge, you say it moves at -0.5c until some time t2, but what is t2? My guess is simply this is just a confusion from measuring the retarded fields.

 

[ilyal]

what transformation do i need to do?

the point p frame and the lab frame where the description of the motion is given
are in the same refernce frame, they are only distant from each other.

can you also please describe how you think the motion would look like from point p?

i am sorry about the t2, it sould be t=2,
the charge moves in a stright line from start to end and then back to start,

the observer in p is located at such location that he is 3 light seconds from both
the start location and the end location

 

[Born2bwire]

I have no idea how the motion would look like from p because I do not understand how p is located. Is the location of p along the dimension of motion or is it offset in space? Is p offset by 3*c from the position of the particle at t=0 and t=1 or t=0 and t=t2? You don't specify what the beginning and end is.

The answer seems weird. It seems to be aware of retarded fields but at the same time ignorant of their consequences. It says that it sees that motion of the field after three seconds, and that would be correct if the point p is offset at a distance of 3*c meters away. But there would be no reason to assume that in the three seconds from the time that it actually started to move until the time that you received the information that the charge was moving at a constant velocity of 0.5*c as the answer assumes. The next measurement is taken at t=4 when you would get the info that it has changed velocities. The jump occurs because of incorrectly assuming the motion of the particle but I do not see why this would be so. If you assume retarded fields then you should know that there is a time lag in the information. In addition, appropriate measurements of the fields that are created by the motion of the particle should give you the appropriate updates of its position. An observer aware of retarded fields and Lorentz transformations should be able to discern what the correct path of the charge was (or at least a solution set of paths).

 

[ilyal]

the start location is the location of the particale at t =0 (and t =2 its the same location), x=0
the end of the path is the location of the particatle at t=1, x=0.5c * 1 = 0.5c

the point p is not along the path, its offset by 3*c from both the start point and the end point, all the three points create a triangle, with the followin edges lenghts;

p - start = 3c
p - end = 3c
start -end = 1c

hopes that clarifys the geomatry.

my problem is the same as yours, i dont understand why you should assume the charge was moving at the same speed for 3 sec after you saw it start moving.

anywhy this is the answer i got from my professor and i still dont understand:

If the question concerned a ball, which you can see with
your eyes, then you would be (pretty much) right. The
observer at P would not see the velocity as constant, however,
but for the geometry given, this effect is small.


This, however, was not the point of the question (though I
agree that your interpretation is reasonable). What they
were asking involved measuring, at the point P, the electric field that the
moving electron produces. Now you know that while the charge is
moving at constant velocity, or until such time as a change is
observed at P, the E of the particle is radial, with the apparent
center being the actual current position of the electron. When
there is a change - that is, acceleration - it takes some time (r/c)
for the news to arrive. This is the reason that the answer is what
is given.

 

[Born2bwire]

Ah, I see. I assume then that you track the position of the particle by the direction of the electric field being measured. As the particle moves, the vector of the electric field will rotate to continue pointing away from the retarded position of the charge. But the problem is that the field lines will propagate since they were emitted by a moving charge. Once the field has been emitted, it will propagate out just as if the charge was still moving. So what happens is that the apparent position due to the field is offset than if the field was emitted by a stationary charge. Basically the retarded field will point to where the charge would have been if it had continued propagating.
You can draw the electric field out....

http://webphysics.davidson.edu...ets/Retard/Retard.html

Choose the SHO to see the field lines for the situation that is described above. There aren't any jumps in the the direction of the field lines. You can see that due to the continued propagtion of the "retarded" charge that the apparent position is offset than what it actually was. The acceleration of the charge also causes distortions to the field lines. Play around with the velocities and switch between the SHO and Inertial settings in that applet to visualize what happens when you accelerate the charge and when you have a moving charge versus stationary.

 

[ilyal]

ok, thx,
that make alot more since,

now, can you please explain why the situation is different for visible light?
or if it isn't, why we don't see the stars jumping in the sky?

 

[Born2bwire]

Originally posted by: ilyal
ok, thx,
that make alot more since,

now, can you please explain why the situation is different for visible light?
or if it isn't, why we don't see the stars jumping in the sky?

It is and it isn't. The problem why the position jumps around is the fact that we are erroneously estimating the position by tracking the direction of the field vectors. I'm sure that if you could instead, using your knowledge of the initial position and path of possible motion, figure out what the actual motion of the particle was by solving for the actual field. The actual field would contain more information than if you were just going by the field vectors (though you would need to be aware of the retarded fields). But this is something that happens whenever you have a source moving at a velocity on the order of the propagation of the fields/waves. Normally, we just take for granted the fact that the speeds of our sources, and the distances from them, are very small compared to the propagation of the signals.

With visible light, and with any moving wave source, we can get the doppler effect as something akin to this. Here, the moving source spaces out the wave fronts differently than if it was stationary. If a source moves towards the observer at a high enough velocity, then the observed frequency of the waves is higher than the frequency when emitted in the moving frame. We see this in our observation of celestial bodies, like quasars. We have a good idea of the actual frequencies of radiation being emitted by a quasar and by measuring the observed frequencies of the received signal we can get an idea on the rate of expansion of the universe. These are the so called "blue" and "red" shifts.

This is all independent of the relativitistic effects that I was talking about earlier. Right now, you are just assuming that the propagation of fields and waves are limited to c, which is inherent in Maxwell's equations. Special relativity expands upon this with the Lorentz transformations. This causes weird things to happen to the fields. For example, when a single charge moves at velocities on the order of c, you get extra magnetic and electric fields in the rest frame. If the charge is moving in a circle, as in a cyclotron, then you can actually get radiation emitted (technically, any accelerating charge emits radiation but when it dos so at relativistic velocities interesting things happen). This electromagnetic radiation is will actually be projected in the direction of the particle's velocity as the speed approaches c. So a very fast electron in a constant magnetic field will emit radiation almost tangential to it's path. The frequency of the radiation and the angle of it's emmission to the path of motion will describe the strength of the magnetic field and the velocity of the particle. This is called synchrotron radiation, first observed in synchrotron accelerators I believe. The Crab Nebula emits synchrotron radiation and from this we can estimate the magnetic field in the nebula.

So those are some of the effects that you can see due to the finite propagation of the fields/waves and due to the consequences from Special Relativity. I would imagine that after doing retarded fields, near the end of the course, you might hit the Special Relativity and the associated Lorentz transformation of the fields. A graduate electrodynamics course will deal with the radiation emission of a moving particle and the resulting relativistic effects.

 

[Biftheunderstudy]

Be careful with the difference between Cosmological redshifts and Doppler shifts. Although they are pretty much similar, Doppler shift is the one you are looking for. Cosmolgical redshifts are caused by the expansion of space "stretching" the wavelengths.
There are some subtle differences between cyclotron and synchrotron radiation as well, as you mentioned a circular moving charge around a magnetic field line is called cyclotron, when that speed becomes relativistic then it is called synchrotron. There are a few accelerators here on Earth that can create synchrotron light.