Understanding the Uniqueness of Retarded Time in Electromagnetism

[arcTomato]

Hello, PFI'm studying electromagnetism, delayed potentials, and I've been wondering about a few things.
My understanding of the delayed potential is that the information from a moving charged particle at a given time travels through space at the speed of light to reach the observation point, so what is actually observed is the potential that deviates from the true position of the charged particle.

My question is: Is it possible for information about a charged particle to be observed from more than one position at the same time?　In other words, is the particle's location information indeterminate?Thank you, have a good day.

 

[A.T.]

My understanding of the delayed potential is that the information from a moving charged particle at a given time travels through space at the speed of light to reach the observation point, so what is actually observed is the potential that deviates from the true position of the charged particle.

If the charge moves at constant velocity, the Coulomb force points towards the current location. See my recent post:

In this applet you can choose "linear" to show a charge at constant velocity. :

https://phet.colorado.edu/en/simulation/radiating-charge

The mechanism is roughly: The field lines emanating from the charge "inherit" its velocity. So if the charge maintains a constant velocity, they will remain straight and radial around the charge. If the charge velocity changes, the field lines just at the charge "inherit" the new velocity, but this disturbance in the pattern propagates at the finite speed c, resulting in "waves".

My question is: Is it possible for information about a charged particle to be observed from more than one position at the same time?

What do you mean by "observe from more than one position at the same time"? You can only have one direction for the Coulomb force at some point in space and time. But if the charge is accelerating, you might not be able to deduce it's position from that measurement.

 

[arcTomato]

Thank you.

What do you mean by "observe from more than one position at the same time"?

What I meant was a situation like where the information of a moving charged particle at time $t_a$ and the information at time $t_b$ is observed simultaneously at time $t_c$.

What exactly are the circumstances, if such a situation could happen?

 

[A.T.]

What I meant was a situation like where the information of a moving charged particle at time $t_a$ and the information at time $t_b$ is observed simultaneously at time $t_c$.

How is the information observed exactly?

 

[arcTomato]

How is the information observed exactly?

Such as for example potential. (That means information propagates at the speed of light.)

 

[A.T.]

Such as for example potential.

How do you observe the potential exactly?

 

[pbuk]

How do you observe the potential exactly?

And how do you define 'simultaneously' for measurement at two different points?

 

[arcTomato]

How do you observe the potential exactly?

Oh, I see. I really didn't think that much about it because this is just a thought experiment of mine. I'm sorry.But how we observe doesn't seem to be that important as far as my question is concerned, I think.

And how do you define 'simultaneously' for measurement at two different points?

Let me clarify my question. (And thank you for answering my question.)

"The potential at time $t_1$ and the potential at $t_2$ are simultaneously observed when a moving charged particle is observed at a single point $M$."

I don't expect to make observations at more than one point.
Is this situation possible?

 

[A.T.]

Oh, I see. I really didn't think that much about it because this is just a thought experiment of mine. I'm sorry.But how we observe doesn't seem to be that important as far as my question is concerned, I think.

it might help you to answer your question, to think about how potentials are related to actual observations,

 

[Dale]

"The potential at time t1 and the potential at t2 are simultaneously observed when a moving charged particle is observed at a single point M."

I am not 100% sure that I am understanding your question. However, if I am understanding correctly then the answer is that the retarded time is unique, so the retarded potential is unique also.

 

[arcTomato]

If I am understanding correctly then the answer is that the retarded time is unique, so the retarded potential is unique also.

Thank you. and sorry for my bad English.
Could you tell me more about it?

Does that mean that the retarded time is uniquely determined because it only depends on the distance between the observation point and the charged particles?

 

[vanhees71]

I am not 100% sure that I am understanding your question. However, if I am understanding correctly then the answer is that the retarded time is unique, so the retarded potential is unique also.

The retarded vector potential in electrodynamics is unique up to a gauge transformation. The retarded vector potential is in the Lorenz gauge, but even within the Lorenz gauge there's some restricted gauge freedom left.

What's, of course, unique is the electromagnetic field $(\vec{E},\vec{B})$ which for the usual physical situation, where you have some localized sources (charge-current distributions), is the retarded solution, which you get by the usual derivative of the vector potential, leading to what's known as Jefimenko equations.

 

[vanhees71]

Thank you. and sorry for my bad English.
Could you tell me more about it?

Does that mean that the retarded time is uniquely determined because it only depends on the distance between the observation point and the charged particles?

You have to view the electromagnetic field as a functional of the charge-current distribution. It's easier to explain if you look at the solution for a point particle. There it's a functional of the particle's world line. This world line is time-like everywhere, and this implies that the retarded time is always uniquely determined for any spacetime point of the observer (the only other solution is the advanced time, which is ruled out by causality reasons).