Which equations indeed describe moving/accelerated charge field?

Question

The thing that I have not understood completely, for 100%, is how does the field of the electric charge is "updated".

1. If to consider a frame of reference where the charge is static, does not move at all, it is simple - the value and the direction of field at point can be calculated from Coulomb's force: $$F = k\frac{q_1 q_2 }{r^2}$$ The important thing is that the value at some point is constant in time (for static charge).

2. When the charge is moving with constant speed, as far, as I understand, because of space contraction, the field is flattened in the direction of the electron move. The results of that contraction in frame of reference are called magnetic field.

   What I do not understand at this point is how does the field of constantly moving charge depends on time?

   If to consider an accelerating charge, the information about the field is propagates with the speed of light, but what about a constant moving charge?

   Say, there is a constantly moving charge and an observer, situated on 300 millions meters from the charge. Will the field at far point be the same as it should be from Coulomb's law (corrected for Lorentz boost), or when the charge will left some point, the far observer still will be "seeing" it there? Is there a "delay" as for accelerated charge? If yes, why?

3. What I do not understand the most in accelerated charge field is why the initial field is "retarded", so it is need a time for new field propagation to reach the observer, but the new "new" field of now already constantly moving charge is changes instantly? Why? Because of what? Or it is not? Maybe even for constant moving charge there is delay, and it has that perpendicular retarded component, but it is tiny?

For example this famous animation, from which equations it was build?

I also intuitively understand but at the same time do not understand why lines at propagation are perpendicular to initial field?

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I will read an entire book, if it will have the answers, but at least point me which one.

Answer 1

In the electrostatic case you have $\boldsymbol{E}\equiv\boldsymbol{E}(\boldsymbol{r})$, but for isotropy actually $\boldsymbol{E}\equiv\boldsymbol{E}(|\boldsymbol{r}-\boldsymbol{r}_{\text{charge}}|)$, where $\boldsymbol{r}_{\text{charge}}$ is the position of the point-charge.

Suppose that $c\to\infty$, so if your charge is moving with velocity $\boldsymbol{v}_{\text{charge}}$ you will have $$ \boldsymbol{E}\equiv\boldsymbol{E}\left(|\boldsymbol{r}-\boldsymbol{r}_{\text{charge}}|\right) $$ at some instant, and you will have $$ \boldsymbol{E}\equiv\boldsymbol{E}\left(|\boldsymbol{r}-\boldsymbol{r}_{\text{charge}}-t\,\boldsymbol{v}_{\text{charge}}|\right) $$ some instant later. The field would depend on $t$, but it's still an electrostatic one and points towards the actual position of the charge.

Suppose now that $c$ is finite, you will have $$ \boldsymbol{E}\equiv\boldsymbol{E}\left(|\boldsymbol{r}-\boldsymbol{r}_{\text{charge}}|\right) $$ at some instant, but now you have to confront $|\boldsymbol{r}-\boldsymbol{r}_{\text{charge}}-t\,\boldsymbol{v}_{\text{charge}}|$ with the distance $tc$. Turns out is not very easy, but at least you could say that, when the field is "updated" in the point $\boldsymbol{r}$, is updated with an "electrostatic field of the past" and points to the direction the charge was in this past, not the direction the charge is now.

Given this different direction you can understand why the field lines get deformed, and this exactly causes the magnetic field to be. In fact magnetic field exists just by the fact $c$ is finite. In a way magnetism exists because special relativity does and is in fact a relativistic phenomenon!

About the third question, I really don't know how to answer that without citing Liénard-Wiechert potential and the associated theory, that you can find in some books like Jackson for example. Hope this helps.

Answer 2

I had the same question in my mind too, but maybe it is clear now. To answer your question, I have added a picture that contains a diagram drawn by me. I would be referring to that picture as an example for giving you an explanation.

The conclusion that I reached probably explains everything including the simulation. One could see from the simulation that after the charge attains a constant velocity, it appears as if the charge is pulling all the electric field lines with it and there is no 'delay'. This is the same thing as saying that an observer moving with the same velocity as the charge would see no difference in his/her observations from the case where he/she was placed next to a static charge. But this may seem counterintuitive and you could think, "Shouldn't there be a 'delay' because no information could travel faster than the speed of light in vacuum? ". The answer to that question is YES. This explanation takes into account the 'delay', and even then, the end result seems to appear as an instantaneous updation in the electric field.

So the explanation is as follows:

Consider the image attached below. (Hope you could see the image, because the image is everything to this answer)

Imagine a charge (indicated by a '+' inside a circle) moving in free space with a constant velocity of 2 m/s, for billions of years, so that we do not have to worry about its initial acceleration that might have been required to attain its current velocity.

Let's say that 45 million years before today, we had set a timer as t=0s. This means that the position of the charge at t= 45 million years marks the present day position of the charge. This is made clear in the diagram.

Also, let's say that we select some points A, B, C and D arranged in a straight line in space, parallel to the direction of motion of the charge, but separated from its line of motion by a huge distance (millions of light years), with every next point separated at a distance of 2m from the previous point. At present, the position of the charge is directly below point A. This arrangement is made clear in the image, and you would soon understand why I chose this setup. Since the charge is moving at 2m/s, and since the present position of the charge (t= 45 million years) is just below point A, for every second that passes, the charge would jump to the position directly below the next point.

The arrows at A, B, C, and D represent the directions of electric field as seen by the charge when it is at its present day position (doesn't mean they belong to the present day charge, infact they belong to the same charge 45 million years ago for this example. This point would be impressed upon later in the answer). The letters x, y and z represents the angles that the electric field makes with the line connecting A, B, C and D.

Now, pay attention closely because I would be discussing about the delay that you had mentioned in your 3rd question.

In the image, the point A was selected such that it takes 45 million years for the electric field information to reach point A from the t=0 position of the charge. Now think about this..... the electric field (magnitude and direction) at point A which would be observed at t= 45 million years (present day), is nothing but the electric field which belonged to the same charge at t=0s, i.e., 45 million years ago.

Hence, one could say that the electric field observed at point A at t= (45 million years) is caused due to the same charge when it was far away at t= 0s. By the same logic, one could also say that the electric field observed at point B at t= (45 million years + 1s) is caused due to the same charge when it was far away at t= 1s. Move forward with the answer only if you got this point.

If the charge sees the electric field at an angle x at A at t= 45 million years, it should see the electric field making the same angle x at B at t= (45 million years + 1s). The only diffrerence is that these two electric fields are created by the charge at t=0s and t=1s respectively. The angle x is same because the geometry is exactly the same, just that the entire system is shifted (now, point B is the same as point A, t=(45 million years + 1s) is same as t= 45 million years, and t=1s is the same as t= 0s, there is no difference in the physics/geometry involved).

What does this mean? This means that as you move with the charge, you measure the same angle x at A, B, C, D etc as long as the charge is below that point at which you measure the electric field. For you, the electric field direction may appear static at a point moving along with the charge.

Also, you may think that the electric field direction changes continuously and instantaneously at a point(for example, from angle y to x at point B as we move from t= 45 million years to t= (45 million years + 1s)), but what actually happens is that this effect is because of the electric field information that continuously reaches this point from the position where the charge was located ages ago.

Instead of doing this excersise for just four points A, B, C and D, do it for infinite points in space. The results agree with the simulation because if you try moving with the charge in the simulation, you won't observe any difference in the electric field's magnitude/direction at some point that also moves with the charge with the same velocity, as long as that point lies inside the EM wave sphere. Hence you wouldn’t observe the 'delay' even though it is at play.

Hope you got my point.