What is duration/extent/pattern of electromagnetic radiation at a point after it has passed that point?

Question

Let us say I have an accelerating charge. At each point x,y,z in its path from my understanding there is a transverse electromagnetic wave being radiated (could also be viewed as a photon). The electric field at any point x1,y1,z1 in the path is disturbed. The moving charge does the same thing all along its path so many electromagnetic waves are created - an infinite amount, one at each point in the path. My question is about what happens to the wave/disturbance created at X1,Y1,Z1 after the charge has passed the point - let's say in some magical way the charge disappears infinitesimally right after X1,Y1, Z1 - there is no more moving charge. Let's also assume that the charge appeared magically an infinitesimal distance from X1,Y1, Z1. So the charge moved a very small distance and then disappeared. What is the fingerprint of the charge after it disappears in the vicinity of X1,Y1, Z1? I am assuming the disturbance spreads out over time from X1,Y1, Z1 - let's assume it's a vacuum that this is happening in. If I put a compass/charge close to X1,Y1, Z1 how long would it show the effect of the moving charge? How would it be calculated? What would happen to compasses/charges put further and further away from X1,Y1, Z1? Would the disturbance die down at all points - some points sooner than others, some points weaker effect than others? One zillion light year away would a super powerful detector of some kind detect the fact that the charge passed X1,Y1, Z1?

As always any feedback is appreciated in advance.

Answer 1

Let us say I have an accelerating charge. At each point x,y,z in its path from my understanding there is a transverse electromagnetic wave being radiated

This is the classical electromagnetic radiation, which works with continua.

(could also be viewed as a photon).

No, photons are part of the quantized theory of electrodynamics, and there is no continuum. It all started with black body radiation, if you remember. The classical produces the ultraviolet catastrophe and the E=hnu existence of a photon was introduced to be able to describe the data. The classical wave emerges from a superposition of photons, but photons have particle interactions and behaviors.

The moving charge does the same thing all along its path so many electromagnetic waves are created - an infinite amount, one at each point in the path.

So there is no infinity in the number of photons or of course in the energy released by the accelerating charge. Photons will be "countable" , i.e generated consecutively by the accelerating interaction, in first order Feynman diagrams.

let's say in some magical way the charge disappears infinitesimally right after X1,Y1, Z1 - there is no more moving charge.

Magic is not part of physics. A charge can disappear, for example an electron can annihilate on a positron. A bunch of electrons on a bunch of positrons. You are describing a pulse.

Let's also assume that the charge appeared magically an infinitesimal distance from X1,Y1, Z1.

No magic allowed. Maybe it was created in a scatter.

So the charge moved a very small distance and then disappeared. What is the fingerprint of the charge after it disappears in the vicinity of X1,Y1, Z1?

There will be a pulse of photons propagating , if enough of them a classical pulse.

If I put a compass/charge close to X1,Y1, Z1 how long would it show the effect of the moving charge?

At the Δ(t) of the passing of the charge, a magnetic field would register, falling with distance according to the classical equations, theoretically , experimentally impossible to measure because of the small signal.

The only effect of a charge having been accelerated will be the radiation emitted due to the acceleration, which might be detected if a photon detector is pointed at the region of the acceleration. Depending on the amount of charge, the photon pulse can be detected. After all astronomers detect gamma ray pulses which happen far away in time and space also possibly due to accelerated charges for small periods.

Answer 2

If you don't mind, I'd like to consolidate your question just a bit as well as phrase it in a bit more concise physics terms. Much of the first part of your question can be paraphrased as:

Suppose a charge appears out of a vacuum at time $t_0$. The charge accelerates a short distance and then disappears at time $t_1$. What is the time-dependent value of the electromagnetic field at an arbitrary point in space? How are these values affected by the fact that electromagnetic waves and even information itself can only travel as fast as the speed of light?

I realize you were hoping for a more intuitive, non-mathematical answer, but a question this intricate needs Maxwell's equations and nothing less. A full solution to this problem using Maxwell's equations is far more than I could ever type on this forum, but Maxwell's equations (and perhaps a computer algebra system such as MatLab or Mathematica) are what you need to answer this question. There's no simple answer that anyone could give in a single sitting.

You also seem to be interested in how things would be affected if the charge were to suddenly appear out of nowhere, accelerate, and then disappear altogether. There are equations called the Lienhard-Wiechart equations that will address this. Once again, solving these equations for an arbitrary point in space is far more than anyone could ever type on this forum, but at least you have a starting point.

You also ask the question:

If I put a compass/charge close to X1,Y1, Z1 how long would it show the effect of the moving charge?

In principle, a frictionless compass would continue to undulate until the charge disappeared. Once the charge disappeared, the compass would stop undulating once all electromagnetic waves (traveling at the speed of light) had moved past the compass. The compass would probably be left in a state of constant rotation after that wave front had passed, or so I guess.

Lastly you ask:

One zillion light year[s] away would a super powerful detector of some kind detect the fact that the charge passed X1,Y1, Z1?

In principle, yes it would. However, it would not begin to show any response until one zillion years after the charge had come into existence (the speed of light is how fast an electromagnetic wave travels of course). Also, its response would be roughly $1/(one-zillion)^2$ times as strong, provided that that your "detector of some kind" were capable of detecting something so weak in the first place.

Answer 3

One zillion light year[s] away would a super powerful detector of some kind >detect the fact that the charge passed X1,Y1, Z1?

More on this: due to quantum mechanics, your detector either would or would not register a very small disturbance of a specific energy at the appointed time. There is no way of knowing beforehand whether this will happen; quantum mechanics only gives us a probability amplitude for the photon showing up. If it does show up, its energy will be $hf$, where $f$ is the frequency and $h$ is Planck's constant.

The charge accelerates a short distance and then disappears at time t1. What is >the time-dependent value of the electromagnetic field at an arbitrary point in >space?

The idea of appearing/disappearing charge is problematic for a few reasons. Here's one:

One of Maxwell's equations goes like this: $$ \nabla \times \mathbf{B} =\mu_0 \left( \mathbf{J} + \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right)$$

The electric field of an electron points radially inward. So if an electron dissapears suddenly, then at that moment the vector field $\frac{\partial \mathbf{E}}{\partial t}$ looks like a lot of very big arrows pointing away from the point where the electron was. Obviously this vector field has a large (maybe infinite) divergence in this region. But its an identity that $\nabla \cdot (\nabla \times \mathbf{B})$ has to equal zero everywhere, and so the divergence on the right hand side must be zero as well. The only scenario that will make this work is if the current density $\mathbf{J}$ is constructed to "cancel out" the divergence created by the disappearing charge. In practice this means that for the charge to change abruptly in some region, an equal amount of charge must be entering or leaving region at the exact same time, so as to account for the difference. So at the moment when the electron vanishes there are some suspiciously electron-like charges departing from its former location.

On top of that, any two electrons are identical to each other. Without going into the full quantum treatment, it would probably work out so that one couldn't tell if the "disappearing" electron had actually made its escape as one of the "bystander" electrons fleeing the scene. And it would probably work out so that not even nature could tell the difference.