Does electric field have a speed itself?

Question

If we bring a charge into the system, it produces electric field around it. I wonder if the propagation of this electric field has the same speed as light. Note that I don't mean electromagnetic field. I mean the charge is brought into the system as stationary. It's logical to me that electric field after some distance from the charge won't be instantly appearing there, so even electric field from the stationary charge also moves outwards. If this wasn't like this, then my mind would blow up with confusion.

So does this electric field also have the speed of light? it should have speed somehow on its own, otherwise, how can $E$ end up being at infinity from charge? it wouldn't just appear there instantly.

Answer 1

I mean the charge is brought into the system as stationary.

This is not possible, it violates the continuity equation. Charge is a conserved quantity so there are only two ways of getting a net charge in a system. One is for the charge to move into the system from the outside and the other is to form a dipole inside the system and then move the opposite charge out of the system.

In the case of the charge moved in from outside the system. If the charge is moving inertially then the field “propagates” at the speed of the charge, not $c$. However, it is very questionable to even use the word “propagate” to describe this. It is not an EM wave and the energy flux is forward rather than outward. Although this is not really “propagation” it is the only sense in which the E field propagates, but note that there is also a B field involved.

In the case of the dipole, the formation of the dipole creates a standard dipole EM wave that propagates outward at $c$. This EM wave has an outward energy flux.

In either case, if the charge accelerates that will produce an EM wave.

how can E end up being at infinity from charge? it wouldn't just appear there instantly.

For the field at infinity, again there are two possibilities.

One is that the charge has always existed. In that case there has been an infinite amount of time for the field to reach infinity at any arbitrary speed, so there is no sense in which that field propagates. It simply always was.

The other possibility is that the charge formed at some specific point in time from a dipole. In that case it does not have any field at infinity. The dipole field is propagating outward at $c$. But the field never has enough time to reach infinity at $c$. The field lines must curve and end at some opposite charge

Answer 2

The other answers and comments are right. It is impossible for a charge to suddenly appear out of nothing.

We are talking about physics here which is a mathematical description of the behavior of the universe. In particular, we are using classical physics. Approximations are made, sometimes in hidden ways. These work well for everyday situations. But sometimes if you look in the corners, you break these approximations.

One approximation is that Euclidean $3$D space is a good model for physical space. It extends forever and has always been there. No need to worry about things like the universe looking very different at the time of the Big Bang, or not knowing anything about what it looked like before that. We do this because Euclidean space behaves very much like physical space for everyday situations. We can simplify life by not worrying about how it might be different far away. If we want to understand the Big Bang, we use more complex versions of physics where space time can be curved.

Sometimes it is good to think about those approximations and understand them. But it is also good to accept them and understand that you get the right answers from them.

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You can have a point charge that has an electric field that extends to infinity. It has always been there. No need to worry about how long it took for the field to reach out infinitely far.

Sometimes people do talk about this kind of situation where a charge suddenly appears, even though it is impossible. I have heard it said that if the sun suddenly disappeared, the Earth would continue in its orbit for 8 minutes. You could ask a similar question about electric field. The answer would be similar.

If you do think in those terms, a charge suddenly appearing creates a disturbance in the electric field not totally different from a charge that has always been at rest suddenly starting to move. This similarity is why people can reasonably say what would happen if a charge appeared.

From other questions you have asked, you are trying to understand electromagnetic waves. E.G. Why does the kink have the following vector direction? That comes down to understanding how this infinite, always-been-there field changes when charges move.

Sometimes people use unrealistic examples to make it understandable. A charge that has always been there suddenly jumps to a new position. This creates a kink in the field. The kink propagates outward at the speed of light. People blithely connect field lines because field lines can never break. You were right to says this doesn't really make sense. It is very like a charge appearing out of nothing, and people sometimes do object to impossible things like that.

Sometimes they don't. Sometimes they may ignore or try to explain them. This often happens when trying to simplify some complicated situation so you can get some intuition about it. In this case, they are trying to make a point about how disturbances in an electric field propagate away from a charge at the speed of light. The simplest possible motion provides the clearest example of this disturbance. The simplest possible motion is an impossible sudden jump that produces an impossible disturbance. The main point is clear. But if you think about the details, it gets confusing.

In general, thinking about these kinds of details is important. You have just picked an unfortunate example to start with. As you continue with physics, you will mostly find that details lead to a logical progression that makes sense. But not always. When you get to relativity and quantum mechanics, things are very different from the classical world. People often try to explain it by comparing it to classical physics. An electron is like a particle and also like a wave. This often leads to explanations that are correct in one way, but wrong in another. Learning to deal with this kind of confusion will be useful.

Answer 3

With your example, you are actually assuming that charge that we bring into the system already has electric field around it to the infinity. But at some very early point, maybe when charge was created(i dont know how), fields must have propagated otherwise how did it end up at infinity in instant time ?

This is the crux of your confusion: It is NOT possible to create charges the way you are thinking of. You can only either move charges inwards from infinity, that already came with E fields all the way to infinity, or if you want to create new charges, they must come in charge neutral pairs, and in so doing, the field pattern that emerges is a dipole pattern that will propagate out at the speed of light. There are no other possibilities.

As for why, the reason is given by Dale. Relativity dictates that only local conservation is possible, and that thus leads to continuity equations.

Answer 4

I don't mean EM disturbances. I am only talking about E field. Does not it have speed ? If not, does E appear at infinity instantly ?

Mathematically, fields aren't things that come into or out of existence. They are always there, everywhere. Even in places where the field takes on a value of zero or is an unchanging value. It's the same as the relationship between free space and gravity waves. Gravity waves are disturbances in space but space is always there, everywhere. Space doesn't appear when the gravity waves arrives. It was always in existence there, ahead of time. Just because the value of the field at a location is zero doesn't mean the field has disappeared from that location in the same way that just because there are no gravity waves passing through or no mass to curve space does not mean space is not there. The fact that the location can take on a value at all means the field exists there.

So technically, the E field did not appear instantly because it did not need to appear. It already existed there for all time, ahead of time before the disturbance in the E-field propagated to that location. Similar to waves propagating through an already existing ocean; The water was already there before the wave ever arrived and will continue to remain there after the wave has gone.

But in normal language when people say E-field they are often using that shorthand for "EM disturbances" or "disturbances propagating in the E-field" because that is what is interesting. Or when they say "a charge sets up an E-field" or "a charge has its own E-field", that is shorthand for a charge has some values it wants to impose on the ever present E-field that already exists.

If you want to talk about space (if you can even call it space) that literally cannot support an E-field, not even a value of zero (like dry land, before the arrival of the water molecules in the aforementioned ocean ever formed), that's starting to get into the formation of the universe and edge of the universe stuff; Such as what may have happened during the Big Bang where space was expanding and the formation(?) of all the fields it supports. I do not believe there are any good answers for that.

Answer 5

The electric field on its own is just an approximation for a physical situation with static or nearly static electric charges. Unfortunately there is no simple condition like $v_{charge} << c$ that satisfies the necessary conditions for this to be a good approximation. Electrons in a conductor are moving extremely slowly but conductors can still act as antennas for electromagnetic radiation, for instance. It is probably better not to talk about a propagation velocity of the field in this case at all and to assume that the change in the electric field or the motion of electric charges does not lead a significant magnetic field while the electric field changes instantaneously (in the approximation) with the change in charge distribution.

The most important consequence is that we have to break relativity to uphold this approximation: the frame in which the charges don't move or only move very slowly becomes the preferred "rest" frame, quite literally.

It also follows that this has to be a short distance approximation because in reality propagation of the em field always takes a finite time that is proportional to the distance, i.e. the further away we go from the charges, the longer we have to wait for the field to settle on its static configuration.