Do moving electrons also create an E-field?

[troglodyte]

Hello guys,

i get a little bit confused about the fact ,that static electrons produces E-Fields and after they moves one speaks only about the magnetic field .But what happens with the E-field in a moving ensemble of electrons?I mean,the E-field should also exists .It doesn't appears,right?.So,in my opinion moving electrons produces also an E-field and indeed a magnetic field at the same time.How do you going to think about this ?

 

[Dale]

So,in my opinion moving electrons produces also an E-field and indeed a magnetic field at the same time.

Yes. You can see this from the Lienard Wiechert potential and their associated fields

 

[Ibix]

The electric field of an electron is different when measured by someone who sees it moving compared to the measurements of someone who sees it as stationary, but it does not vanish, no.

 

[Delta2]

The moving electron indeed has both electric and magnetic field around it. The spatial and temporal evolution of the fields are according to Maxwell's equations or Jeffimenko's Equations(which are derived from Maxwell's equations) where we need to specify the charge density and the current density for the case of a moving electron as
$$\rho(\vec{r},t)=e\delta(\vec{r}-\int \vec{v}(t)dt)$$
$$\vec{J}(\vec{r},t)=\rho (\vec{r},t)v(t)$$
where $$\vec{v}(t)$$ the velocity of the electron and $e$ the charge of electron.

https://en.wikipedia.org/wiki/Jefimenko's_equations

 

[PeroK]

Hello guys,

i get a little bit confused about the fact ,that static electrons produces E-Fields and after they moves one speaks only about the magnetic field .But what happens with the E-field in a moving ensemble of electrons?I mean,the E-field should also exists .It doesn't appears,right?.So,in my opinion moving electrons produces also an E-field and indeed a magnetic field at the same time.How do you going to think about this ?

It's clear that the E-field of a static electron and a moving electron may be different, but is it physically reasonable that the E-field disappears altogether as soon as the electron moves? The E-field for an electron drifting slowly must physically be close to that of an electron idealised to be at rest. Otherwise, there would be very little evidence for electric fields, as things are rarely completely at rest with respect to each other.

 

[troglodyte]

Thank you all for this very informal answers.But this question wasn't that trivial as it maybe seems on the first look.

troglodyte

 

[vanhees71]

It's non-trivial. The most simple case is a point particle moving with constant velocity. Then you don't need to use the somewhat complicated retarded-potential (or equivalently the Jefimenko) equations but you can use the relativistic covariance of electrodynamics to derive the field of the moving particle from the static case.

So let the electron be at rest in the reference frame $\Sigma'$. Let's located it in the origin of the coordinate system. Then the electromagnetic potentials are (in Lorenz gauge)
$$\Phi'(t',\vec{r}')=A^{\prime 0}=\frac{q}{4 \pi |\vec{r}'|}, \quad \vec{A}'(t,\vec{r}')=0.$$
Now we try to express this with four-vectors. The only four vectors present in this problem is obviously the spacetime four-vector $x'=(c t',\vec{r}')$ and the four-velocity of the charge, which here is $u'=(1,0,0,0)$. The potentials are also components of a four-vector $A^{\prime \mu}=(\Phi',\vec{A}')$.

Now $\vec{A}'=0$, and thus we can write $A^{\prime \mu}=u^{\prime \mu} \Phi'$. That's already a covariant way to write the four-vector potential.

The only thing that's not written in a covariant form is the argument of $\Phi'$, but that's also easy to achieve. As mentioned before, we have $x^{\prime \mu}$ and $u^{\prime \mu}$. From these we can build the non-trivial Lorentz scalars $x^{\prime \mu} x_{\mu}'=c^2 t^{\prime 2}-\vec{r}^{\prime 2}$ and $u^{\prime \mu} x_{\mu}'=c t'.$ Now what we need is
$$\vec{r}^{\prime 2}=(u^{\prime \mu} x_{\mu}')^2-x^{\prime \mu} x_{\mu}'.$$
So we finally have
$$A^{\prime \mu}(x)=u^{\prime \mu} \frac{q}{4 \pi \sqrt{(u^{\prime \mu} x_{\mu}')^2-x^{\prime \mu} x_{\mu}'}}.$$
But this is now entirely written in four-vectors and Minkowski products! Thus we can simply use it in the reference frame $\Sigma$, where the particle is moving with constant velocity $\vec{v}=c \vec{\beta}$. In this reference frame the four-velocity is $u^{\mu}=(\gamma,\gamma \vec{\beta})$, where $\gamma=1/\sqrt{1-\vec{\beta}^2}$.

We get
$$u_{\mu} x^{\mu}=\gamma (c t-\vec{\beta} \cdot \vec{r})$$
and thus
$$(u_{\mu} x^{\mu})^2-x_{\mu} x^{\mu} = \gamma^2 (c t-\vec{\beta} \cdot \vec{r})^2 - c^2 t^2 +\vec{r}^2.$$
So finally you have
$$A^{\mu}=\frac{q}{4 \pi \sqrt{ \gamma^2 (c t-\vec{\beta} \cdot \vec{r})^2 - c^2 t^2 +\vec{r}^2}} \gamma \begin{pmatrix} 1 \\ \vec{\beta} \end{pmatrix}.$$
Then the really tedious work left is to get the electromagnetic field components from
$$\vec{E}=-\vec{\nabla} A^0 - \frac{1}{c} \partial_t \vec{A}, \quad \vec{B}=\vec{\nabla} \times \vec{A}.$$
After some algebra you get
$$\vec{E}=\frac{q}{4 \pi} \frac{\vec{x}-\vec{v} t}{\sqrt{\gamma^2 (\vec{r}_{\parallel}-\vec{v} t)^2 + \vec{r}_{\perp}^2}}, \quad \vec{B}=\vec{\beta} \times \vec{E}.$$
Here
$$\vec{r}_{\parallel}=\vec{\beta} \frac{\vec{\beta} \cdot \vec{r}}{\beta^2}, \quad \vec{r}_{\perp}=\vec{r}-\vec{r}_{\parallel}$$
are the projections of the position vector in direction of the electron's velocity and perpendicular to it.

As you see, the split of the electromagnetic field in electric and magnetic components is observer dependent: An observer in the rest frame of the electron only finds an electrostatic Coulomb field, while an observer in the frame, where the electron moves with constant velocity gets electric as well as magnetic field components.

In the more complicated case, where an electron moves non-uniformly, i.e., is accelerated you get of course more complicated fields, and the most important addition is that it's not just this "Lorentz boosted Coulomb field" we have just calculated but there's also a part that describes electromagnetic waves, i.e., an accelerated charge generates electromagnetic waves. This field, however, you can only calculate using the retarded potentials or Jefimenko equations.

 

[troglodyte]

As you see, the split of the electromagnetic field in electric and magnetic components is observer dependent

If I understand it correctly is this the deep connection/explanation for the observed magnetic field after the electron begins to move in his reference frame ?Maybe the magnetic field exist before it moves(no movement ) but we can't recognise it?!Could this be a correct interpretation?
.The question about the roots of the magnetic field rises parallel to that I have stated in this thread.It seems a bit obscure to me that the constant movement of the electron "creates" a magnetic field.Newton makes no distinction between a constant motion and stand still. Compared to the not moving electron the effect of creating a magnetic field by the constant moving electrons is pretty enormous .I think,no matter whether classical (retarded time) or relativistic view both descriptions leads to a reference frame problematic and leads approx to the same results(pls,correct if I am wrong ).Maybe through the relativistic case one gets a more elegant way to describe this situation.

 

[PeroK]

If I understand it correctly is this the deep connection/explanation for the observed magnetic field after the electron begins to move in his reference frame ?Maybe the magnetic field exist before it moves(no movement ) but we can't recognise it?!Could this be a correct interpretation?
.The question about the roots of the magnetic field rises parallel to that I have stated in this thread.It seems a bit obscure to me that the constant movement of the electron "creates" a magnetic field.Newton makes no distinction between a constant motion and stand still.Vompared to the not moving electron the effect of creating a magnetic field by the constant moving electrons is pretty enormous .I think,no matter whether classical (retarded time) or relativistic view both descriptions leads to a reference frame problematic and leads approx to the same results(pls,correct if I am wrong ).Maybe through the relativistic case one gets a more elegant way to describe this situation.

"It is known that Maxwell’s electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbour-hood of the magnet an electric field with a certain definite energy, producinga current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electro-motive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case."

ON THE ELECTRODYNAMICS OF MOVING BODIES By A. EINSTEIN June 30, 1905

 

[Ibix]

It seems a bit obscure to me that the constant movement of the electron "creates" a magnetic field.

It's a mistake to think of electric and magnetic fields as being different things. They aren't. They are both parts of the electromagnetic field - your reference frame just defines how you end up describing which parts are electric field and which parts are magnetic field. It isn't fundamentally more mysterious than how come I, standing in front of a rectangular table, call it long and narrow, while you, standing at the side, call it short and wide. The different perspective means we label different aspects of the same thing "length" and "width". There's no strange length-turning-into-width going on - we just have different descriptions of the same thing.

That's why I phrased my first post in this thread the way I did - in terms of the state of motion of the observers measuring the field, not in terms of the motion of the electron. There is no implication of an electron being in some special state when it is at rest - rather, whenever it is at rest with respect to you then you will measure an electromagnetic field that is purely electrostatic. I, moving with respect to you, would see both electric and magnetic components.

 

[vanhees71]

If I understand it correctly is this the deep connection/explanation for the observed magnetic field after the electron begins to move in his reference frame ?Maybe the magnetic field exist before it moves(no movement ) but we can't recognise it?!Could this be a correct interpretation?
.The question about the roots of the magnetic field rises parallel to that I have stated in this thread.It seems a bit obscure to me that the constant movement of the electron "creates" a magnetic field.Newton makes no distinction between a constant motion and stand still. Compared to the not moving electron the effect of creating a magnetic field by the constant moving electrons is pretty enormous .I think,no matter whether classical (retarded time) or relativistic view both descriptions leads to a reference frame problematic and leads approx to the same results(pls,correct if I am wrong ).Maybe through the relativistic case one gets a more elegant way to describe this situation.

To think about the electromagnetic field as split into an electric and magnetic field is the wrong way to think about it. The physical entitity is the electromagnetic field as a whole. It's split in electric and magnetic components is dependent on the inertial reference frame relative to which the observer is at rest.

 

[Dale]

I think,no matter whether classical (retarded time) or relativistic view both descriptions leads to a reference frame problematic and leads approx to the same results(pls,correct if I am wrong )

Yes, you are wrong on this point. The whole point of developing relativity was to fix the problem with the previous reference frames. It did so exceptionally well.

 

[vanhees71]

It's also important to note that the Maxwell equations in fact are a relativistic field theory. Of course, Maxwell didn't know about relativity. It was however immediately clear that the Maxwell equations are not Galilei invariant.

At this time the physicists took this rather as a feature to solve the problem about how to determine physically absolute space and time and an inertial reference frame in the sense of Newtonian mechanics. Thus they took the Maxwell theory of electromagnetism as an opportunity to do so, i.e., they assumed that the electromagnetic waves and the dynamical electromagnetic fields, following as a simple consequence from Maxwell's equations, are in fact vibrational states of a medium (similar to water waves being vibrations of water as a fluid), called "the aether", and then the preferred inertial reference frame is the restframe of the unexcited aether.

The trouble, however, still was to establish a theory of electrodynamics in reference frames moving against the aether and thus "the electrodynamics of moving bodies". There were theories by H. Hertz and Lorentz's theory of electrons, and in the beginning they looked pretty promising. Today we know that these theories are correct as an approximation for relative motions to the aether at order $v/c$, where $c$ is the light speed in a vacuum.

On the other hand it was pretty clear with the advent of experiments which were sensitive to phenomena at a precision of order $(v/c)^2$. The most famous one is the Michelson Morely experiment to determine the "aether wind" due to the motion of the Earth around the Sun. From their point of view it seemed to be a complete failure, because they couldn't find any such effect though their apparatus for sure was able to reach this order-$(v/c)^2$ accuracy. The conclusion thus was that there is no preferred frame of reference nor an aether, and several physicists thought about possible effects. One is the attempt by FitzGerald and Lorentz to save the aether theory by the assumption that lengths along the direction of the aether velocity shrink by a factor $\sqrt{1-v^2/c^2}$. It was also known that Maxwell's equations are invariant under alternative transformations since the 1880ies, when Voigt found such a transformation, which is equivalent to the Lorentz transformation of special relativity, but this was taken rather as a mathematical curiosity first.

Famously it was Einstein who thought about the problem in a different way, i.e., he thought about the "asymmetries" that only occur in the interpretation of Maxwell's equations involving moving bodies, which are not apparent in the phenomena. That was a very new concept of thinking, namely the thinking in terms of symmetries. So Einstein came to the conclusion that one has to redefine the description of space and time such that the special principle of relativity is valid not only for mechanics but also for electrodynamics, and this lead him to special relativity. The math was already there (Lorentz, Poincare, FitzGerald, Heaviside), but the physical interpretation was entirely new, and the way of thinking in terms of symmetries has been the most successful concept for the development of the physics of the 20th century.

The result also is that there's no aether but that the electromagnetic field is itself a dynamical quantity as is matter, and all these dynamical entities are much better described in a theory which obeys the spacetime symmetry of special relativity (i.e., Minkowski space) rather than the Newtonian one.

The spacetime structure of Minkowski space changes the notion about the cause-and-effect relation of events as compared to Newtonian mechanics with its absolute time, and the consequence is that there's a "limiting speed", i.e., there cannot be causes by signals propagating faster than this limiting speed. Empirically it seems as if this limiting speed is indeed the speed of light, which occurs in the fundamental Maxwell equations of electromagnetism (in units which are not hiding their physical structure as unfortunately the SI units do), i.e., the speed of light in a vacuum. This is reflected by the retarded propagator of the d'Alembert operator, $1/c^2 \partial_t^2-\Delta=\partial_{\mu} \partial^{\mu}$ which describes the causal generation of electromagnetic waves by the charge-current distribution.

That was however refined only 10 years later in 1915, when Einstein after 10 years of struggle with a relativistic description of the gravitational interaction (at the same time as Hilbert) discovered general relativity, which is even more general: There are no more preferred types of reference frames, and the inertial frames are only choosable for a sufficiently small part of spacetime as particular frames, where the physical laws take the form of special relativity.

 

[troglodyte]

What a mind blowing lesson to me.I have no words for such a great engagement in explaining physics from all of you.I just feel a bit like the platonian cavemans who looks in the "light" for the first time.

Thank you to all of you!