Traveling electric and magnetic fields

[letsenibeh]

Hi this is my first thread since joining the website, so let me know if I am violating any rules here. I read a bunch of rules to follow in the website but don't think I can remember everything.

I am reading chapter 18 on Feynman's Lectures on Physics volume 2 (electromagnetism). It talks about how electric and magnetic fields can propagate in a free space.

Here is the situation: There's an infinite sheet of charge at the origin in y-z plane. Axis z points out of the page, x-axis points towards right, and y-axis points up. The attached figure will help understanding the situation.

Then the sheet of charge accelerates to a certain velocity towards positive y direction (upward) and maintains the velocity. There is also sheet of opposite charge at the origin to cancel any electrostatic effect.

Once the acceleration starts, B field is generated as such in the figure due to modified Ampere's Law (One of Maxwell's equations). While B is changing, there will be electric field generated due to Faraday's Law.

However, once the sheet of charge reaches a certain velocity, its velocity will stay constant, meaning although the B field generated due to Ampere's Law will be maintained, the E field generated due to Faraday's Law, (curl E = rate of change of B) will be eliminated due to the constant value of B.

The book, however, assumes that E field stays, taking a part in the electromagnetic wave propagating through space. I wonder if Feynman's assumption is wrong, or what prevents the E field being eliminated.

 

[WannabeNewton]

You seem to be under the mistaken assumption that a charged body moving at a constant velocity through an inertial frame has a static magnetic field. This is certainly not true. The magnetic field of a charged body moving at uniform speed can certainly have time dependence.

 

[letsenibeh]

What does that mean, time dependence?

 

[WannabeNewton]

$\vec{\nabla}\times \vec{E} = -\partial_t \vec{B}$ can only vanish if $\vec{B}$ is not a function of time. But the magnetic field of a charged body moving at uniform speed will in general have $\vec{B}$ as a function of time; the same goes for $\vec{E}$. Just take any point in space; if the charged plate is moving towards that point and eventually away from that point then certainly the values of the electric and magnetic fields of the charged plate evaluated at that point have to change in time. The same logic applies to any other point in space.

 

[humbleteleskop]

You seem to be under the mistaken assumption that a charged body moving at a constant velocity through an inertial frame has a static magnetic field. This is certainly not true. The magnetic field of a charged body moving at uniform speed can certainly have time dependence.

I'm not sure what "inertial frame" has to do with it, but it looks like Biot–Savart law says otherwise. Quote: - In the case of a point charged particle q moving at a constant velocity v, Maxwell's equations give the following expression...

http://en.wikipedia.org/wiki/Biot-savart_law

 

[WannabeNewton]

The wiki article exactly agrees with my statement. Where is the contradiction? $\vec{E}$ depends on time through $\hat{r}$, $\theta$, and $r$. Read the article again. This is a straightforward computation from the Lienard-Wiechert potentials.

 

[Born2bwire]

I'm not sure what "inertial frame" has to do with it, but it looks like Biot–Savart law says otherwise. Quote: - In the case of a point charged particle q moving at a constant velocity v, Maxwell's equations give the following expression...

http://en.wikipedia.org/wiki/Biot-savart_law

Biot-Savart Law is only valid in magnetostatics, where we have a constant current. A finite charged body moving at a constant velocity through space is not a constant current (i.e. a single electron moving in space). For it to be a constant current, we would need an infinite body (line current) or a body that has a changing velocity so that we have a closed loop.

So if we have a single charge (or a body of charges) moving at a constant velocity, it will radiate electromagnetic waves. Jackson has a treatment on this but that is a rather advanced text on electrodynamics.

 

[humbleteleskop]

The wiki article exactly agrees with my statement. Where is the contradiction? $\vec{E}$ depends on time through $\hat{r}$, $\theta$, and $r$. Read the article again. This is a straightforward computation from the Lienard-Wiechert potentials.

You are talking now about electric field. I was talking about magnetic field in response to what you said about magnetic field, not electric field.

 

[humbleteleskop]

Biot-Savart Law is only valid in magnetostatics, where we have a constant current. A finite charged body moving at a constant velocity through space is not a constant current (i.e. a single electron moving in space). For it to be a constant current, we would need an infinite body (line current) or a body that has a changing velocity so that we have a closed loop.

The equation I gave is actually for point charges. The equation for el. current is just an integral of that equation over line or loop. It doesn't matter if a charge is traveling inside a wire or as a free electron in space.

So if we have a single charge (or a body of charges) moving at a constant velocity, it will radiate electromagnetic waves. Jackson has a treatment on this but that is a rather advanced text on electrodynamics.

Yeah, like this:

http://en.wikipedia.org/wiki/Lorentz_force

And if you want to calculate trajectories of those charges influenced by a magnetic field you can use Biot-Savart Law and Lorenz force equations.

 

[WannabeNewton]

You are talking now about electric field. I was talking about magnetic field in response to what you said about magnetic field, not electric field.

The magnetic field is given by the cross product of the electric field with the velocity so if the electric field depends on time then so does the magnetic field...

I'm still waiting to see how this contradicts what I said.

 

[humbleteleskop]

Note bold text:

You seem to be under the mistaken assumption that a charged body moving at a constant velocity through an inertial frame has a static magnetic field. This is certainly not true. The magnetic field of a charged body moving at uniform speed can certainly have time dependence.

 

[WannabeNewton]

Again, what in the wiki article contradicts that?! It clearly writes down the Lienard-Wiechert fields for a classical charged point particle for which it is evident that the magnetic field has time dependence.

 

[Born2bwire]

The equation I gave is actually for point charges. The equation for el. current is just an integral of that equation over line or loop. It doesn't matter if a charge is traveling inside a wire or as a free electron in space.




Yeah, like this:

http://en.wikipedia.org/wiki/Lorentz_force

And if you want to calculate trajectories of those charges influenced by a magnetic field you can use Biot-Savart Law and Lorenz force equations.

If you read the article carefully, it states that the Biot-Savart Law is only valid for magnetostatics. Which means that it only applies for constant, non-time varying currents. But we do not use Biot-Savart to calculate the field of the electron in a cyclotron. We use it to calculate the force of a preexisting static field on the electron. We readily know that cyclotron motion radiates, another topic in Jackson, in the form of electromagnetic radiation. In addition, we know that a single electron can't work because then we do not have a constant current. The current density is a point source. Only when we have a significant distribution of electrons such that we see a uniform field will we have a constant current. So we know from observation and deduction that we can't use the Biot-Savart to find the fields due to the cyclotron motion of the electron itself.

Edit: The aforementioned Lienerd-Weichert potentials are what we use to find the fields due to point charges in motion.

Edit edit: Another good thought exercise. We have a loop of constant current. We can use the Biot-Savart law to calculate the magnetic field from this loop since it is magnetostatic. However, we now move this loop along its axis of symmetry. In the inertial frame of the loop, Biot-Savart still is true. However, in the inertial frame where it is translating, this is not true. Now we have our body of charges moving at a constant velocity. We know that the magnetic field must change in the lab frame and thus we now have an electric field and thus electromagnetic radiation. In fact, we can easily calculate the EM field by applying the Lorentz transformation on the magnetic field we calculated for the loop's rest frame. Yes, this is a topic in Jackson but you can find this in undergraduate texts like Griffiths or Purcell.

 

[humbleteleskop]

Again, what in the wiki article contradicts that?! It clearly writes down the Lienard-Wiechert fields for a classical charged point particle for which it is evident that the magnetic field has time dependence.

Did you say charged body moving at constant velocity has no magnetic field?

 

[Born2bwire]

Did you say charged body moving at constant velocity has no magnetic field?

No, he is stating that it is not true to assume that it has a static magnetic field as it can have a time-varying magnetic field instead.

 

[humbleteleskop]

Which means that it only applies for constant, non-time varying currents.

Yeah, the equation I posted is for constant velocity.

But we do not use Biot-Savart to calculate the field of the electron in a cyclotron. We use it to calculate the force of a preexisting static field on the electron.

Indeed. That thing on the photo is "teltron tube", a type of cathode ray tube used to demonstrate the properties of electrons. Not cyclotron, no. I'm talking about constant velocity in response to post #2.

 

[humbleteleskop]

No, he is stating that it is not true to assume that it has a static magnetic field as it can have a time-varying magnetic field instead.

With constant velocity magnetic field is constant.

 

[WannabeNewton]

With constant velocity magnetic field is constant.

No it isn't! $\hat{r}$, $r$, and $\theta$ all depend on time. I'm getting tired of repeating the same thing over and over again and I'm sure Born2bwire is as well.

 

[Nugatory]

With constant velocity magnetic field is constant.

The charged particle is a million kilometers away, moving towards you. Wait a while, and it's zooming by, right under your nose. Wait a bit longer and the charged particle has passed you and is now a million kilometers away and moving farther away all the time. Is the magnetic field at the tip of your nose the same at all three times?

 

[Born2bwire]

Yeah, the equation I posted is for constant velocity.




Indeed. That thing on the photo is "teltron tube", a type of cathode ray tube used to demonstrate the properties of electrons. Not cyclotron, no. I'm talking about constant velocity in response to post #2.

But there are many systems that have constant currents with constant velocity, like the moving current ring I mentioned above, that have EM fields. There are systems of charges that move at constant velocity but are not constant current, like the moving sheet of charge described by Feynman in the OP, that have EM fields. These systems are not valid for Biot-Savart. And there are systems that have bodies of charges that are constant current but not constant velocity like a loop current that are static magnetic field.

The difference is whether or not we have electrostatic, magnetostatic, or neither. If neither, then there must be an electromagnetic field. There is the quasi-static case where the fields are decoupled but that is actually an approximation. Regardless, full Maxwell's Equations treatment is always correct. So Jefimenko's Equations or the Lienard-Weichart potentials are always valid and can show these relationships. See in Jefimenko's Equations how Biot-Savart falls out when the current is time-indepedent.

Edit:

With constant velocity magnetic field is constant.

Wait wait wait... I got a handle on your confusion now. This is not a constant magnetic field. This is the magnetic field spatial distribution for a given position of the electron in a snapshot in time. But the electron is moving. You, the observer, would see a time-varying electromagnetic field as the electron flew by you as Nugatory stated above. Now if you observed it in the frame of the electron, all you would see is the Coulombic electric field. The magnetic field comes about via Lorentz transformation when we see it from a moving inertial frame. If you calculate the Poynting vector you can see the direction that energy is radiated via the EM radiation.

 

[HomogenousCow]

The field generated must reflect on the symmetry (or lack thereof) of the source.
Since the current density is not constant in time how could any field it generates be.
To elaborate, since we are talking about a point particle the current density only exists at the position of the particle at any time, the position of the particle is changing with time.

 

[humbleteleskop]

No it isn't! $\hat{r}$, $r$, and $\theta$ all depend on time. I'm getting tired of repeating the same thing over and over again and I'm sure Born2bwire is as well.

Biot-Savart equation above is just a magnetic field potential equation. It is really decoupled from Lorenz force equation (below) where its result applies to that other charge. Yes, unit vector and distance can vary with time, but this is just a proxy field potential equation to get the force vector acting on that other charge in this Lorenz force equation below. And then you do the same vice versa to get the force on this first charge.

 

[humbleteleskop]

The charged particle is a million kilometers away, moving towards you. Wait a while, and it's zooming by, right under your nose. Wait a bit longer and the charged particle has passed you and is now a million kilometers away and moving farther away all the time. Is the magnetic field at the tip of your nose the same at all three times?

http://en.wikipedia.org/wiki/Biot-savart

If the charge on my nose has constant velocity, then the magnetic field potential of the charge on my nose depends only on its velocity and charge magnitude. Cross product with unit vector over distance should really be in the Lorenz force equation.

 

[humbleteleskop]

But there are many systems that have constant currents with constant velocity, like the moving current ring I mentioned above, that have EM fields. There are systems of charges that move at constant velocity but are not constant current, like the moving sheet of charge described by Feynman in the OP, that have EM fields. These systems are not valid for Biot-Savart. And there are systems that have bodies of charges that are constant current but not constant velocity like a loop current that are static magnetic field.

The difference is whether or not we have electrostatic, magnetostatic, or neither. If neither, then there must be an electromagnetic field. There is the quasi-static case where the fields are decoupled but that is actually an approximation. Regardless, full Maxwell's Equations treatment is always correct. So Jefimenko's Equations or the Lienard-Weichart potentials are always valid and can show these relationships. See in Jefimenko's Equations how Biot-Savart falls out when the current is time-indepedent.

Let's solve the OP problem with both "your" and "my" equations and see if we get the same results. And if not, then we'll see whose equations are better. Do you accept the challenge?

Wait wait wait... I got a handle on your confusion now. This is not a constant magnetic field. This is the magnetic field spatial distribution for a given position of the electron in a snapshot in time. But the electron is moving. You, the observer, would see a time-varying electromagnetic field as the electron flew by you as Nugatory stated above. Now if you observed it in the frame of the electron, all you would see is the Coulombic electric field. The magnetic field comes about via Lorentz transformation when we see it from a moving inertial frame. If you calculate the Poynting vector you can see the direction that energy is radiated via the EM radiation.

Please place your bets. I'm warming my calculator for the duel of equations.

 

[jtbell]

With constant velocity magnetic field is constant.

Imagine that a charged object flies past you, traveling in a straight line at constant velocity. It approaches from a distance, e.g. from the east, passes a short distance in front of you, and then recedes into the distance in the west. You are holding magnetic-field and electric-field sensors.

The magnitudes of the fields registered by the sensors increase as the object approaches you, reach a maximum value when it is directly in front of you, and decrease as the object recedes from you. The direction of the electric field also changes.

 

[vanhees71]

For a generally accelerated charge you need the retarded potentials (Lienard-Wiechert potentials). For a charge moving with constant velocity you just need to do a Lorentz boost.

The four-potential for a point charge at rest can be written in the Lorenz gauge (in Heaviside-Lorentz units) as
$$A^{\mu}(t,\vec{x})=\frac{q}{4 \pi r} (1,0,0,0), \quad r=|\vec{x}|.$$
Now for a reference frame, where the charge is moving with the constant velocity $\vec{v}=v \vec{e}_1$ we have $x'^{\mu}={\Lambda^{\mu}}_{\nu} x^{\nu}$ with the appropriate Lorentz-boost matrix. In components it reads
$$c t'=\gamma(c t + \beta x), \quad x'=\gamma(x+v t), \quad y'=y, \quad z'=z$$
with $\beta=v/c$ and $\gamma=1/\sqrt{1-\beta^2}$. Since the vector potential is a vector field we have
$$A'^{\mu}(x')={\Lambda^{\mu}}_{\nu} A^{\nu}(x).$$
Then you get
$$\vec{E}'(t',\vec{x}')=-\partial_{t'} \vec{A}(t',\vec{x}')-\vec{\nabla}_{\vec{x}'} A'^0(t',x'), \quad \vec{B'}(t',\vec{x}')=\vec{\nabla}_{\vec{x}'} \times \vec{A}'(t',\vec{x}').$$
After all the algebraic dust has settled you find
$$\vec{E}'(t',\vec{x}')=\frac{Q}{4 \pi r^3} \gamma \begin{pmatrix} x'-v t' \\ y' \\ z' \end{pmatrix}, \quad r=\sqrt{\gamma^2(x'-v t')^2+y'^2+z'^2}$$
and
$$\vec{B}'(t',\vec{x}')=\vec{\beta} \times \vec{E}'(t',\vec{x}').$$
All this you get as well directly by using the Lienard-Wiechert formula for the special case of a uniformly moving point charge.

For the problem in Feynman's book, you have to use the retarded potential for the moving sheet of charge. This is, however very artificial, because it's just suddenly set into motion to a constant speed, i.e., with a singular acceleration. Nevertheless you'll then see that you do not get only a boosted electrostatic field (having time dependent electric and magnetic components in the frame, where the charge distribution moves at constant velocity) but also an radiation field, whose front velocity is the speed of light.

 

[humbleteleskop]

Imagine that a charged object flies past you, traveling in a straight line at constant velocity. It approaches from a distance, e.g. from the east, passes a short distance in front of you, and then recedes into the distance in the west. You are holding magnetic-field and electric-field sensors.

The magnitudes of the fields registered by the sensors increase as the object approaches you, reach a maximum value when it is directly in front of you, and decrease as the object recedes from you. The direction of the electric field also changes.

I have no argument. I can only point at those equations, which I already did, and say that I have used them with great success in n-body real-time simulation of moving charges, like free electrons and positrons, as well as el. current in wires. It could simulate Ampere's force law for example, or bend electron beams like in that 'teltron tube'.

Here is Biot–Savart law and Lorentz force in one equation, plus integrated over line.

http://en.wikipedia.org/wiki/Ampère's_force_law

I use point charge equations for everything and I get above type of integral integrated automatically by itself as a function of real-time and as it gets animated in real-time. Wires attract, wires repel, electron beams bend, and stuff like that. Could also trace trajectories of electrons and positrons spiraling around in a magnetic field, kind of like bubble-chamber experiments.

 

[vanhees71]

The Biot-Savart Law is wrong for time-dependent sources and have to be replaced by the ones you obtain from the retarded potentials.

In Heaviside-Lorentz units the solution for the wave equation
$$\left (\frac{1}{c^2} \partial_t^2-\Delta \right) A^{\mu}=\frac{1}{c} j^{\mu}$$
reads
$$A^{\mu}(t,\vec{x})=\frac{1}{4 \pi c} \int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x}' \frac{j^{\mu}(t'-|\vec{x}-\vec{x}'|/c,\vec{x}')}{|\vec{x}-\vec{x}'|}.$$
Taking the derivatives leads to the electric and magnetic field components
$$\vec{E}=-\frac{1}{c} \partial_t \vec{A}-\vec{\nabla} A^0, \quad \vec{B}=\vec{\nabla} \times \vec{A}.$$
This leads to the generalization of the static Coulomb and Biot-Savart Laws. For some reason, the corresponding equations are known as Jefimenko's Equations although the retarded potential and the corresponding field solution are already found by Lorenz (the Danish physicist, written without a t, not the more famous Dutch physicist Lorentz). You find these equations (in SI units) nicely explained in the Wikipedia:

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

 

[Born2bwire]

Let's solve the OP problem with both "your" and "my" equations and see if we get the same results. And if not, then we'll see whose equations are better. Do you accept the challenge?




Please place your bets. I'm warming my calculator for the duel of equations.

Why don't we keep it simple?

An electron is at x=-100 m at t=0 and moves with a constant velocity of 90 m/s in the positive x direction. What is its position as a function of time? What is its electromagnetic field as a function of time and position of observation? We can assume non-relativistic approximations and ignore time retardation for this case..

Then, what is the exact answer if the speed is 0.99c?

 

[humbleteleskop]

Why don't we keep it simple?

An electron is at x=-100 m at t=0 and moves with a constant velocity of 90 m/s in the positive x direction. What is its position as a function of time? What is its electromagnetic field as a function of time and position of observation?

Position in the algorithm is always calculated as a function of time interval that is the difference between 'current time' and the time when position was calculated the last time, i.e. delta time or time-step. Time variable can be synchronized with actual real-time or scaled arbitrarily: fast, slow, forward, backward. Algorithm is expression of classic equations of motion with uniform acceleration.

The third one, and this is the last step. First velocity needs to be re-calculated as a function of force, but there are no external fields so the velocity stays constant. Electric filed potential is calculated from Coulomb's law and magnetic field potential is calculated from Biot-Savart law.

Neither are function of time or function of position of observation. For that we need at least one more charge so this one has something to be relative to. There is distance r in both of those equations, but there is no other charge q2. These are proxy equations, decoupled from their force equations, that is Coulomb force and Lorentz force. E and B potentials are about how some other charge q2 feels in the field of q1, not how q1 feels in its own field.

We can assume non-relativistic approximations and ignore time retardation for this case..
Then, what is the exact answer if the speed is 0.99c?

You say non-relativistic and then you say 0.99c. The answer is given by those two equations above. For E it's all constants, and for B multiply 0.99c with those constants there and that's the answer. These are classic equations, I don't know what did you expect. The real question here is are they good enough for the problem in the opening post, and whether are your equations indeed any better, and really, just how much.

What results do you get?

 

[mattt]

With constant velocity magnetic field is constant.

Are you serious or just trolling? :-)

 

[mattt]

For example, imagine a xyz-inertial frame. With respect to this frame, we have a point charge whose equation of motion is:

$$(-t, 1, 0)$$

That is, the point charge is moving along the line of (implicit) equations $$y=1, z=0$$ (which is parallel to the x-axis, and at a distance=1 ) and the vector velocity of the point charge is

$$\vec{v}=(-1,0,0)$$

Can you compute $$\vec{B}(x,y,z,t)$$?


If you compute it (using that Biot-Savart formula), you'll get:

$$\vec{B}(x,y,z,t) = \frac{\mu_0}{4\pi}\frac{q}{((t+x)^2 + (y-1)^2 + z^2)^{\frac{3}{2}})}(0, z, 1-y)$$


That is


$$\vec{B}(x,y,z,t) = \frac{\mu_0}{4\pi}\frac{q}{((t+x)^2 + (y-1)^2 + z^2)^{\frac{3}{2}})}(z\vec{j} + (1-y)\vec{k})$$


At any given point $$P(x_0,y_0,z_0)$$ (not in the line y=1, z=0), the $$\vec{B}$$ field at that point mantain the same direction and sense for all time, but its magnitude obviously depends on time.

That is why, at any given point $$P(x_0,y_0,z_0)$$, the magnitude of the vector $$\vec{B}(x_0,y_0,z_0,t)$$ goes to zero when the point charge is too far away from this point (in the distant past and in the distant future), and this magnitude reaches its maximum value when the point charge is in the same yz-plane that the point P (i.e. when the point charge is closest to the point P).

 

[vanhees71]

This is obviously wrong, as you can see by comparing with the correct solution given in #26!

The reason is very simple! The charge and current density are given by
$$\rho(t,\vec{x})=q \delta^{(3)}(\vec{x}-\vec{v} t), \quad \vec{j}(t,\vec{x})=q \vec{v} \delta^{(3)}(\vec{x}-\vec{v} t), \quad \vec{v}=\text{const}.$$
Thus you have to use the retarded potentials (or Jefimenko's equations) to evaluate the fields. The result is, of course, the same as the one obtained in #26 via a Lorentz boost from the rest frame of the particle to the frame, where it moves with the constant velocity $\vec{v}$. The way over the retarded potentials or Jefmenko's equations are tedious compared to the elegant way via the Lorentz transformation!

Again, this is an example for the harm done by textbooks not emphasizing the relativistic covariance of electrodynamics and overemphasizing statics!

 

[mattt]

I meant that even using "his" Biot-Savart formula for that situation, $$\vec{B}(x,y,z,t)$$ depends explicitly on time.

 

[humbleteleskop]

Are you serious or just trolling? :-)

I didn't invent that equation and I have nothing to say about it except that it works as a part of Lorenz force equation. By itself, it's incomplete. What that means and how it relates to whatever you are talking about, I don't know.

I notice now there is no actually any second object in the opening post, no interaction, no forces acting except within the sheet itself. Those are kinematic equations, and if the answer is not about calculating forces, then I was wrong to say those equations apply. I'm sorry for the confusion.