# Traveling electric and magnetic fields

1. Jun 1, 2014

### 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.

2. Jun 1, 2014

### 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.

3. Jun 1, 2014

### letsenibeh

What does that mean, time dependence?

4. Jun 1, 2014

### 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.

5. Jun 1, 2014

### humbleteleskop

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

6. Jun 1, 2014

### 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.

7. Jun 1, 2014

### Born2bwire

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.

8. Jun 1, 2014

### humbleteleskop

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.

9. Jun 1, 2014

### humbleteleskop

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.

10. Jun 1, 2014

### WannabeNewton

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.

11. Jun 1, 2014

### humbleteleskop

Note bold text:

12. Jun 1, 2014

### 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.

13. Jun 1, 2014

### Born2bwire

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.

Last edited: Jun 1, 2014
14. Jun 1, 2014

### humbleteleskop

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

15. Jun 1, 2014

### Born2bwire

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.

16. Jun 2, 2014

### humbleteleskop

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.

17. Jun 2, 2014

### humbleteleskop

With constant velocity magnetic field is constant.

18. Jun 2, 2014

### WannabeNewton

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.

19. Jun 2, 2014

### Staff: Mentor

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?

20. Jun 2, 2014

### Born2bwire

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:
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.

Last edited: Jun 2, 2014