Questions on moving charge(s)

[Jacques]

Hi

I was thinking of two experiments and I cannot figure what would be the results.

The experiment are done in a vacuum isolated from any outside electric or magnetic field.

A moving charge will generate a magnetic field (it will move the needle of a compas), but if I follow that charge I will not be able to detect a magnetic field. Does the magnetic field exist or not ?

The second experiment is 2 equal charge q1 and q2 are moving in the same direction with speed v. If v is greater than a value, I will see the distance between the charge decreasing because of the z-pinch effect. But if I move in the same direction and speed what will I see ?

[timo]

A moving charge will generate a magnetic field (it will move the needle of a compas), but if I follow that charge I will not be able to detect a magnetic field. Does the magnetic field exist or not ?

It's (vaguely) akin to asking whether an object moves to the left or not. In different reference frames the velocity to the left has different values. I do not know what answer to the question "does an object move to the left or not" you prefer, but similarly to the motion of an object to the left, the magnetic field created by a charge depends on the reference frame (as you already noted) and I think the answer is pretty much the same for the motion-to-the-left question and yours.

[Jacques]

Yes and that's why I thaught of the second experiment

An observer with a relative velocity will see the charges getting closer to each other, but a second observer without relative velocity will see the charge getting farther away from each other... Can you see the paradox I see ?

[timo]

No, I don't see it. Mostly because I have no idea what the z-pinch effect is. I'm currently not fit (and not interested, either) to really read the WP article or even make some serious approaches understanding it (looking up literature or simply doing a calculation of my own). Spontaneously, I see nothing that states that there is an effect in the order of magnititude such that it would prevent two electrons to repell each other. I don't know to what extend the effect is tied to plasma conditions, either - two electrons certainly do not make up a plasma.

[Jacques]

May be someone else can answer that question ?

[NeonBlack]

If v is greater than a value, I will see the distance between the charge decreasing because of the z-pinch effect

 

You are almost correct. However, if you tried to calculate the speed at which the pinch force overcomes the electrostatic repulsion, you would find that "a value" is the speed of light.

Edited August 11, 2008 by NeonBlack

[swansont]

The reason it works for parallel conductors is that when you are moving at v, it looks like positive charges moving in the opposite direction. For bare charges you need to look at the relativistic form of Maxwell's equations to see what's going on.

Edited August 11, 2008 by swansont

[Jacques]

Thanks for your answers!

Ok it is impossible that for two moving charges, the pinch force overcomes the electrostatic repulsion (no matter can go at c), but the pinch force goes agains the repulsion, so it is substracted from the electrostatic force. So the paradox still apply. One observer will see the same charges separating at a different rate...

 

The reason it works for parallel conductors is that when you are moving at v, it looks like positive charges moving in the opposite direction.

That doesn't explain the phenomena. Positive charge moving in the opposite direction in both conductor will also repel. The explaination I learned is that a wire generate a magnetic field around it, and the magnetic around each a wire are attracted by the magnetic field of the other.

For bare charges you need to look at the relativistic form of Maxwell's equations to see what's going on.

NeonBlack did you used the relativistic form of Maxwell's equations to calculate the speed at which the pinch force overcomes the electrostatic repulsion ?

For bare charge, I am not 100% sure but I think that the pinch effect is used in particle accelerator to keep the beam collimated. I don't know if the effect is enought to overcome the electro static repulsion ?

Edited August 12, 2008 by Jacques
multiple post merged

[NeonBlack]

I am not sure what you mean. I have never heard of a "relativistic form of Maxwell's equations." The usual Maxwell's equations are already consistent with relativity. I used the Lorentz transformations for fields in the calculation, if that's what you mean.

 

That the charges separate at different rates in different frames is no paradox. Simply put, say the rate at which the charges move with respect to each other is dy/dt. Observers in different frames will see different dt's so this should not be surprising.

[swansont]

I am not sure what you mean. I have never heard of a "relativistic form of Maxwell's equations." The usual Maxwell's equations are already consistent with relativity. I used the Lorentz transformations for fields in the calculation, if that's what you mean.

 

That the charges separate at different rates in different frames is no paradox. Simply put, say the rate at which the charges move with respect to each other is dy/dt. Observers in different frames will see different dt's so this should not be surprising.

 

http://en.wikipedia.org/wiki/Formulation_of_Maxwell's_equations_in_special_relativity

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

 

It's been a long while since I've looked at this, but IIRC there's a correction for e.g. charge density from length contraction that isn't present in Maxwell's equations.

 

That doesn't explain the phenomena. Positive charge moving in the opposite direction in both conductor will also repel. The explaination I learned is that a wire generate a magnetic field around it, and the magnetic around each a wire are attracted by the magnetic field of the other.

 

Conductors are electrically neutral, so there is no electrostatic repulsion. There will always be charge motion in any reference frame. This is different from free charges in a plasma, for which, frankly, I don't have a good physical grasp of the phenomenon.

Edited August 12, 2008 by swansont
multiple post merged

[NeonBlack]

Swanson, I did not read the wiki pages in entirety, but they just look like an alternate treatment of electromagnetism using 4-vectors and treating magnetism as a relativistic effect. I still haven't seen anything suggesting a relativistic form of Maxwell's equations. Still the only thing I can think of is the field transformations http://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#Lorentz_transformation_rules_for_fields

Did you mean this:

http://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#Covariant_formulation

?

[swansont]

It's the relativistic correction for fields and charge distribution, IIRC.

[Pete]

The explaination I learned is that a wire generate a magnetic field around it, and the magnetic around each a wire are attracted by the magnetic field of the other.

The force on one conductor is due to the charges in that conductor moving through the magnetic field generated by the other conductor. The motion of the charge in wire which is parallel to the other wire is that of charge moving perpendicular to the magnetic field generated by the other wire. In such case the charges experience a force which ends up causing the wire itself to experience a force on it.

 

You might also want to learn about the 4-current 4-vector. It is used to determine the charge and current densities in other frames when they are specified in one frame.

Conductors are electrically neutral' date=' so there is no electrostatic repulsion.

[/quote']

If a wire is electrically neutral in frame S then it won't neccesarily be neutral in another frame S' which is moving relative to S. E.g. if there is a neutral current carrying wire in the inertial frame S which is lying on the x-axis and S' is moving at constant velocity in the x-direction then the wire will be charged as observed in S'. For proof please see the section labeled Charged Density on a Moving Wire in my web site under http://www.geocities.com/physics_world/em/rotating_magnet.htm

 

Pete

[Jacques]

Pete thanks for your answers, I went on your site and was not sure about the first sentence:

In the following page we find the both the volume and ...

Since the wire is electrically neutral (i.e. r = 0)

I thaught that wire carrying electricity where charged. If I touch a wire I will get socked

and all the "-" charges are equally spaced a distance L = L- apart as well.

The negative charge are moving at v , to get L- maybe we need to apply Lorentz contraction to L ? That would leave a negative charge to the wire...

 

Anyway I am sure that you know your physic better than me, but if you could explain where is my mistake it will help me.

 

Neonblack

if you tried to calculate the speed at which the pinch force overcomes the electrostatic repulsion, you would find that "a value" is the speed of light.

Do you have references where I can see how you did the calculation ?

[swansont]

If a wire is electrically neutral in frame S then it won't neccesarily be neutral in another frame S' which is moving relative to S. E.g. if there is a neutral current carrying wire in the inertial frame S which is lying on the x-axis and S' is moving at constant velocity in the x-direction then the wire will be charged as observed in S'. For proof please see the section labeled Charged Density on a Moving Wire in my web site under http://www.geocities.com/physics_world/em/rotating_magnet.htm

 

Pete

 

Which was the point of the second link in my previous post, that said you have to relativistically account for the charge distribution. This isn't present in a classical treatment, where one would see the wire as neutral.

[Pete]

I thaught that wire carrying electricity where charged. If I touch a wire I will get socked

In the rest frame of the wire it is assumed the wire is uncharged. That doesn't mean you won't get zapped when you touch a wire. In order to be zapped you have to form a ciruit with your body. I.e. one finger reaches inside a light socket while your foot touches the ground. A current is then set up your body due to the difference in potential of the wires.

The negative charge are moving at v , to get L- maybe we need to apply Lorentz contraction to L ? That would leave a negative charge to the wire...

Sorry. That is a typo in the web page. The "-" is supposed to be a subscript to indicate the distance between negative charges.

 

I should also have pointed out that this is an ideal case. In the real world there is actually a non-zero charge density on the surface of the wire which is non-uniform. This surface charge distribution creates an electric field inside the wire and it is that field that pushes the current along. That charge density is extremely small though. I doubt that the electrostatic attraction due to this charged can be measured in practice since it is that small.

This isn't present in a classical treatment' date=' where one would see the wire as neutral.

[/quote']

Classical em has this as a flaw. I.e. if one uses classical EM (i.e. non-relativistic) then one cannot explain the phenomena. Even for very very very slowly moving charges there is a definite and significant electric field around the wire in the moving frame. With this electric field the charge density can be calculated using Gauss's law and is then found to me non-zero!

 

Pete

[NeonBlack]

Do you have references where I can see how you did the calculation ?

 

I will post something tomorrow. I will make some pretty pictures (okay, maybe not so pretty) pictures and doing latex takes a long time for me.

 

In the mean time, take a look at the field transformations, which were mentioned earlier by Pete.

http://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#Lorentz_transformation_rules_for_fields

We can treat electric and magnetic fields as different aspects of the same thing, in much the same way as space and time.

[NeonBlack]

For simplicity, we're going to do this calculation without 4-quantities

These are the lorentz transformations for electromagnetic fields if the motion is in the direction of the x-axis. (I lifted these from wiki)

[math]E'_x = E_x[/math]

[math]E'_y = \gamma \left ( E_y - v B_z \right )[/math]

[math]E'_z = \gamma \left ( E_z + v B_y \right )[/math]

[math]B'_x = B_x[/math]

[math]B'_y = \gamma \left ( B_y + \frac{v}{c^2} E_z \right )[/math]

[math]B'_z = \gamma \left ( B_z - \frac{v}{c^2} E_y \right )[/math]

 

Now take a look at the picture- I warned you it wasn't going to be pretty. We have two positive charges some distance d apart in the y-axis. We'll let frame S be the frame in which they are at rest. In frame S', they are moving together with some speed, v.

 

Let's calculate the electric and magnetic fields at the point of the top charge in frame S. (The bottom charge will see the same thing in the opposite direction)

[math]\vec{B}=0[/math]

[math]\vec{E}=\frac{1}{4 \pi \epsilon_0} \frac{q}{d^2} \hat{y}[/math]

We can also write [math]E=E_y[/math]

 

Now, let's use the field transformations to calculate E and B in the S' frame as they pass the origin.

[math]E_y'=\gamma E_y[/math]

[math]B_z'=\gamma \frac{v}{c^2}E_y[/math]

We can calculate the force on the top charge, [math]\vec{F}=q(\vec{E}+\vec{v} \times \vec{B})[/math]

With the quantities that we know,

[math]F_y=q(\gamma E_y - \gamma \frac{v^2}{c^2} E_y)[/math]

To find the speed at which the magnetic attraction overcomes the electrostatic repulsion, we can let F=0, and we find v=c.

 

Every step of this is not explicit, but I think it should be enough for you to be able to follow. Also, please let me know if you see any mistakes.

[booker]

Re Z-pinch. This is about plasma; electrically neutral, or nearly neutral stuff. It's made of ions plus their stripped electrons.

 

So like two electrically neutral wires in parallel, sending a current through the plasma causes it to pinch.

 

Besides that, if one inertial observer sees two charges drawing together, so will another.

 

http://en.wikipedia.org/wiki/Formulation_of_Maxwell's_equations_in_special_relativity

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

 

It's been a long while since I've looked at this, but IIRC there's a correction for e.g. charge density from length contraction that isn't present in Maxwell's equations.

 

Swansont, I'm afraid you've misremembered. No additional terms are required, that's what manifestly covariant is all about!

 

If you're thinking of the term added to the current density, that's the charge density scaled by the speed of light. It's not an additional term. Charge density and current density are combined into a single unit, conveniently reducing the number of Maxwell's equations by one.

Edited August 17, 2008 by booker
multiple post merged

[swansont]

Re Z-pinch. This is about plasma; electrically neutral, or nearly neutral stuff. It's made of ions plus their stripped electrons.

 

So like two electrically neutral wires in parallel, sending a current through the plasma causes it to pinch.

 

Besides that, if one inertial observer sees two charges drawing together, so will another.

 

Yes, a net neutral plasma will behave much like chatges in a conductor. I thought this effect has been observed in charged beams as well; that's the scenario I couldn't understand.

[Jacques]

Thanks NeonBlack for your calculations, but what you computed isn't it the force of a moving charge with repect to a non-moving charge (S frame vs S' frame) ? In the problem the charges are not moving relative to each-other...

[Pete]

Thanks NeonBlack for your calculations, but what you computed isn't it the force of a moving charge with repect to a non-moving charge (S frame vs S' frame) ? In the problem the charges are not moving relative to each-other...

It is instructive to try this without relativity by working as follows: Calculate the force on a charged particle which is moving through the magnetic field. Since the force remains invariant in a Galilean transformation the force has the exact same value in the rest frame of the charge. By definition - The force due to the charge of a particle which is at rest in frame S is an electric force. This force can now be used to calculate the electric field measured in that frame. Once that field is determined one can then use Gauss's law to determine the charge denisty of the wire as measured in frame S.

 

Pete

[booker]

Yes, a net neutral plasma will behave much like chatges in a conductor. I thought this effect has been observed in charged beams as well; that's the scenario I couldn't understand.

 

This is actually a quesiton I asked as a freshman. Two like-charged beams will still repell, but repell less with an increased velocity of the charges.

 

The charges in each beam repell.

 

In the lab frame, the magnetic field generated by one beam acts to attract the other beam. I believe that at v --> c the electric and magnetic forces are equal and opposite. There isn't any other critical velocity, so hueristically at least, it makes sense.

 

Secondly, in the lab frame the relativistic masses of the charged particles increase as well, slowing divergence of the beams.