2 charges and the speed of light

[abro]

Consider two equal and opposite charges moving in the same direction and velocity v₀, and are separated a distance R. There will always be an attractive electromagnetic force F_el=q²/(4πεR²), but when they are moving, they will produce a magnetic field B=μqv/(4πR²), and thus a repulsive magnetic force F_B=Bqv.
I tried finding the velocity at which these forces are balanced;
F_el=F_B
q²/(4πεR²)=μq²v²/(4πR²)
v₀=1/√(ε₀μ₀), which turns out to be the speed of light.
I find this result very surprising; the forces will be canceled out when only the velocity is the speed of light, and the charge and distance can be anything. Does this mean like light has two very small charges with equal spins (1/2+1/2=1) and no mass and thus can interact electromagnetically?

Edit: Here is a picture to clarify what I mean.

 

[Drakkith]

No, light is simply a wave in the electromagnetic field. The wave has no charge, but the direction of the force the wave applies to electrical charges alternates back and forth. This is why the electrons in an antenna move back and forth in response to a radio signal and allows you to listen to a song on the radio.

 

[Dale]

Consider two equal and opposite charges moving in the same direction and velocity v₀, and are separated a distance R. There will always be an attractive electromagnetic force F_el=q²/(4πεR²), but when they are moving, they will produce a magnetic field B=μqv/(4πR²), and thus a repulsive magnetic force F_B=Bqv.
I tried finding the velocity at which these forces are balanced;
F_el=F_B
q²/(4πεR²)=μq²v²/(4πR²)
v₀=1/√(ε₀μ₀), which turns out to be the speed of light.

Hi abro, welcome to PF!

This is a very interesting result, which I am glad to hear that you discovered on your own. It turns out that this is not a co-incidence, but is a reflection of the fact that magnetism can be seen as a consequence of the combination of Coulomb's law and relativity:
http://physics.weber.edu/schroeder/mrr/MRRtalk.html

In relativity there is a phenomenon called time dilation, which means that a moving clock ticks more slowly than a stationary clock of the same construction. If you have two charges at rest and they attract each other through some force then you can use that as a kind of clock which "ticks" when they collide. In a frame where they are moving they "tick" slower due to relativity, which is interpreted as a magnetic field opposing the electrical force and thereby reducing the net force.

 

[Born2bwire]

I would point out that since they are moving in the same direction and speed, that we know already that the magnetic field is not going to have a net effect at that instance. If we observe in the frame of the charges, so that they are stationary, then there is no magnetic field, just the coulombic field. So whatever force we see from magnetic field in a lab frame should not affect the net attraction. If we get a magmetic force that affects the charges, then there must be an associated change in the electric field. This can change once we allow the charges to move towards each other.

This all ties into what Dalespam has already stated above.

 

[Dale]

So whatever force we see from magnetic field in a lab frame should not affect the net attraction.

But it does affect the net attraction. If it didn't then the electrostatic attraction would be too large to account for the observed acceleration. It is essential to have the magnetic field and to have it affect the attraction.

If we get a magmetic force that affects the charges, then there must be an associated change in the electric field.

Yes, because the charges are moving dE/dt is non-zero at each point.

 

[Born2bwire]

But it does affect the net attraction. If it didn't then the electrostatic attraction would be too large to account for the observed acceleration. It is essential to have the magnetic field and to have it affect the attraction.

Yes, because the charges are moving dE/dt is non-zero at each point.

I'm talking about that instance in time where the charge are moving at the same velocity, up until we allow the charges to move freely. In this case, we have a frame where the two charges are at rest with respect to each other and so the force in that frame is the coulombic attraction. If we look in the lab frame, I would expect the net force should still be the same by extension of case between two current carrying wires.

The OP assumes that the electric and magnetic fields have a cancelation. That is not always the case. If the two charges lie colinear to the velocity, then the charges lie in the null of each other's magnetic field. In this case, the magnetic field never affects the charges. If we allow the axis between the charges diverge from the direction of the velocity, then the magnetic field contributes a force. But now the Lorentzian boost to the electric field counteracts this force.

My intuition is that it would be that the Lorentzian boost to the electric field is canceled by the boosted magnetic field. In the case where the velocity is perpendicular to the line between the charges, we can describe the system dynamically in the rest frame of the charges. In that case, the charges come together under Coulombs law as the magnetic field they make does not interact with the other charge since the position and velocity are colinear. The length contraction does not affect the transformed case because this is perpendicular to the motion of the frame.

But now that I think about it, you still have time contraction between the two. So perhaps the force is not the same then. Interesting problem to work out on paper if I had the time.

EDIT: Ah, I've got it now. If the two charges lie along the velocity, the force is not the same between the two frames because now the distance between the two charges undergoes a Lorentz contraction. So the electric field does not transform, but the distance between the two is contracted which affects the resulting coulombic force between the two.

 

[Dale]

I'm talking about that instance in time where the charge are moving at the same velocity, up until we allow the charges to move freely. In this case, we have a frame where the two charges are at rest with respect to each other and so the force in that frame is the coulombic attraction.

Yes, this is exactly the scenario I am considering also.

If we look in the lab frame, I would expect the net force should still be the same

But the net force is not the same in the lab frame. Kinematically, it cannot be the same due to time dilation and length contraction. Forces are not invariant.

Dynamically, the magnetic field given by Maxwell's equations is the force which ensures that the kinematics work out correctly. The magnetic field does indeed exert a force on the charge and that force is the required force to "correct" the electric force relativistically.

 

[abro]

I don't really see how relativity applies to this scenario, because we are talking about well defined charges moving in the same direction. If we move along the charges, we don't see a magnetic field thus no repulsion, but when we see them moving we see a magnetic field and thus a repulsion. That's impossible, everyone has to agree on the fact there's a force acting on them. Kind of vague to be honest, I hope the picture I just uploaded will clarify things a bit.

 

[pervect]

I don't really see how relativity applies to this scenario, because we are talking about well defined charges moving in the same direction. If we move along the charges, we don't see a magnetic field thus no repulsion, but when we see them moving we see a magnetic field and thus a repulsion. That's impossible, everyone has to agree on the fact there's a force acting on them. Kind of vague to be honest, I hope the picture I just uploaded will clarify things a bit.

Normally, relativistic effects are important only at high velocities. The magnetic field is an interesting example of where relativistic effects are important even at low velocities.

The "order" of the relativistic corrections are not any different mathematically. It's just that the electric force is so strong that the tiny relativistic corrections are noticeable.

That's impossible, everyone has to agree on the fact there's a force acting on them.

Everyone does agree that there is a force acting between the charges. What they don't agree on is whether the force is purely an electric force, or whether the force has both electric and magnetic components.

The force is called the "electromagnetic force" precisely because of this ambiguity. Everyone who uses the appropriate relativistic transformation laws for the force agrees on the magnitude of the force. What they don't agree on is whether this force is purely electric or whether the force has both electric and magnetic components.

 

[Dale]

If we move along the charges, we don't see a magnetic field thus no repulsion, but when we see them moving we see a magnetic field and thus a repulsion. That's impossible, everyone has to agree on the fact there's a force acting on them.

Your math and reasoning are both correct. If you assume that the charges are moving at c then the net force is 0, and everyone has to agree whether or not the net force is 0. Therefore the assumption that the charges could move at c must be incorrect.

Due to relativity the charges can move very fast, arbitrarily close to c, but cannot reach c. So the invalidity of the assumption is consistent with relativity and what we know of the laws of physics.

Note that at any speed less than c the net force is attractive, and everyone agrees on that fact.

 

[ChrisVer]

Well, for me I would say that the net force is zero, because the EM force travels with the speed of light.
So if you have two charges separated by some distance L, and make them move with the speed of light, it sounds natural they won't feel any kind of force, except for L=0.
Because suppose you have the charge 1 at position $\vec{x}_{1}$ and the particle 2 at position $\vec{x}_{2}= \vec{x}_{1}+\vec{L}$ and they move at c, they won't ever be casually connected.

However I am not sure in case they move closer to each other... any idea anyone?

 

[Dale]

Since they cannot move at c it is one of those "what would the laws of physics say if the laws of physics were wrong" questions. Basically, all you can do is to say you will get a contradiction.

To me, what is more interesting is how the forces differ for v<c.

 

[KatamariDamacy]

Fel=FB
q2/(4πεR2)=μq2v2/(4πR2)
v0=1/√(ε0μ0)

Very interesting, and I'd say profoundly so. Have you searched the internet to see if anyone else ever spoke about this relation and its meaning before?

Shouldn't there also be a rotational component to that magnetic field, so that there is a force in the direction perpendicular to the screen, making them actually orbit around their middle point and propagate like a double helix spiral?

Does this mean like light has two very small charges with equal spins (1/2+1/2=1) and no mass and thus can interact electromagnetically?

"Spin" is far more complex than that, it's a whole new three-dimensional magnetic moment on its own. Magnetic dipole axis, or north-south bearing, of this spin has orientation, so depending on the relative angle between the two spin magnetic dipole axis those charges are going to end up attracting or repulsing after all. But which will it be, attraction or repulsion, and to answer that it seems we first need to know how this spin magnetic moment interacts with its own Lorentz force magnetic moment and that of the other charge, before we look into how spin magnetic moments interact with each other.

Smallest negative and positive charge are electron and positron, so how did you come up with "no mass"? Basically what you seem to be asking is could photons actually be electron-positron pairs. It is then interesting to note how electron-positron "annihilation" process produces photons, and in what is called "pair production" process we have photons split up into electrons and positrons. I'd say you're onto something.

 

[KatamariDamacy]

I would point out that since they are moving in the same direction and speed, that we know already that the magnetic field is not going to have a net effect at that instance. If we observe in the frame of the charges, so that they are stationary, then there is no magnetic field, just the coulombic field. So whatever force we see from magnetic field in a lab frame should not affect the net attraction. If we get a magmetic force that affects the charges, then there must be an associated change in the electric field. This can change once we allow the charges to move towards each other.

What equations are you referring to and what net force result do you get?

There is so called "pinch" or "z-pinch" effect which is exactly due to magnetic fields and Lorenz force between parallel moving charges. It is responsible for holding electrons tightly together in lightning bolt "plasma streams", and electron beams, otherwise they would fly apart. It works just like magnetic attraction between two parallel wires when both have electrons flowing in the same direction. It's Lorentz force, also called "Ampere's force law" in the case of current carrying wires.