Lorentz Force Paradox

[eram]

I give up. You need to retake your class (if you ever had one).



You have it exactly backwards.

All i want is some clarification.

I may be uninformed, but you don't have to insult me. I had hoped you would be more understanding.

Also my class didn't cover relativistic lorentz forces, as i've mentioned before.



So you say that we must take into account the relative motion between the charge and the B field, and that is what v stands for in q(Bxv), not the relative velocity between Bob and the charge.

Unfortunately the wiki article is quite ambiguous as to what v stands for exactly. but i'll just follow your lead.



However, when I learnt that a moving charge generates a magnetic field, aren't we taking into account the velocity between Bob and the charge, assuming there is no magnet nearby? Or have I learnt it wrong?


I may be confused at times, but well, its all part of the learning process...

 

[Tach]

So you say that we must take into account the relative motion between the charge and the B field, and that is what v stands for in q(Bxv), not the relative velocity between Bob and the charge.

yep.

 

[eram]

ok, that was fine but a bit too brief.


what about the point that i mentioned about a moving charge creating a magnetic field with no other magnets or charges nearby?



I'm afraid we may be arguing till kingdom come.

 

[OnlyMe]

ok, that was fine but a bit too brief.


what about the point that i mentioned about a moving charge creating a magnetic field with no other magnets or charges nearby?



I'm afraid we may be arguing till kingdom come.

Your initial situation suggested a field originating from a strong magnet not the charged ball suspended in the field created by the magnet.

A very powerful magnet is floating next to Sam, at rest. There is a stationary charged ball floating in the magnetic field. Since both the magnet and the charged ball are at rest w.r.t. Sam, the charged ball experiences no force, and it just floats there.

It does not sound like the discussion has stayed consistent with the initial description.

 

[Tach]

ok, that was fine but a bit too brief.

It was just enough to get you to think about what you are asking next:

what about the point that i mentioned about a moving charge creating a magnetic field with no other magnets or charges nearby?

What about it? Does the charge move with respect to the field it generates?

I'm afraid we may be arguing till kingdom come.

I doubt it.

 

[przyk]

but aren't we supposed to take into account the velocity of the charged particle w.r.t. Bob's frame?

Yes, but you are also supposed to take into account the velocity of the magnet with respect to Bob's frame. Ultimately the resolution to your paradox is that the electric and magnetic fields are frame-dependent quantities. The magnet may produce a purely magnetic field in Sam's frame, but in Bob's frame the magnet will actually be surrounded by both nonzero magnetic and electric fields, and it'll work out in such a way that the net Lorentz force on the charged ball is zero.

 

[Aqueous Id]

Unfortunately the wiki article is quite ambiguous as to what v stands for exactly. but i'll just follow your lead.

Aha. Problem solved.

v is the velocity vector, and v x B is not multiplication, but the vector cross product. You would need to have a little background in linear algebra to understand this.

So v is the velocity, in some coordinate system, in directions x, y and z. And B is the magnetic circuit (flux) vector, also having components in the x, y and z directions. v x B is a vector cross product, which is another way of saying that it only multiplies the amount of velocity that crosses the field at a right angle. That's the amount that matters. In addition, it gives a vector result, which is perpendicular to both v and B. (in the right hand sense. See right hand rule.) The force produced by dragging a charge across a B field, or vice versa, has a direction and it's going to act on these two bodies (the ball and the thing producing the B field). It's going to push and pull on them.

However your charge is stationary, so there will be no force induced since the velocity is zero.

You might want to try to find some sample problems to work. They will help you better understand this. Good luck.

 

[eram]

Your initial situation suggested a field originating from a strong magnet not the charged ball suspended in the field created by the magnet.



It does not sound like the discussion has stayed consistent with the initial description.

yeah. I brought it up to discuss what velocities we should take into account.


according to Tach, even from Bob's frame we have to take into account the relative velocity between the charge and the magnet.

and an alternative example by przyk is that using Farday's Law of EMI combined with relativistic electromagnetism changes the fields around the moving magnet (w.r.t. Bob)


Considering that magnetism is apparently a relativistic effect of the electric field, it still needs relativistic correction.
Confusing, eh?

 

[OnlyMe]

yeah. I brought it up to discuss what velocities we should take into account.


according to Tach, even from Bob's frame we have to take into account the relative velocity between the charge and the magnet.

and an alternative example by przyk is that using Farday's Law of EMI combined with relativistic electromagnetism changes the fields around the moving magnet (w.r.t. Bob)


Considering that magnetism is apparently a relativistic effect of the electric field, it still needs relativistic correction.
Confusing, eh?

I may just be confused, but it seems that Bob is nothing more than an observer with a velocity relative to the magnet, its magnetic field, the charged ball and Sam. Sam, the magnet, its magnetic field and the charged ball all share an inertial frame, essentially at rest relative to oneanother.

There does not seem to me to be anything in the hypothetical that is both moving relative to any magnetic or electric field, and interacting with them. As far as Bob is concerned, as an inertial external observer, he sees a magnet, a ball and Sam, comoving relative to himself.

How does an external observer who is not interacting, change anything in the system he observes? If he represented a conductor or charged object moving relative to the "stationary" fields there would be an interaction, but that was not the stated case.

I think I must be missing something or just may not have read or kept up with the discussion.

 

[Tach]

Considering that magnetism is apparently a relativistic effect of the electric field, it still needs relativistic correction.
Confusing, eh?

Not at all.

1. Sam's frame
The charged particle has zero speed $$(v_x,v_y,v_z)=(0,0,0)$$
The Lorentz force is $$\vec{F}=q \vec{v} x \vec{B}=0$$

2. Bob's frame
Bob moves with speed $$(V,0,0)$$ wrt Sam.
The force transforms according to the rules:

$$F'_x=F_x-\frac{V}{c^2} \frac{v_yF_y+v_zF_z}{1-\frac{v_xV}{c^2}}=0$$
$$F'_y=\sqrt{1-\frac{V^2}{c^2}}\frac{F_y}{1-\frac{v_xV}{c^2}}=0$$
$$F'_z=\sqrt{1-\frac{V^2}{c^2}}\frac{F_z}{1-\frac{v_xV}{c^2}}=0$$

 

[eram]

Not at all.

1. Sam's frame
The charged particle has zero speed $$(v_x,v_y,v_z)=(0,0,0)$$
The Lorentz force is $$\vec{F}=q \vec{v} x \vec{B}=0$$

2. Bob's frame
Bob moves with speed $$(V,0,0)$$ wrt Sam.
The force transforms according to the rules:

$$F'_x=F_x-\frac{V}{c^2} \frac{v_yF_y+v_zF_z}{1-\frac{v_xV}{c^2}}=0$$
$$F'_y=\sqrt{1-\frac{V^2}{c^2}}\frac{F_y}{1-\frac{v_xV}{c^2}}=0$$
$$F'_z=\sqrt{1-\frac{V^2}{c^2}}\frac{F_z}{1-\frac{v_xV}{c^2}}=0$$

so it seems that there is no net force acting on the charge in Bob's frame. Now if the charged ball was moving in the magnetic field relative to both Bob and Sam, then it would perform complex helical manoeuvres.

How does an external observer who is not interacting, change anything in the system he observes?

yeah, this is the weird part. As the charge is moving with a certain velocity relative to Bob's frame and it is in a magnetic field, it should experience a lorentz force, or at least i thought so.

it all boils down to what v stands for in F= q(B x v), something which i'm still not very certain about.

 

[Tach]

so it seems that there is no net force acting on the charge in Bob's frame.

Yes. Now you understand.

 

[eram]

the confusion stems from what v stands for in the formula q (B x v)

from the relativistic transformations that Tach provided in #30, there should be no force on the ball in both frames.

My assumption was that any charged object that moved created a magnetic field, and that the magnetic field strength was dependent on the relative velocity to an observer.

Hence, observers in different frames would observe a moving charged object produce magnetic fields of different strengths.

An example is two positively charged balls moving together. If Sam moves with those balls, they would have zero velocity relative to him and produce no magnetic field. They still electrically repel each other.

For Bob, he observes both balls magnetically attracting each other, since they move with a velocity relative to him and generate a magnetic field.

The magnetic attraction causes both balls to repel more slowly. I read that there was actually no Lorentz force, but Time Dilation that created the illusion of a magnetic force.

For this thread, i was hoping for a relativistc explanation.

The apparent solution is that i was wrong about what v stands for, or that the moving magnet in Bob's frame creates an electric field that balances the Lorentz force.

I dont know where time dilation comes into the picture, but oh well.

 

[Tach]

For this thread, i was hoping for a relativistc explanation.

You got a relativistic explanation. The fact that you don't understand it is another issue.
Since you didn't understand the explanation, here is another one: In one frame, there is no motion between the line of force of the magnetic field and the test charge. In a frame moving with speed $$\vec{V}=(+V,0,0)$$ wrt. the first frame BOTH the test charge and the lines of force move with speed $$\vec{V}=(-V,0,0)$$, so, there is NO relative motion between them, therefore NO Lorentz force.

I dont know where time dilation comes into the picture, but oh well.

It doesn't.

 

[eram]

You got a relativistic explanation. The fact that you don't understand it is another issue.
Since you didn't understand the explanation, here is another one: In one frame, there is no motion between the line of force of the magnetic field and the test charge. In a frame moving with speed $$\vec{V}=(+V,0,0)$$ wrt. the first frame BOTH the test charge and the lines of force move with speed $$\vec{V}=(-V,0,0)$$, so, there is NO relative motion between them, therefore NO Lorentz force.




It doesn't.

goes back to the question of what v stands for. Whether a magnetic field is generated depends on the relative velocity between charged particle and observer.

I remember reading that the Lorentz force between two charges moving parallel to each other is caused by time dilation in the Cartoon Guide to Physics by Gonick & Huffman.

And time dilation is dependent on the relative velocity.


Sounds kinda stupid quoting a comic book lol. I have also seen this on some sites, though i can't seem to find them at the moment.

 

[Tach]

goes back to the question of what v stands for.

For the third (and last) time , $$\vec{v}$$ is the RELATIVE velocity between the test probe and the magnetic field.

 

[eram]

For the third (and last) time , $$\vec{v}$$ is the RELATIVE velocity between the test probe and the magnetic field.

what about that book by gonick? There should be a pdf of it somewhere.

But I'll just go with the flow for now.

 

[OnlyMe]

what about that book by gonick? There should be a pdf of it somewhere.

But I'll just go with the flow for now.

I think the problem you are having is that the charged ball is not moving relative to the magnetic field in any frame of reference, because the ball and magnetic field are at rest relative to each other, in "their" common frame of reference.

If you move a wire through a magnetic field you get a current in the wire. Same thing if the wire is standing still and the magnetic field moves relative to it.

In your example, it is only an external observer who moves relative to the shared frame of reference of the charged ball and magnetic field.

I don't even think your example designates the observer as moving through the magnetic field. Unless the observer is a charged particle or conductor and moves through the magnetic field, the motion of the observer has no affect on anything other than what things look like to the observer.

 

[eram]

I think the problem you are having is that the charged ball is not moving relative to the magnetic field in any frame of reference, because the ball and magnetic field are at rest relative to each other, in "their" common frame of reference.

If you move a wire through a magnetic field you get a current in the wire. Same thing if the wire is standing still and the magnetic field moves relative to it.

In your example, it is only an external observer who moves relative to the shared frame of reference of the charged ball and magnetic field.

I don't even think your example designates the observer as moving through the magnetic field. Unless the observer is a charged particle or conductor and moves through the magnetic field, the motion of the observer has no affect on anything other than what things look like to the observer.

but what I read about the Lorentz force being an effect of time dilation sounds contrary. I'm still thinking it through.

 

[OnlyMe]

but what I read about the Lorentz force being an effect of time dilation sounds contrary. I'm still thinking it through.

I think you are mixing up a Lorentz force and Lorentz transfomation, or not.

The problem is that in the example you gave in the opening post below, the charged ball is not moving reltive to the magnetic field of the magnet.., so there is no Lorentz force.

... imagine two people, Sam and Bob, who are floating in deep space. Both are in an inertial frame.

A very powerful magnet is floating next to Sam, at rest. There is a stationary charged ball floating in the magnetic field. Since both the magnet and the charged ball are at rest w.r.t. Sam, the charged ball experiences no force, and it just floats there.

The last part of the example, below is where the problem begins...

However, Bob is whizzing past Sam at a high velocity. The charged ball has a velocity w.r.t. to Bob's inertial frame. Bob would see that a Lorentz force is acting on the charged ball, and even see the ball move around in circles.

Bob is moving relative to both the charged ball and the magnetic field, he sees them both just setting there in space, as he whizzes by. If he and Sam are looking at their watches, they may notice that their watches don't match.., i.e. some time dilation.., but neither Bob or Sam see the charged ball moving through a magnetic field, so no Lorentz force...

From Wiki,
... the Lorentz force is the force on a point charge due to electromagnetic fields. If a particle of charge q moves with velocity v in the presence of an electric field E and a magnetic field B, then it will experience a force ...​

The point charge.., the charged ball in your example is at rest relative to the magnetic field, as seen by both Bob and Sam.