Lorentz Force Paradox

[eram]

t's 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.


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.


Imagine the awkward conversation between Bob and Sam afterwards.

Bob: Hey did you see that charged ball whipping about in a flurry!?

Sam: I don't know what you're talking about. :crazy: That ball was perfectly still. Are you high? :m:

Bob: I swear! I saw it with my own two eyes! :wtf:

Sam: I'm going to get you a pair of spectacles. :facepalm:

 

[Aqueous Id]

Does the ball have mass and is it acted on by gravity? Suppose this magnetic field is induced by an electromagnet, and the power is switched off. Will the charge move? Why or why not? What are the implications of having a charge float motionless in a magnetic field?

In order to induce a Lorentz force a charge needs to be moving with respect to a B field. How does this rule apply here?

 

[eram]

In order to induce a Lorentz force a charge needs to be moving with respect to a B field. How does this rule apply here?

umm...didn't i write it in?

 

[origin]

Why would the ball have a velocity relative to the magnetic field? Wouldn't bob see ball and magnetic field (if you could see the field) moving at the same velocity - in other words there would not be a relative motion between the ball and the field from either perspective.

 

[Tach]

Why would the ball have a velocity relative to the magnetic field? Wouldn't bob see ball and magnetic field (if you could see the field) moving at the same velocity - in other words there would not be a relative motion between the ball and the field from either perspective.

Velocity is not frame invariant in relativity. Zero velocity in "Sam frame" transforms in non-zero velocity in "Bob frame" .

 

[Tach]

t's 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.


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.


Imagine the awkward conversation between Bob and Sam afterwards.

Bob: Hey did you see that charged ball whipping about in a flurry!?

Sam: I don't know what you're talking about. :crazy: That ball was perfectly still. Are you high? :m:

Bob: I swear! I saw it with my own two eyes! :wtf:

Sam: I'm going to get you a pair of spectacles. :facepalm:

There is no paradox. Your error is that you are using the classical definition of the Lorentz force, you need to use the relativistic definition. When you do that, you will find out that the equations of motion are the same in both frames.

 

[eram]

There is no paradox. Your error is that you are using the classical definition of the Lorentz force, you need to use the relativistic definition. When you do that, you will find out that the equations of motion are the same in both frames.

heh didnt learn about that in my relativity course. what about at low velocities?

 

[Tach]

heh didnt learn about that in my relativity course. what about at low velocities?

relativity works just the same at ALL velocities, didn't you learn that in class?

 

[eram]

well the effects should be much less pronounced.

so according to theory, what should Bob observe?

 

[eram]

unfortunately, what i learned about relativity didn't involve matrices and tensors. so i'm not sure how to interpret it.

 

[Tach]

well the effects should be much less pronounced.

so according to theory, what should Bob observe?

Simple, if Bob moves along the x axis with speed +v, both the magnetic lines and the charged particle move with speed -v. There is no circular motion in Bob's frame, as you seem to believe.

 

[eram]

Simple, if Bob moves along the x axis with speed +v, both the magnetic lines and the charged particle move with speed -v. There is no circular motion in Bob's frame, as you seem to believe.

so to Bob both the charged particle and magnet move together in an inertial frame?

but aren't we supposed to take into account the velocity of the charged particle w.r.t. Bob's frame? And as long as it moves in a magnetic field, it experiences a Lorentz force?

Excuse me if i seem impolite, its just the way it sounds over the internet.

 

[origin]

Velocity is not frame invariant in relativity. Zero velocity in "Sam frame" transforms in non-zero velocity in "Bob frame" .

Agreed, I was trying to make the same point that you made, which is, regardless of which frame you are using there will be no relative velocity between the magnetic field and the ball in this scenario.

 

[Tach]

so to Bob both the charged particle and magnet move together in an inertial frame?

Yes. You said that you had a relativity class or am I misquoting you?

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

How about Bob's motion wrt the magnet, aren't you supposed to consider that?

And as long as it moves in a magnetic field, it experiences a Lorentz force?

Does it move? Are you sure abbout that?

 

[Tach]

Agreed, I was trying to make the same point that you made, which is, regardless of which frame you are using there will be no relative velocity between the magnetic field and the ball in this scenario.

yes, of course.

 

[eram]

Yes. You said that you had a relativity class or am I misquoting you?

yeah i did have a class. though you might still consider me a noob.


How about Bob's motion wrt the magnet, aren't you supposed to consider that?

Yeah i thought about that.

but i've always thought that when calculating the lorentz force we have to take into account the relative velocity of the charge and forget about the relative velocity of the magnetic field.

The equations state q(Bxv) where B is the flux density and v is the relative velocity of the charge. No mention of the relative velocity of the magnet.

Unless its part of relativistic lorentz force, in which case i am clueless.

 

[Tach]

Yeah i thought about that.

but i've always thought that when calculating the lorentz force we have to take into account the relative velocity of the charge and forget about the relative velocity of the magnetic field.

There isn't any relative motion.

 

[eram]

The equations state q(Bxv) where B is the flux density and v is the relative velocity of the charge. No mention of the relative velocity of the magnet.

Unless its part of relativistic lorentz force, in which case i am clueless.

unless you're stating that v stands for the relative velocity between the charge and the magnet.

but i've always learned that a moving charge creates a magnetic field, and the flux density of this field depends on the relative velocity of the charge, even when there is no magnet around.

 

[eram]

There isn't any relative motion.

isn't there relative motion between Bob and the charged ball?

as i previously stated, shouldn't we take into account only the relative motion of Bob and the charge, and not the relative motion between the charge and the B field?

 

[Tach]

isn't there relative motion between Bob and the charged ball?

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

as i previously stated, shouldn't we take into account only the relative motion of Bob and the charge, and not the relative motion between the charge and the B field?

You have it exactly backwards.