# A question about forces in different references of frame

1. May 8, 2007

### henry_arsenal

Hello Everyone, I'm new to this forum, I've to say here is one of the greatest physics communities in the net and I'm really glad I have the opportuninty to use the huge amount of information presented here.

By the way, I have a question about Electromagnetics:

Consider two stationary charges, +Q1 and +Q2 and an observer beside them. This observer measures a columbian force on both of the charges and nothing else.

Now, imagine an observer getting close to the charges at a relative speed of V. He should measure an additional force (rather than the former columbian force) produced by the magnetic field which exists due to the fact the the moving observer finds out that the charges are getting close to him at a speed of -V, of course from his point of view.

Both of the observers obey the laws of physics and maxwell's equations correctly, but they measure different forces on the charges. What is the correct answer to this problem?

I guess it should be related to the special relativity transformations, but I don't know how. I'll be thankful if you help me understanding how to solve this problem.

Thanks

Arman.

2. May 8, 2007

### Meir Achuz

Force can be Lorentz transformed by introducing the "Minkowski force", which is a 4-vector. If you define force as $${\bf F}=\frac{d\bf p}{dt}$$,
then the Minkowski force is given by
$${\cal F}^\mu=\left[\gamma\frac{dE}{dt},\gamma{\bf F}\right].$$
Transform this as a 4-vector, and then you can identify F in the new system.

Last edited: May 8, 2007
3. May 8, 2007

### Xezlec

Notice that the magnetic force in the moving frame is always in the opposite direction to the electric force. But, the distance between the charges appears less (because of length contraction), so their electric force is a little bit stronger. So everything's fine.

EDIT: I think my above explanation is wrong. If you picture the case where the span between the charges is perpendicular to their motion, then there is no length contraction, but the problem still happens. Does the force really change?

Meir Achuz: I think force is invariant in this case... am I missing something? I have no clue what Minkowski force is.

Last edited: May 9, 2007
4. May 10, 2007

### Meir Achuz

Forget the name, the definition is in the equation I wrote.
"Force" is not a Lorentz invariant.
Read an intermediate EM or relativity book.