Understand Electromagnetic Force in Different Inertial Frames

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Discussion Overview

The discussion revolves around understanding the electromagnetic force in different inertial frames, particularly focusing on the effects of length contraction on positive and negative charges in a conductor. Participants explore the implications of charge distribution and neutrality in various frames, examining both conceptual and technical aspects of the problem.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about the lack of length contraction for negative charges in a specific inertial frame, questioning why it does not lead to an electric force in that scenario.
  • Another participant argues that for the conductor to remain electrically neutral, the distances between charges must be arranged such that the negative charges are farther apart in their rest frame than the positive charges are in theirs.
  • A participant seeks clarification on how the arrangement of charges achieves neutrality in the context of length contraction, suggesting that equal distribution of charges should imply equal separation.
  • It is noted that the assumption of equal distribution of charges is frame dependent, and that neutrality in one frame may not translate to neutrality in another.
  • Participants discuss the possibility of achieving neutrality by adjusting the velocities of the charges, indicating that there are multiple arrangements that can satisfy the condition of neutrality.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the implications of charge distribution and length contraction across different frames. There are competing views on how neutrality is maintained and the assumptions underlying charge arrangements.

Contextual Notes

The discussion highlights the dependence of charge distribution on the chosen inertial frame and the dynamic nature of the situation, indicating that assumptions made in one frame may not hold in another. There are unresolved questions regarding the specific arrangements of charges and their implications for neutrality.

MathewsMD
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I've attached an image to a conceptual problem I'm having a little trouble understanding. It shows part a, b, c, and d (but d is cut off). a and b are one inertial frame while c and d are another. c and d makes sense, but I'm having a little trouble with a and b. Based on the image for c, there is length contraction for the + charges in the wire causing there to be more localized charge and thus an electric field. This makes sense since the + charges are moving in this inertial frame. Now I don't quite see why this doesn't occur in part a/b (i.e. why is there no length contraction of the - charges? If the + charges are at rest while the - charges are moving at speed v in this case, shouldn't there be a localization of - charges also causing for an electric force AND magnetic force in a/b? Any clarification on why the length contraction and thus electric force is not present in a and b would be very helpful!
 

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This does happen, it is just that in order for the conductor to be electrically neutral in the original system, then the minus charges must be the same distance from each other as the positive ones (it is clearly possible to arrange for this to be the case). This of course means that there is more distance between the minus charges in their rest frame than there is between the positive charges in their rest frame.
 
Orodruin said:
This does happen, it is just that in order for the conductor to be electrically neutral in the original system, then the minus charges must be the same distance from each other as the positive ones (it is clearly possible to arrange for this to be the case). This of course means that there is more distance between the minus charges in their rest frame than there is between the positive charges in their rest frame.

Okay, so length contraction does occur in case a/b? I may be misinterpreting something here since I don't see how the particular arrangement given in the attachment achieves this.

I agree the conductor should be electrically neutral in this case for there to be no net electric force, but I am just a little confused on how this is achieved in case a/b if there is length contraction...wouldn't this cause unequal separation?

For the conductor to be electrically neutral, shouldn't the charges in the conductor be moving +v/2 and -v/2 respectively (i.e. of equal speed) in the inertial frame?
 
MathewsMD said:
wouldn't this cause unequal separation?

Only if the distance between the positive charges in their rest frame is the same as the distance between the negative charges in theirs. Your problem seems to be that you are implicitly assuming that this is the case.
 
Orodruin said:
Only if the distance between the positive charges in their rest frame is the same as the distance between the negative charges in theirs. Your problem seems to be that you are implicitly assuming that this is the case.

Yes, I was assuming that. If you don't mind explaining, why isn't this the case? Since there are an equal amount of + and - charges in the wire, can't we assume equal distribution? Thus, if the both charges travel at speed v in the other charge's inertial frame, wouldn't this assumption be true?

I think I'm starting to get what you're saying (i.e. this possible arrangement can be achieved) but I'm just failing to see why it is achieved in the particular situation given.
 
MathewsMD said:
Since there are an equal amount of + and - charges in the wire, can't we assume equal distribution?

This statement is frame dependent. You have to define in which frame you want to have as many + and - charges. Why would it be the case? The entire situation is dynamic and I am suspecting the idea from the beginning was to have an electrically neutral wire transmitting a current. There are several ways of doing that. The main point is to realize that making it neutral in one frame will result in it being charged in another.
 
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Orodruin said:
This statement is frame dependent. You have to define in which frame you want to have as many + and - charges. Why would it be the case? The entire situation is dynamic and I am suspecting the idea from the beginning was to have an electrically neutral wire transmitting a current. There are several ways of doing that. The main point is to realize that making it neutral in one frame will result in it being charged in another.

Okay. So we're assuming that in the inertial frame given for case a/b that this wire HAS to be neutral and is thus equally distributed in this particular frame. That makes sense. Thank you for all the help! Please feel free to add on anything you think I may still be misinterpreting. :)
 
MathewsMD said:
Okay. So we're assuming that in the inertial frame given for case a/b that this wire HAS to be neutral and is thus equally distributed in this particular frame

Correct. You can also achieve this by, as you said, having the + charges going with velocity +v/2 and the - charges with velocity -v/2. The overall charge and current would be the same. But if you want the wire to be neutral in the rest frame of the + charges, then there is only one way of arranging that.
 

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