Surface Integrals in Gauss's Theorem of Charge in Motion

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Discussion Overview

The discussion revolves around the application of Gauss's theorem to charge in motion, specifically focusing on how to properly evaluate the surface integral involved. Participants explore the implications of time dependence and the nature of the charge considered in the integral.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • Some participants suggest that the surface integral can be taken over a period of time, while others argue it should be evaluated at a single instant.
  • One participant states that any Gaussian surface can be chosen, independent of the motion of the charge.
  • There is a proposal that the charge used in the integral should be the one present inside the surface at the moment of calculation, assuming no relativistic speeds are involved.
  • Another participant challenges the necessity of the assumption regarding relativistic speeds, noting that Maxwell's equations account for relativistic effects.
  • A later reply mentions the possibility of considering a time-average of the charge if integrating over time, but emphasizes that Gauss's theorem applies at every instant.
  • Concerns are raised about the implications of a moving Gaussian surface on the evaluation of the integral.

Areas of Agreement / Disagreement

Participants express differing views on whether the surface integral should be evaluated over time or at a single instant, indicating a lack of consensus on this aspect of Gauss's theorem in the context of charge in motion.

Contextual Notes

There are unresolved questions regarding the assumptions made about the charge and the treatment of relativistic effects, as well as the implications of a moving Gaussian surface on the integral.

BIT1749
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gauss's theorem is also applicable to charge in motion.but how the surface integral has to be taken??
 
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However you like. You can choose any gaussian surface, it does not need to follow the charge in any way.
 
i have read in a book that the surface integral has to be taken over a period of time.but what value should we put in place of charge??
 
you use the charge that was inside of the surface at the moment you chose to do the calculation (assuming no charges with relativistic speed are present)
 
What book was this? Gauss' law is true instantaneously.
 
BIT1749 said:
i have read in a book that the surface integral has to be taken over a period of time
No, the integral is a 2D integral over a spatial surface defined at a single instant of time.

BIT1749 said:
.but what value should we put in place of charge??
huh? You put the charge in place of the charge. You can't put anything else there.
 
dauto said:
(assuming no charges with relativistic speed are present)
I don't think that is a necessary assumption. Maxwell's equations are fully relativistic already.
 
BIT1749 said:
i have read in a book that the surface integral has to be taken over a period of time.but what value should we put in place of charge??
In that specific example, they may have been considering a time-average... But in general as others have said, Gauss' theorem works at every instant of time. So you can integrate over time and then divide by the time interval if you want to get a time average.

edit: p.s. be careful in cases where the Gaussian surface is also moving.
 

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