Are Gauss's Law and Coulomb's Law Equivalent?

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SUMMARY

Gauss's Law and Coulomb's Law are fundamentally equivalent, with Gauss's Law being derivable from Coulomb's Law. The discussion highlights that while Coulomb's Law is often introduced first in electromagnetism textbooks, it is possible to derive Gauss's Law using the concept of electric flux through a closed surface. The equivalence is demonstrated through the application of the divergence theorem, showing that the flux through any surface containing a point charge remains constant regardless of the surface shape. This principle extends to any charge distribution through the method of superposition.

PREREQUISITES
  • Understanding of Gauss's Law in electromagnetism
  • Familiarity with Coulomb's Law and electric fields
  • Knowledge of electric flux and divergence theorem
  • Basic concepts of charge distributions and superposition principle
NEXT STEPS
  • Study the derivation of Gauss's Law from Coulomb's Law in detail
  • Explore the application of the divergence theorem in electromagnetism
  • Investigate different charge distributions and their effects on electric fields
  • Learn about the superposition principle in electrostatics
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Students of electromagnetism, physics educators, and anyone seeking to deepen their understanding of the relationship between Gauss's Law and Coulomb's Law.

Enjolras1789
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Gauss's law may be used to derive Coulomb's law. I've never seen it done the other way around. Is this possible? Because I have not seen it, I infer that these forms are not logically equivalent.
 
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They are the same. Take a point charge. Since the Coulomb field is everywhere divergenceless except at the position of the charge, you can show it doesn't matter what shape your surface containing the charge is, the flux through any surface containing the charge is the same. You can easily evaluate the flux using a sphere with the charge at the origin. This gives Gauss' law for a point particle, but you can extend it to any charge distribution using superposition.
 
Almost every intro textbook in EM (at least the more popular ones) introduce Coulomb Law and use it to derive the Gauss Law. It's usually done for the case for the point-particle, and then generalised to a continuous charge distribution.
 

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