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What I think is that Gauss's law is an independent law of nature which shouldn't change on modifying Coulomb's law.Am I correct?

Well the relation between Gauss's law and Coulomb's law is very strong but I can't strictly say that they can be derived from each other just like I can't say that law of conservation of linear momentum can be derived from Newton's 3rd law.

Edit

It's my gentle request that if one of them is a theorem then please name one of them as a theorem as it makes them quite complicated.

Qmechanic
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Shreyansh Pathak
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    No they are not independent. Gauss' law can be mathematically derived from Coulomb's law, and if you're careful about the details they can be shown to be equivalent. – doetoe Sep 09 '19 at 14:32
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    The equivalence implies that you can postulate either as a law of nature, making the other a consequence (or a theorem). Once you have equivalence, I don't think it is useful to distinguish which is the law and which is the theorem. – doetoe Sep 09 '19 at 14:44
  • Maybe this helps? https://en.wikipedia.org/wiki/Coulomb%27s_law#Deriving_Gauss's_law_from_Coulomb's_law – infinitezero Sep 09 '19 at 15:16
  • Related: https://physics.stackexchange.com/q/93/2451 , https://physics.stackexchange.com/q/47084/2451 and links therein. Are you assuming 3 spatial dimensions, or can the number of spatial dimensions also vary? – Qmechanic Sep 09 '19 at 15:34
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    Concerning the edit: the names are traditional. Not only is [physics.se] not in charge of them, but it would take significant time for even a concerted effort by the global physics community to effect a change. And to relatively little gain. – dmckee --- ex-moderator kitten Sep 09 '19 at 22:48
  • @Qmechanic♦the no. of spatial dimensions also vary. – Shreyansh Pathak Sep 10 '19 at 07:19
  • "if one of them is a theorem then please name one of them as a theorem" ─ what if the laws are equivalent, and they can both be derived from each other? (Or maybe both can be derived from each other with some reasonable additional assumptions, as is the case here?) Which one is the "law" and which one is the "theorem"? It sounds like Feynman's Messenger Lecture The Relation of Mathematics and Physics is pretty close to required viewing for you at this time. – Emilio Pisanty Sep 13 '19 at 11:50

1 Answers1

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No, Coulomb's law can be derived directly from Gauss law in three dimensions and vice-versa (assuming electric fields are governed by linear equations). Changing one implies that the other changes too.

However, if one measured the Coulomb's law to go as $\frac{1}{r^n}$ this could be a direct consequence of the world being $(n+1)$ dimensional with the Gauss law intact.

Akerai
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  • Well,Gauss's law is a law,we must not say that it can be derived. – Shreyansh Pathak Sep 09 '19 at 14:27
  • Nomenclature aside, they are equivalent if electric fields linearly add. The answer stands unchanged. – Akerai Sep 09 '19 at 14:30
  • Nomenclature in physics is not a layman's nomenclature. – Shreyansh Pathak Sep 09 '19 at 14:31
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    It is pointless to go down this rabbit hole of a discussion. Assuming Coulomb's law and the superposition principle - one can show that the Gauss law is a natural mathematical consequence. – Akerai Sep 09 '19 at 14:36
  • Read the edited version of the question. – Shreyansh Pathak Sep 09 '19 at 14:37
  • "this could be a direct consequence of the world being (n+1) dimensional with the Gauss law intact". Could you please explain this statement in more detail? – user600016 Sep 09 '19 at 15:56
  • I will start building up the intuition. In a 2D world one would obtain the Coulomb law by enclosing a point charge by a circle - whereas the divergence would be the electric field flowing through the circumference of the circle - this would lead to a $1/r$ dependence.

    In a 3D world one encloses a charge by a spherical shell and integrates the flow of electric field through the enclosing surface yielding $1 / r^2$ dependence.

    In a n-D world one encloses a point charge by a shell of an n-sphere, yielding the Coulomb law of $1/ r^{n-1}$. Of course, here I am assuming non-compact dimensions.

     – Akerai Sep 09 '19 at 17:46
  • Therefore if the world had $n+1$ extended spatial dimensions and the Maxwell's equations held true - this would lead to a Coulomb law scaling as $1/r^n$. Interestingly, for n>3 there would be no stable gravitational or electrostatic orbits. See https://en.wikipedia.org/wiki/Bertrand%27s_theorem – Akerai Sep 09 '19 at 17:55