If Coulomb's were to depend on $\frac{1}{r^n}$ instead of $\frac{1}{r^2}$ where $n\neq 2$,would Gauss's law still be the same?

Question

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.

Akerai · Answer 1 · 2019-09-09T14:28:40.393

3

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.

edited Sep 09 '19 at 14:28

answered Sep 09 '19 at 14:24

Akerai

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If Coulomb's were to depend on $\frac{1}{r^n}$ instead of $\frac{1}{r^2}$ where $n\neq 2$,would Gauss's law still be the same?

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