Question

Would Gauss’s law remain valid if Coulomb’s law were replaced by an inverse cube law? anysuitable example? is fair to say that Coulomb’s law and Gauss’s law are equivalent statements?

Answer 1

what do you tink?!

Answer 2

no idea :(

Answer 3

Coulombs law can be derived from Gauss' law, so in that way it can be regarded as a special case of Gauss' law. However I would not say that they are equivalent, but rather that one implies the other.
I have don't know what you mean about your inverse cube question.

Answer 4

There is no inverse cube relationship (1/x^3) in Coulombs law, just an inverse square relationship (1/x^2)

Answer 5

you can force an inverse cube-looking equation if you remove the denormalize unit vector\[\vec F=\frac1{4\pi\epsilon_0 r^2}\hat r=\frac1{4\pi\epsilon_0 r^3}|\vec r|\]but this is somewhat circular and pointless, and still results in an inverse square effect in terms of experienced force

Answer 6

if you denormalize the .... *

Answer 7

One of the tests for the closeness to -2 of power of r is how well Gauss's law is satisfied. 1/r^3 would not.

Answer 8

@TuringTest

I think the question assumes what if our universe behaved in such a way, that electric force went down as 1/r^3.. which i know is pretty stupid, because the composer of the question is only trying to test the mathematical reasoning behind the derivation of Gauss law and not the physical aspects of it

one way i can think is what if we had dipoles everywhere, only electric dipoles (of infinitesimally small size) and no mono poles just like magnetism :D, then we can get 1/r^3 relationship, and Gauss law would have RHS = 0

Answer 9

First read this simple explanation of Virial theorem.
http://math.ucr.edu/home/baez/virial.html

and then read this one. This is a good way to reach the Coulombs law.
www7b.biglobe.ne.jp/~kcy05t/viriproof.html

Answer 10

@sinaphysics
That is by far the most unnecessary and most non related replies ever xD

Answer 11

it would not change the Gauss's law

Answer 12

well i read in feynman lectures on physics that gauss law is valid only coz inverse square law is valid
couloumb's law and gauss law can be derived from each other
you can read about it in volume 2 chapter 4 of feynman lectures