# Coulomb's law and Gauss' Law

Which of these laws is more fundamental or forms the basis of electrostatics? I started off with Coulomb's law and then I studied Gauss' law. I was wondering which one is more universal?

My professor derived Gauss' law using Coulomb's law but didn't do it the other way, so is Coulomb's law more fundamental? And can Gauss' law be used to prove the other?

• Just apply Gauss' law to a point charge. – ACuriousMind Mar 1 '15 at 18:38
• @ACuriousMind That was quite silly of me . Thanks . Got the derivation of each of them using the other one . What about the more fundamental one ? – Klosew Mar 1 '15 at 18:44
• With the addition of $\vec{F}_E = q \vec{E}$ you can derive either from the other. They are formally equivalent and choosing one to be "more fundamental" is a philosophical statement. – dmckee --- ex-moderator kitten Mar 1 '15 at 18:44
• Klosew, I assume you didn't read this before posting your question? en.wikipedia.org/wiki/Gauss%27s_law#Relation_to_Coulomb.27s_law – Alfred Centauri Mar 1 '15 at 18:52
• @Klosew, I would say that Gauss' law is more general than Coulomb's law which is what I think you mean by "more fundamental". – Alfred Centauri Mar 1 '15 at 19:22

Because Gauss's law applies for both moving and stationary charges, while Coloumb's law applies only for stationary charges, Gauss's law can be considered more fundemental. This is why Gauss's law is one of the four Maxwell equations. The derivation of Gauss's law from Coloumb's law only works for stationary charges; for moving charges the derivation is invalid yet Gauss's law still holds. However, Gauss's law along with the information from Maxwell's third equation that the $curl E = 0$ for stationary charges (since then $B$ will be constant), can be used to derive Coloumb's equation. In short, Gauss's law can be considered more fundemental because it applies to both stationary and moving charges, while Coloumb's law applies only to stationary charges.

If we are only considering three spatial dimensions, then Coulomb's and Gauss's law are completely mathematically equivalent and there is no basis to consider either to be more fundamental than the other. But in a number of dimensions other than three, they are no longer equivalent, and when theorists consider generalizing electromagnetism to other numbers of dimensions, they almost always keep Gauss's law the same and modify Coulomb's law. So in that very weak sense, one could consider Gauss's law to be more fundamental. I discuss here why they do so. At the end of the day it basically boils down to a philosophical preference for mathematical elegance; until we actual find a universe with a different number of dimensions, there is no "right" answer.

From the Feynman Lectures on Physics (I would have made this a comment but I don't have enough points)

From our derivation you see that Gauss' law follows from the fact that the exponent in Coulomb's law is exactly two. A $1/r^3$ field, or any $1/r^n$ field with $n≠2$, would not give Gauss' law. So Gauss' law is just an expression, in a different form, of the Coulomb law of forces between two charges. In fact, working back from Gauss' law, you can derive Coulomb's law. The two are quite equivalent so long as we keep in mind the rule that the forces between charges are radial.

• So both can be called equally fundamental ? – Klosew Mar 1 '15 at 18:42
• They are in a sense equivalent. They describe the same phenomenon from different points of view. Similar to the different formulation of classical mechanics. I guess you could call them equally fundamental, but I don't really like the term. – Gonate Mar 1 '15 at 18:45
• Isn't Gauss's law true for moving charges, unlike Coloumb's law, which works solely for stationary charges. – user70720 Mar 1 '15 at 18:50
• @Andy, No, Gauss Law is true for all charges. What's more, Gauss's Law is just the Divergence Theorem applied to the electric field. The Divergence Theorem is a the 3D equivalent of the Fundamental Theorem of Calculus and Green's Theorem. The general formulation is called Stoke's Theorem. – Gonate Mar 1 '15 at 19:02
• @Gonate I ment that Gauss's Law works for all charges, moving and stationary, while Coloumb's law works solely for stationary charges. – user70720 Mar 1 '15 at 19:07