# Why does Coulomb's law hold only for two point charges? [closed]

What is the condition for validation of Coulomb's Law?

## closed as off-topic by AccidentalFourierTransform, Cosmas Zachos, WillO, stafusa, ZeroTheHeroSep 16 '18 at 0:02

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## 2 Answers

Coulomb's Law holds for any static charge configuration. In general, it is better expressed as $$\mathbf{E}(\mathbf{r})= \frac{1}{4 \pi \epsilon_0} \int \frac{(\mathbf{r} - \mathbf{r'})}{|\mathbf{r} - \mathbf{r'}|^3} dq(\mathbf{r'})$$ where $$dq$$ is a infinitesimal charge element, $$\mathbf{r}$$ is the position at which the field is being calculated, and $$\mathbf{r'}$$ is the location of the charge element.

• Actually it also holds in the quasi-static regime, where $v/c\ll 1$ for the moving charges. – ZeroTheHero Sep 16 '18 at 0:02

Because the distribution of charges in finite size conductors changes with distance between them (due to their influence on each other) and that breaks the relationship.

If the distance between the conductors is much greater than their size, the formula works fine.

• I didn't again got why don't this law work if distance is lesser than 10^-15, why doesn't it hold? Will you please explain? – user206730 Sep 15 '18 at 15:26
• @user206730 The law holds for point charges. Since truly point charges don't exist, it works for chrges that are much smaller than the distance between them, so that they could be approximated as point charges. Obviously, for distance of 10^-15, it couldn't be the case. – V.F. Sep 15 '18 at 18:00
• What could not be the case and why? You again did not tell why this law is not valid for very small distance (I mean nuclear distances).. – user206730 May 8 at 6:15
• "What could not be the case and why?" We cannot have charges small enough relative to the distance between them, if the distance is 10^-15, because the size of an electron is greater than 10^-15. – V.F. May 8 at 15:55
• "We cannot have charges small enough relative to the distance between them" Did you mean size of charges should not be smaller than distance between them? – user206730 May 9 at 0:18