# Maxwells' equations and Coulomb's law

Coulomb's law and Maxwell's equations should be consistant as one can be derived from the other.

Say we have a point charge with such a charge that $-kq=1$, meaning that at any point the electric field will have a magnitude of

$$|E|=\frac{1}{r^2}$$

where $r$ is the distance from the origin (were we place our charge), and the vectors point towards the origin at all point. This would be equivalent to the following in cartesian co-ordinates:

$$E=-\hat{x}\frac{x}{(x^2+y^2+z^2)^{3/2}}-\hat{y}\frac{y}{(x^2+y^2+z^2)^{3/2}}-\hat{z}\frac{z}{(x^2+y^2+z^2)^{3/2}}$$ We can verify that

$$|E|=\frac{1}{\sqrt{x^2+y^2+z^2}}$$

Gauss's law in its differential form allows us to calculate a charge distribution that would give rise to such a electric field using the divergence operator:

$$\nabla \cdot E=-\frac{1}{x^2+y^2+z^2}$$

from wolfram alpha: one, two

Which absolutely doesn't make sense to me! Intuitively I would think it would be zero everywhere except (0,0,0). Or at least not to go of to infinity at any point.

Could somebody please explain what's going on?

## 2 Answers

Actually if you calculate $\boldsymbol{\nabla} \cdot \mathbf{E}$, you get zero except at the origin, where you get infinity. So you can do it more precisely and obtain a delta function. I suspect an error in your calculation.

• The error is he says $|\mathbf{E}|$ goes like $1/r$. Commented Jan 31, 2014 at 0:36
• In other words, he should have done this calculation instead. Adding up gives a denominator of $2x^2 + 2y^2 + 2z^2 - (y^2 + z^2) - (x^2 + z^2) - (x^2 + y^2) = 0$ Commented Jan 31, 2014 at 17:17
• You meant numerator, but yes I agree with you. Commented Jan 31, 2014 at 17:26

What is the problem? Any charged object has a volume, so the denominator never can be zero. So E will never be infinite. The law is $$E = 1 / r^2$$ is correct. But always $$\vec \nabla \cdot \vec E = q/\epsilon_0$$ because kq=1 yuo can immediately calculate: $$k= 1/ (4 \pi \epsilon_0 )$$ $$q = 4 \pi \epsilon_0$$ and verify with $$\vec \nabla \cdot \vec E = \int^0_r\int^0_{2\pi}\int^0_\pi E r \sin(\theta) d r d \theta d \phi= E 4\pi r^2$$ $$E 4\pi r^2 = q /\epsilon_0$$ $$4\pi = q /\epsilon_0$$ $$q = 4 \pi \epsilon_0$$