Why Does Gauss's Law Give Zero Flux for a Point Charge Inside a Sphere?

[eok20]

i know that the flux through a sphere with a point charge at the center is non-zero $$\left( \frac{q}{\epsilon_0}\right)$$ but if I wanted to calculate this using Gauss's law I would take the divergence of $$E=\frac{q}{4\pi\epsilon_0r^2}\hat{r}$$ which is 0 so I would get the flux to be 0. What am I doing wrong?

Thanks.

 

[robphy]

http://forums.wolfram.com/mathgroup/archive/1997/Mar/msg00049.html

 

[Claude Bile]

The problem lies in the fact that you are using a point charge. Discontinuities such as these are commonly represented by the Dirac delta function that is defined as follows (for 3 dimensions).

$$\int_{V} \delta(r) d\tau = 1$$

where V is any volume that contains the origin. Also;

$$\delta(r) = 0$$ for r not equal to 0
$$\delta(r) = \infty$$ for r equal to 0

The problem is that when you calculate the divergence, it does not include the origin (since at the origin you are effectively dividing by zero). When the charged sphere has a finite radius, this is not a problem, because the contribution from the origin is infinitesimally small. In the case of the point charge however, the entire contribution is coming from the origin, hence the original error.

To fix this, you need to include the Dirac delta function when you calculate the volume integral.

Claude.