Solving Maxwell's Equations: A Challenge

[101111001101]

Hello all... I have been working on this problem that I just am not being able to solve.

I've been spending my spare time learning some vector calculus and non-euclidean geometry (my aim is to be able to finally tackle relativity). After learning some basic things about the del function, I found that I had sufficient mathematical knowledge to be able to derive Maxwell's equations (well I thought I did).

I had a go at the first of the four. The way I am trying to do it is by taking Coulomb's law, writing it as a vector in three dimensions and then taking the divergence. I am hoping to get this from it:

$$\nabla{\mathbf{.E}} = \frac{\rho}{\epsilon}$$

But I keep on getting 0. And I have no idea why...
Here is a scanned copy of my working, I would be very grateful if you could point out my errors (other than the minus sign maybe) to me.

Cheers.

http://postimage.org/image/q44ym4ro/

 

[MisterX]

For the hypothetical situation you described, with a single point charge at (0,0,0), the charge density should be zero everywhere except at (0,0,0).

The charge density would be q times a 3 dimensional (generalized) Dirac delta. This way the volume integral of $\rho$ for a volume containing (0,0,0) would be equal to q.

Perhaps it would be simpler to just use the integral forms when dealing with point charges.

 

[K^2]

If you mean taking divergence of E from point charge, you have to use Cauchy Formula, in which case you will get exactly q δ(x)/ε = ρ/ε.

 

[Tosh5457]

Definition of divergence.

If you assume the charge has a continuous distribution over space, the divergence won't be 0. So if you put q = ρV in Coulomb's law, where ρ is the charge density and V is the volume of some arbitrary region, you'll be able to derive the equation. The volume V must be a function of x, y, and z of course.

 

[101111001101]

So, the problem lies in not the mathematics but my interpretation of it?

Why does a point charge yield a zero divergence but a charge enclosed in finite volume (say a sphere of radius R) yield finite divergence when the electric fields generated by them at point satisfying $$x^2 + y^2 + z^2 \geq R^2$$ is the same. An imaginary sphere around the point charge has field lines leaving the volume enclosed through the surface, and the field lines generated by those two configurations of charge are the same at a large distance.

 

[101111001101]

Also, is there no other way to deal with it (other than using dirac delta function)? Does this mean that there is a singularity there? How does classical electromagnetism deal with this singularity? Or are the equations inadequate (meaning that point charges are the wrong way to look at what is really going on)?

 

[vanhees71]

If you misinterpret the $\delta$ distribution as a function, there's a singularity, but it's not a function but a functional on the space of sufficiently smooth and sufficiently quickly falling test functions, and that's the adequate description of a point particle in a field theory.