Gauss' Differential law intuition

[THE HARLEQUIN]

Hi ,
while trying to prove gauss law for electric field from gauss theorem i came up with this problem .
as Gauss' differential law expresses
∇⋅E = ρ/εο

what i get from that is ... the divergence of electric field which is the flux density is related to the charge density ...but i can't get the epsilon part ( like the physical relation of its with divergence ) .
i tried to prove it without using solely the understanding of divergence but failed .. but i believe its possible to come up with a logical explanation .
Any mathematical proof with physical explanation will be appreciated ( i don't want to see Gauss differential law proved from Gauss integral law btw :3 )
thanks in advance .

 

[andrewkirk]

It sounds like what you are after is a derivation of Gauss's Law from Coulomb's Law, since Coulomb's Law gives the official definition of $\epsilon_0$. There's such a derivation here (you need to click on the 'show' link on that page).

 

[THE HARLEQUIN]

It sounds like what you are after is a derivation of Gauss's Law from Coulomb's Law, since Coulomb's Law gives the official definition of $\epsilon_0$. There's such a derivation here (you need to click on the 'show' link on that page).

thanks andrewkirk ...
i have seen this already .. but the problem is i am not familiar with the sifting property ... think you have any simpler process of deriving this ?

 

[jtbell]

i can't get the epsilon part

It's basically just a proportionality constant that comes in because of the units that you're using (SI a.k.a. MKS). In Gaussian units, the right-hand side is 4πρ.

i am not familiar with the sifting property

Sifting property?

 

[THE HARLEQUIN]

hmmm i have derived a proof.. but don't know if its ok :/

btw it would be great if u could explain this "sum of divergence of the electric field throughout the volume is sum of the flux through the surfaces " with some animation or something. i know they will be the same but having trouble visualizing this. @jtbell

 

[andrewkirk]

hmmm i have derived a proof.. but don't know if its ok :/

I can see problems with it. Firstly, the divergence operator can't pass through the $\frac{1}{r^3}$ factor without justification, as $r$ depends on $x, y, z$. Secondly, it's not clear what the role of $r$ is, as there are no words explaining the meaning of the various variables. $\vec{r}$ appears intended to be a location vector, but the location of what, relative to what?

It looks to me as though in the last step you may be essentially paralleling the Divergence Theorem, which is what's used to get from the integral form to the differential form.

The most natural way to prove Gauss's Law, it seems to me, is to use Coulomb's Law to prove the Integral Form of Gauss's Law and then use the Divergence theorem to prove the Differential Form from that.

If you don't want to go that way, I think you will have to come to terms with the Dirac Delta, because the charge is supposed to reside in a zero-dimensional point, which is exactly the sort of thing Dirac Delta is designed to cope with. The sifting property referred to in the proof is simply that, if $a\leq c\leq b$, then:
$$\int_a^b f(x)\delta(x-c)dx=f(c)$$