Proof of Gauss's Law

Swapnil

I was wondering, how would you prove Gauss's law (either mathematically or intuitively)? I mean, I know that sometimes people take it as the fundamental law (one without proof) but how would you derive such a law from Coulomb's law?

Any help would be extremely appreciated. I have been fussing over this topic for a month now.

leright

http://www.rpi.edu/dept/phys/Dept2/p...uss/proof.html

Swapnil

Thank you very much. That was a good website that gave an intuitive feel for the Gauss's law. Is it possible for you (or someone else) to give me a link that explains the mathematical derivation of Gauss's law (possible using Stoke's theorem or something)?

leright

That IS the mathematical derivation of Gauss's law. Stoke's theorem has nothing to do with gauss's law. However, we can go from the integral form of gauss's law to the differential form of gauss's law using the DIVERGENCE theorem, but the application of the divergence theorem in order to do this is quite trivial.

Stokes theorem is used to transform the integral forms of maxwell's third equation (Faraday's law) and fourth equation (ampere's law) into their differential forms.

Swapnil

Oh yeah. Sorry about that. I was thinking Divergence theorem but I wrote down Stokes' theorem.

Wait. Is that what Divergence theorem is used for, just to tranform Gauss' law from integral form to differential form and vice-versa? This whole time I was thinking that Divegence theorem was used to derive Gauss' law from Coulomb's law.

So in the end there is no rigorous proof of Gauss' law (the fact that it works for ANY closed surface)?

Meir Achuz

A simple derivation uses the definition of differential solid angle:
d\Omega=r.dA/r^3. Then \int qdA.r/r^3=4\pi q.
For q outside the closed surface, the sign of the solid angle has to be used.

rbj

the divergence

[tex] \nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} [/tex]

is saying the same thing as

[tex] \oint_S \mathbf{E} \cdot d\mathbf{A} = \int_V \frac{\rho}{\epsilon_0} dV = \frac{Q}{\epsilon_0} [/tex]

one is the differential form, the other is the integral form.

there are plenty of decent E&M fields books that prove it rigorously. can you check one out and look it up?

Swapnil

I have read/looked-over a lot of EM books, but none of 'em offer a satisfying proof of Gauss' law. The most common response is in terms of field lines, which I think is just a clever trick to avoid answering the question.

Galileo

Here's the idea of the proof in short, you can work out the details yourself:

(I): Show that [itex]\nabla \cdot \vec F =0[/itex] if F has the form: [itex]\hat F=K\hat r/r^2[/itex] ([itex]r\neq 0[/itex]).

(II)The divergence theorem holds for simple solid regions, but can be extended to hold for regions that are finite unions of simple solid regions. Any volume region with a 'hole' in it can be viewed as a union of two simple solid regions. (For example: A solid sphere with a cavity in it, the outer surface is the surface of the sphere and the inner one is the surace of the cavity.)
(III) Show that the flux through such a surface is zero (if the inner surface surrounds the origin). This means the flux through both surfaces is equal. Since the surfaces were arbitrary, it means the flux through any connected closed surface that surrounds the origin has the same value. It thus suffices to calculate the flux through a sphere.

quasar987

Do you call a proof that uses

[tex] \nabla \cdot \frac{\vec{r}-\vec{r}'}{|\vec{r}-\vec{r}'|^3} = 4\pi \delta^3(\vec{r}-\vec{r}') [/tex]

rigorous? Just curious.

rbj

Okay, let's begin with the concept of flux and the natural justification of the inverse-square law in our existance in 3 spatial dimensions.

As an illustrative example, let's think about the concept of power and intensity. At the beach, when you feel the sun's radiation on your skin, what you are measuring is how much energy (in a particular band of frequencies) is impinging upon your skin in a particular amount of time. If you're out in the sun longer, more energy falls on your skin (this is the concept of power: how much energy per unit time) and, in the same intensity of sunlight, the bigger you are, the more of that sunlight you will scoop up: more area exposed to the sun, more power that falls on that area. So, whether it is light or sound waves (or something else), intensity is amount of power that falls on a unit area that is positioned perpendicular to the oncoming wave. it's watts/m². double the area, you get twice as much power. Double the intensity of radiation, you also get twice as much power.

Okay, consider a [itex]P_0[/itex] watt light bulb and assume we're measuring all of the radiation (infared, too) at a distance of [itex]r[/itex] meters. Imagine a sphere of radius of [itex]r[/itex] surrounding and centered at the light bulb. Conservation of energy says that the entire [itex]P_0[/itex] watts is distributed over the entire surface area of that sphere , which is [itex]4 \pi r^2[/itex] and the argument of symmetry says that every one of those m² gets the same amount of power. And every little snippet of area is directly facing the light bulb and is perpendicular to the wave front. So, at that distance the power, [itex]P_A[/itex], falling on an area of [itex]A[/itex] is

[tex] P_A = \frac{P_0}{4 \pi r^2} A [/tex]

the factor

[tex] I = \frac{P_0}{4 \pi r^2} [/tex]

is the intensity of radiation and is proportional to the power of the point source, as one would expect. Note also, that with a constant source power, the intensity is also proportional to the reciprocal of the square of the distance from the source. A natural inverse-square behavior. Note also that if one were to integrate the intensity (which is [itex] \frac{P_0}{4 \pi r^2} [/itex] over the entire spherical surface area (which is [itex] 4 \pi r^2 [/itex]) i would be adding up all the little snippets of power falling on the little snippets of area and that would add to the total power [itex] P_0 [/itex] emitted by the point source. Doesn't matter how big the sphere is. If the sphere is twice as big, it has 4 times the surface area, but the intensity is ¹/₄ as large and it comes out to be the same total power. Adding up all of the little snippets of power impinging upon the surface of the sphere equals the power emitted by the source inside and we can thank the Conservation of Energy law for that simple fact.

Now here's a little example:

Someone, somewhere measured the intensity of radiation from the sun (the power falling on a square meter facing the sun out in a vacuum in Earth orbit) to be: 1360.77 W/m². We also know that we are 149.6 x 10⁹ meters (about 93 million miles) from the sun. From that we can calculate how big of a light bulb that the sun is:

[tex] I = \frac{P_0}{4 \pi r^2} [/tex]

[tex] 1360.77 \ \mbox{W/m}^2 = \frac{P_0}{4 \pi (149.6 \times 10^9 \ \mbox{m})^2} [/tex]

results in:

[tex] P_0 = 1360.77 \ (\mbox{W/m}^2) 4 \pi (149.6 \times 10^9 \ \mbox{m})^2 = 3.827 \times 10^{26} \ \mbox{W} [/tex]


Now this is intensity, not electric field, but both share the same mathematics. Both are proportional to the source agent (power of point source vs. electric charge of point source) and both are a inverse-square relationship with distance. Now, instead of calling this quantity "intensity" which is power per unit area, for electric charges this natural inverse-square quantity is called "flux density" or "electrostatic flux density", and if we were talking about this other inverse-square phenomenon called gravity, it would be "gravitational flux density". These are those field lines that they like to draw and you don't like. But it doesn't matter, if experimental evidence shows that the electrostatic force follows the inverse-square law precisely (and it does, experiment confirms the exponent of [itex]r^{-n}[/itex] to be be [itex]1.99999999 < n < 2.00000001[/itex], then whether or not you like it, that density of lines of field model is valid, even if it comes from our imagination. If the E field is inverse-square, we can imagine a point source radially emitting a bunch of field lines (a.k.a. "total flux"), the quantity of which is proportional to the amount of charge, and the density of those field lines crossing a perpendicular surfaces (a.k.a. "flux density") is proportional to the number emitted and inversely proportional to the square of the distance.

Now we're moving toward proving:

[tex] \oint_S \mathbf{E} \cdot d\mathbf{A} = \frac{Q}{\epsilon_0} [/tex]

but I want you to get used to these first concepts before moving on. If you don't like any of the above, I ain't gonna waste time continuing.

Another installment to follow.

Meir Achuz

Is anything misssing?

Swapnil

You mean [itex]\nabla \cdot \vec F \neq 0[/itex], right?

quasar987

(No, he meant =0)

leright

No, he meant = 0. The only point where there is a non-zero divergence in that particular vector field is where r = 0, but he excluded that point. Everywhere else the divergence is zero.

Swapnil

I did the calculation for [tex] \vec{r} = x\hat{i} + y\hat{j} + z\hat{k}[/tex] and I get [tex]\frac{1}{x^2+y^2+z^2}[/tex].

Also, why should the divergence be zero, don't we have a radial field?

Galileo

Well, that's one of the special properties of inverse square field. It may look like it has a divergence, but it doesn't and it's the underlying reason for Gauss' law.

You could do it in cartesian coordinates:
[tex]\vec F(x,y,z)=K \frac{\hat r}{r^2}=K \frac{\vec r}{r^3}=K \frac{x\hat{i} + y\hat{j} + z\hat{k}}{(x^2+y^2+z^2)^{3/2}}[/tex]
but that's really a bit of a hassle.
Try it in spherical coordinates, whhich makes it trivial.