Personally, I like Gauss' law better if the ε0 is moved over to the other side, under the integral. That way, it still works if the permitivity isn't constant; you just replace ε0 with a function ε --AxelBoldt
What do you mean if εo isn't constant? Do you mean for cases where there is matter between the surface and the charge, and thus you need to account for a drop in the E field due to the permittivity of that matter? --BlackGriffen
- Yes, that's what I meant. If there's matter with varying permittivity ε around, you need to integrate ε multiplied with E in Gauss's law. --AxelBoldt
- Ok, I'm adding a note about it now.
- Upon further reflection, that is wrong. The dielectric constant of matter doesn't just magically reduce the electric field. The dielectric constant (I had the wrong name previously) is a measure of how easy it is to separate the molecules of matter in to a dipole. To show why is relatively easy (now that I think about it properly). Consider a small positively charged sphere. The electric field outside this sphere is is the same is if it was a point charge: kq/r2. Stick a neutrally charged spherical shell around it. The electric field of the sphere creates dipoles within the shell that surrounds it. The net effect is like two thin shells of charge have formed; a negative one on the inner surface of the shell and a positive one on the outer surface. The charges of these shells have to be precisely equal due to conservation of charge. The net effect? Everywhere but on the inside of the walls of the neutral shell, the electric field still looks like kq/r2. Within the walls of the shell the electric field is weaker, but as long as the surface entering that region removes no net charge, the decrease in the electric field is compensated for by two factors: a change in the area of integration, the fact that the charge shells are approximations of microscopic dipoles means that there is still a net surface charge that compensates for the inner charge. Even in the limiting case, metals, where the surface charges are great enough to reduce the electric field in the body of the metal to zero, Gauss's law holds. --BlackGriffen
Is it worth mentioning that the elegant formulations of Maxwell's equations were not developed by Maxwell, but by another man ? Maxwell had the right idea, but he was definitely not elegant in his math.... I've done a bit of web-searching to validate this idea and currently cannot vouchsafe it. I recall a history of science teacher describing it in great detail when I was younger, but have no way to verify/validate it.
Okay, I found a page that claims: 1884: Oliver Heaviside expresses Maxwell's Equations as we know them today ie: http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Heaviside.html therefore this validates my recollection. (it said Maxwell's equations were originally 20 equations in 20 variables instead of two equations in two variables) Now I can go to sleep...
I deliberately left the history empty because I did not know it. By all means, add a history section. I do know that the wave equations for light that can be derived from them led to relativity. (the term describing the velocity of the wave was 1/(μoεo).5 which is equal to c, and the fact that it didn't contain a term for the velocity of the observer is what sparked Einstein's imagination/lead to his postulate that c is a constant to any observer.
Also, 4 Maxwell's equations with 4 variables (time, charge density, the electric field, and the magnetic field). Where do you get two? --BlackGriffen
I think it would be nice to mention in the main article how εo, μo and c are related, so that people realize that the speed of light occurs in Maxwell's equations and that therefore the conjecture that electromagnetic radiation is light is not too abstruse. --unknown
Nice idea, but this article needs to remain focused on these equations because that is all it is for. A better place for that connection would be in an electromagnetic radiation/waves/light article. --BlackGriffen
Oh, there's also a minor oversight: ε is used as the permittivity and also as the electromotive force around a loop. --unknown
EMF is supposed to be a scripty E. Anyone know how to do one of those? --unknown
I understand that, but there are only so many symbols in the english language. I used ε instead of ΔV or ΔφE for three reasons: first, φ and/or V are used in electrostatics to represent the electric potential as a scalar function in space, and any closed loop integral over a continuous scalar function in space has to be zero; second, ε is the closest thing (almost exactly the same, in fact, to the scripty thing described above); and third, the limited number of symbols means that what the symbol represents has to be labeled each time anyway. To give you another couple of overloaded characters in physics: p represents both momentum and pressure (in mathematics p also represents the period of the wave); v is used for velocity, volume, and voltage, velocity is generally lower case, volume is upper, and voltage is usually upper if it's constant and lower if it's time varying. I've really beaten that horse to death, but I wanted to make it crystal clear that I had considered the conflict when I wrote the article. --BlackGriffen
And one last thing: I don't quite understand why the last paragraph mentions cgs versus mks units? How could the units possible change the equations? --AxelBoldt
If you use kg for mass, m/s2 for acceleration, and lbs for force, Newton's second law takes on the form F=kma, k a constant. Choosing a better system makes k go away, simplifying the equation. It's the same deal with CGS and MKS, a lot of the constants go away in the former system. --Unknown
Precisely, I'll add more to the main page presently, but it's all about clairity. --BlackGriffen