Electric Field: Q, Epsilon 0, and the Inverse Square Law

[fdarkangel]

electric field is written as q/4*pi*r²*e0.
i guess that 1/4*pi*r² stands for the decrement of flux density, with area of sphere (then it looks like force = "flux per unit area").

i need clarification about the constant epsilon 0. I'm not asking definitions such as "it's electrical permitivity of space". what's the physical meaning of it?

also, if the inverse square law can be generalized for all forces in universe using the description based on sphere, why are we writing GmM/r² and not mM/4*pi*r².g0, where g0 stands for a constant such as "gravitical permitivity of space"?

 

[Crosson]

Coloumbs law is, in theory, sufficient to determine the static electric field in any situation. This is despite that fact that it only applies to point charges (you are probably learning to "add up" coulomb charges to form any shape, ala vector integration).

As it turns out, coloumbs law is not a very convenient way to look at electric fields in general (other than point charges). Perhaps you have already been exposed to the integral form of Gauss's law? If so, then you know it is an extremely nice way to find electric fields in situations of high symmetry. Gauss's law is one of what are called "Maxwell's Equations", which are considered to be a complete, concise, practical, and elegant formulation of electricity and magnetism.

The bottom line is that the mks system prefers to have the 4 pi appear in coloumbs law then in Gauss's law. Gauss's law is certainly more fundamental.

The permitivity of free space is the ratio between the charge inside of a closed surface (a sphere for example) and the flux of electric field through the surface.

Part of an interesting analogy in mechanics, epsilon zero is the term that corresponds to the spring constant, or the bulk modulus (something having to do with "tension", as opposed to mu zero having to do with "inertia").

 

[kevinalm]

It's really just a scaling factor, needed because of the arbitary nature of systems of units. It relates the E field intensity units to the unit of force and the unit of charge. The numeric value depends on the system of units chosen. (MKS, ft-lb-slug, etc.)

>>edit I should clarify what I said. I should more properly "needed in part because" I intended my post to elaborate on what was already said by Crosson, but may have given the impression that I was contradicting. The analogy to a "spring constant" is very good.

 

[fdarkangel]

The permitivity of free space is the ratio between the charge inside of a closed surface (a sphere for example) and the flux of electric field through the surface.

thanks, that clarified things a bit.
what about gravitational force? similar Gaussian surfaces can be chosen for gravitation field as in electric field. having known the relation between Coloumb & Gauss law, I've started to think that we can choose a more fundamental law for gravity. in order to make such step, we first need to define a term such as "gravity flux" and choose such Gaussian surface that gravity field is constant (preferably, surface vector is perpendicular to g vector): a shpere.

so, where's 4 pi of gravity?

 

[kevinalm]

Gravity's 1/(4*pi) *g0 =G , it's embedded in the gravitational constant. Gravitation flux doesn't have anything similar to em induction (relating a changing E field to an B field and vice versa as in Maxwells equations) so it isn't particularly useful to write it out that way.

 

[fdarkangel]

thanks, i just had some doubts about gravity law, about having a spherical flux density decrement. i was assuming that $$G = \frac{1}{4\pi g_0}$$ just like $$k = \frac{1}{4\pi \varepsilon_0}$$.

however, i'd just like to remind how moving masses affect the time-space grid, albeit it's not very similar how magnetic field is formed from the moving electiric-charged particles. but i guess having a term "gravity-flux" won't be convenient at this point.

again, thanks for clarifying.

 

[kevinalm]

I almost mentioned general relativity, but I didn't want to confuse the issue. In em the form of Maxwell's equations makes e0 useful. There is no incentive for g0, in fact I've never seen it outside of some tinkering I did when I first asked the same question you did.

 

[rbj]

I almost mentioned general relativity, but I didn't want to confuse the issue. In em the form of Maxwell's equations makes e0 useful. There is no incentive for g0, in fact I've never seen it outside of some tinkering I did when I first asked the same question you did.

i think you need to Google the term "GravitoElectroMagnetism" or "GravitoElectroMagnetic" or "GEM". they are derived from G.R. but they look almost identical to Maxwell's Eqs except that

$$\epsilon_0 = - \frac{1}{4 \pi G}$$

and charge density is replaced by mass density. this should be satisfying since both E&M and gravitation are inverse-square forces that has effect traveling at the same speed of c.

certainly the concept of flux can be applied to any inverse square law and, when you do that, the $\frac {1}{4 \pi}$ factor ends up in the inverse square law to get it out of the more fundamental field equations.

r b-j

 

[kevinalm]

Sorry, I should have been more clear. The point I was trying to get across was in Newtonian physics there is no particular incentive to write it in the form 1/(4*pi*g0), so the original poster won't find a g0 in any textbook he is likely to be studying. Gauss's law is inherent in the form of the equation F=G * m1*m2 /r^2.

 

[fdarkangel]

The point I was trying to get across was in Newtonian physics there is no particular incentive to write it in the form 1/(4*pi*g0), so the original poster won't find a g0 in any textbook he is likely to be studying.

i'm studying physics 2 & modern physics this term, and indivudually studying general relativity, and i have never came across $$G = \frac{1}{4\pi g_0}$$ or similar thing. this is why i was in doubt.

 

[rbj]

i'm studying physics 2 & modern physics this term, and indivudually studying general relativity, and i have never came across $$G = \frac{1}{4\pi g_0}$$ or similar thing. this is why i was in doubt.

google the terms. not implying that the textbooks at that level do anything about it, but Gauss's Law is just as applicable to gravitation as it is to E&M. you will find that if you apply it to the gravitational fields surround some body or collection of bodies, that the result will be [itex] 4 \pi G M [/tex] where [itex] M [/tex] is the total mass contained in the surface of integration.

if you defined graviational flux to be the G field divided by [itex] 4 \pi G [/tex], then, what you find after the surface intral is just [itex] M [/tex]. so to get field from flux, you multiply by [itex] 4 \pi G [/tex] which is the same as [itex] \frac {1}{\epsilon_0} [/tex] in E&M.

r b-j

 

[Crosson]

If you study field theory and green functions, you know Newton's theory is equivalent to the following partial differential equations:

$$\nabla^2\phi = 4\pi G \rho$$

$$a = \nabla\phi$$

a is acceleration, rho is mass density, phi is gravitational potential. These were known in the 18th century.

This is identical with Gauss' law for voltage (poisson's equation):

$$\nabla^2 V = \Frac{\rho}/{\epsilon_0}$$


So there was a compelling reason to change G to 1\4piG or something

 

[fdarkangel]

no, I'm not studying field theory and green functions (yet), as far i know, field theory is covered in quantum physics. i'll have wait next year for them. however, being familiar with electromagnetic theory , this last post clarified it all.
huge thanks!

 

[kevinalm]

I think you misundersood me. The question was asked why gravitation uses a composite G variable while em uses 1/(4*pi*e0). I essentially answered that it was convenient for em to do so while in gravitation it isn't particularly helpfull as you just demonstrated. In em e0 is used for relating a time variant E field to an equivalent displacement current density, and hence to a del X B . Newtonian gravity doesn't have anything like that.

 

[rbj]

I think you misundersood me. The question was asked why gravitation uses a composite G variable while em uses 1/(4*pi*e0). I essentially answered that it was convenient for em to do so while in gravitation it isn't particularly helpfull as you just demonstrated. In em e0 is used for relating a time variant E field to an equivalent displacement current density, and hence to a del X B . Newtonian gravity doesn't have anything like that.

why do you keep repeating the same fallacy, even after it's been pointed out that it's fallacious? did you try googling "GEM" or "gravitoelectromagnetism"?

try these for starters.

http://arxiv.org/PS_cache/gr-qc/pdf/9912/9912027.pdf

http://www.iop.org/EJ3-Links/26/B2PcnrMQ9Qr,dG8lppV,HA/q01911.pdf

http://arXiv.org/PS_cache/gr-qc/pdf/0207/0207065.pdf



in each of these papers, you will see a set of equations that look exactly like Maxwell's Eqs. (with a minus sign, because like masses attract, not repel) but they are about gravity, not E&M. it is only a matter of common practice that "they" use the composite $G$ rather than some $1/(4 \pi g_0)$ or similar. gravitation and E&M both propagate at the speed of $c$. gravitation and E&M both are, in a static situation, inverse-square laws. gravitation is different from E&M, but, for reasonably flat space-time, gravitational radiation works like E&M, except you replace $\epsilon_0$ with $-1/(4 \pi G)$. if you want to get rid of some $4 \pi$ factors, you might want to define a $g0 = 1/(4 \pi G)$ and your equations get simpler.

 

[SpaceTiger]

Why is so much thought being wasted on this trivial point? One can often look more silly defending one's honor (or attacking another's) than just letting it go.

 

[kevinalm]

I don't know. I was explaining a minor point of _Newtonian_gravitation_ and found myself dodging black holes, parahelion precession and gravitational radiation. Geez. Sorry if I offended anyone. I hope in all this mess the op found a satisfactory answer to his original question(s).