Why Are There Two Forms of the Poisson Equation?

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Discussion Overview

The discussion revolves around the two forms of the Poisson equation as they relate to Gauss's law in differential form. Participants explore the differences between the equations, particularly focusing on the presence of the factor of 4π in one version and the implications of using different unit systems (MKS vs. CGS).

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant notes the two forms of the Poisson equation: del2V(r) = -p(r)/e and del2V(r) = -4*pi*p(r)/e, questioning the difference and the origin of the 4π factor.
  • Another participant suggests that the first equation is in MKS units, while the second may correspond to CGS units without the permittivity term.
  • A participant mentions encountering the equation with "e" in a specific book and does not view it as a mistake, indicating it is built upon in the text.
  • Further clarification is provided that the form with "e" is the CGS version of the macroscopic Poisson equation for a medium with permittivity, assuming a linear medium.
  • One participant expresses relief at understanding the connection between the equations after the discussion.

Areas of Agreement / Disagreement

Participants express differing views on the interpretation of the equations and their unit systems, indicating that multiple competing perspectives remain without a clear consensus.

Contextual Notes

There are unresolved questions regarding the mathematical equivalence of the two forms and the implications of using different unit systems, which may depend on specific assumptions about the medium and context.

mysubs
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Hello everybody

I've been searching this today but I am a bit lost now. I've encountered two forms of Gauss law in its differential form, Poisson equation :

del2V(r) = -p(r)/e

del2V(r) = -4*pi*p(r)/e

where V:e.potential, p:charge density, e:permivity

Now, what's the difference between these two /or/ where does the (4pi) in the second one comes from?

Mathematically they are not equivalent, but they are presented as such. Any opinions?
 
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hello,

your first equation looks like Poisson eq. in MKS units. The second eq. confuses me too. Without the "e" in denominator, this equation would become Poisson eq. in cgs units.

Check: http://scienceworld.wolfram.com/physics/PoissonsEquation.html"

cheers
 
Last edited by a moderator:
Hi snapback,

I've already came across this link during my search, but the equation in mind was with "e".

I encountered the equation in Holst book of poissn-bltzman eq., I can't see it as a mistake as he built upon it later.

Thanks for the reply
 
Hi mysubs,

do you mean http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.9.9570" by "Holst" ?

Yes, he is indeed using the form with "e" rather extensively. Now, I think this is the CGS version of the the macroscopic form of Poisson equation for a medium with permittivity "e". "Macroscopic" means the version of Poisson eq. where you have actually grad(e grad V(r))=... and it is assumed that it is a linear medium, so that "e" is a scalar.

I haven't seen this version before, and will add it as another chapter in "my personal book of annoyance with different unit systems in electrodynamics" ;-)

cheers
 
Last edited by a moderator:
Yes, that one. I thought it might have something to do with this but couldn't think of any connection mathematically.

Now it all works out, and I feel like a complete human being again. Thanks for the help!
 

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