Why Are There Two Forms of the Poisson Equation?

[mysubs]

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?

 

[snapback]

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

 

[mysubs]

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

 

[snapback]

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

 

[mysubs]

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!