Why Are There Two Forms of the Poisson Equation?

In summary, the conversation was about two forms of Gauss law in its differential form - the first being Poisson equation in MKS units and the second being the CGS version of the macroscopic form of Poisson equation for a medium with permittivity "e". The difference between the two equations was discussed, with one containing the factor of 4*pi. The conversation also mentioned a link that explains Poisson equation and the book by Holst that extensively uses the form with "e". The conversation concludes with the clarification that the second equation is indeed the CGS version and a sense of relief for understanding the connection between the two equations.
  • #1
mysubs
4
0
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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  • #2
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
 
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  • #3
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
 
  • #4
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
 
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  • #5
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!
 

Related to Why Are There Two Forms of the Poisson Equation?

1. What is the Poisson equation?

The Poisson equation is a mathematical equation that describes the distribution of electric potential in a given region, based on the distribution of electric charge within that region.

2. Why is the Poisson equation important?

The Poisson equation is important because it allows scientists and engineers to predict and understand the behavior of electric fields in various systems. It has many applications in fields such as electromagnetism, electrostatics, and fluid dynamics.

3. How is the Poisson equation derived?

The Poisson equation is derived from the more general Maxwell's equations, which describe the behavior of electric and magnetic fields. By taking the divergence of both sides of the Maxwell's equations, the Poisson equation can be obtained.

4. What are the boundary conditions for the Poisson equation?

The boundary conditions for the Poisson equation depend on the specific problem being solved. However, in general, the boundary conditions specify the values of the electric potential at the boundaries of the region in which the equation is being solved.

5. How is the Poisson equation solved?

The Poisson equation can be solved using various numerical and analytical methods, depending on the complexity of the problem. Some common methods include finite difference, finite element, and Green's function methods.

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