We all know that for the gravitational field we can write the Poisson Equation:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\nabla^2\phi=-4\pi G\rho[/itex]

But I was wondering if, mathematically, we can write the same equation for a scalar field which scale as [itex]r^{-2}[/itex].

Here is the thing. When you deal with gravity, the Poisson equation is derived from the Gauss's law for gravity:

[itex]\int_{\partial V}\dfrac{GM}{r^2}\cdot d\vec{S}=4\pi G M[/itex]

Then we apply the Gauss's law and we get the differential form of the Poisson equation:

[itex]\nabla\cdot\vec{f}=4\pi G\rho[/itex]

My question is: suppose that we have a scalar field

[itex]p=\dfrac{L}{4\pi r^2}[/itex]

Can we make an analogy between this field and the gravitational force and write a Poisson equation for this field in the following form?

[itex]\nabla\cdot \vec p=l[/itex]

where [itex]L=\int l dV[/itex]

My question might also be interpreted as: can we apply the Gauss's theorem to a scalar field?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Poisson Equation for a Scalar Field

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**