# Poisson Equation for a Scalar Field

by matteo86bo
Tags: guass' law
 P: 57 We all know that for the gravitational field we can write the Poisson Equation: $\nabla^2\phi=-4\pi G\rho$ But I was wondering if, mathematically, we can write the same equation for a scalar field which scale as $r^{-2}$. Here is the thing. When you deal with gravity, the Poisson equation is derived from the Gauss's law for gravity: $\int_{\partial V}\dfrac{GM}{r^2}\cdot d\vec{S}=4\pi G M$ Then we apply the Gauss's law and we get the differential form of the Poisson equation: $\nabla\cdot\vec{f}=4\pi G\rho$ My question is: suppose that we have a scalar field $p=\dfrac{L}{4\pi r^2}$ Can we make an analogy between this field and the gravitational force and write a Poisson equation for this field in the following form? $\nabla\cdot \vec p=l$ where $L=\int l dV$ My question might also be interpreted as: can we apply the Gauss's theorem to a scalar field?

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