# Poisson and Laplace Charge or Boundaries

The electrostatic scalar potential ($$V$$) arises in a static situation because the electric field ($$\vec{E}$$) is curl-free, allowing $$\vec{E}$$ to be expressed as the negative gradient of the potential ($$\vec{E}=-\nabla V$$). When this relationship is incorporated into Gauss's law, it reveals that $$V$$ must satisfy Poisson's equation ($$\nabla^2 V=-\frac{\rho}{\varepsilon\_0}$$), where $$\rho$$ is the charge density. The governing equation fundamentally determines the potential's nature: when the field is sourced by internal charge density ($$\rho$$), Poisson's equation applies, yielding a radial potential that decays outward from the source. If the region is charge-free ($$\rho=0$$), the potential is governed by Laplace's equation, forcing the solution to be entirely constrained by external boundary conditions, acting as a smooth, time-independent interpolation that averages the fixed potential values on the edges.

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