📢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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Poisson and Laplace Charge or Boundaries | Audiosviadean on Notion