# Visualize solutions to Poisson's Equation and Laplace's Equation

The two electrostatics visualizations reveal that the nature of the scalar potential field $$V$$ is fundamentally determined by the governing equation: Poisson's Equation is used when the field is sourced by internal charge density ( $$\rho$$ ), resulting in a radial potential that decays outward from the source, as seen in the point charge demo. In contrast, Laplace's Equation governs the potential in a charge-free region ( $$\rho=0$$ ) and is entirely constrained by external boundary conditions, forcing the potential to be a smooth, time-independent interpolation that averages the fixed potential values on the edges, as demonstrated by the square plate visualization.

### Brief audio

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### Condensed Notes

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