# 电磁学 Electromagnetism PHYS201

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The special case of a point charge at the origin, for which $\rho=q \delta(\mathbf{r})$ and $\mathbf{E}=q\left(\mathbf{r} / r^{3}\right)$, shows that $\nabla \cdot\left(\mathbf{r} / r^{3}\right)$ acts as if
$$\nabla \cdot \frac{\mathbf{r}}{r^{3}}=4 \pi \delta(\mathbf{r})$$
yields an equation for the electrostatic potential $\phi:$
$$\nabla \cdot \mathbf{E}=-\nabla \cdot \Gamma \phi=4 \pi \rho \quad \text { or } \quad \nabla^{2} \phi=-4 \pi \rho .$$
This is known as Poisson’s equation. In a portion of space where $\rho=0$, becomes
$$\nabla^{2} \phi=0$$

## PHYS201COURSE NOTES ：

$$\psi_{0}(\mathbf{x}, 0)=e^{i \mathbf{k}{0}-\left(\mathbf{x}-\mathbf{x}{0}\right)} h\left(\mathbf{x}-\mathbf{x}{0}\right)+\text { c.c. }$$ where $$h\left(\mathbf{x}-\mathbf{x}{0}\right)=\int d \mathbf{q} a(\mathbf{q}) e^{i \mathbf{q} \cdot\left(\mathbf{x}-\mathbf{x}_{0}\right)}$$