Poisson Equation for Dirac Delta and Point Charge Potential

Symbol Definitions

Symbol	Meaning
	Position vector
	Three-dimensional Dirac delta function
	Volume
	Scalar potential
	Scale factors in curvilinear coordinates
	Unit normal vector

Overview

디락 델타 함수와 포아송 방정식의 관계를 이해하고, 점전하의 정전기 퍼텐셜이 어떻게 유도되는지 다룬다.

Key Points

1. 라플라시안과 디락 델타 함수

\nabla^2 \left(\frac{1}{r}\right) = \nabla \cdot \left(\nabla \frac{1}{r}\right) \tag{2}

\nabla \frac{1}{r} = -\frac{1}{r^2}\hat{r} \tag{3}

\nabla \cdot \left(\frac{\hat{r}}{r^2}\right) = 4\pi \delta^3(x,y,z) \tag{4}

\int_V \nabla \cdot \left(\frac{\hat{r}}{r^2}\right) dV = \oint_{S=V\text{의 경계}} \frac{\hat{r}}{r^2} \cdot \hat{n} , da = \frac{1}{R^2} \oint |da| = \frac{4\pi R^2}{R^2} = 4\pi \tag{5}

\int_V \nabla \cdot \left(\frac{\hat{r}}{r^2}\right) dV = 4\pi \tag{6}

\nabla \cdot \vec{A} = \frac{1}{h_1 h_2 h_3} \left{ \frac{\partial}{\partial x_1}(A_1 h_2 h_3) + \frac{\partial}{\partial x_2}(A_2 h_1 h_3) + \frac{\partial}{\partial x_3}(A_3 h_1 h_2) \right} \tag{7}

\nabla \cdot \vec{A} = \frac{1}{r^2 \sin\theta} \left{ \frac{\partial}{\partial r}(r^2 \sin\theta A_r) + \frac{\partial}{\partial \theta}(r A_\theta) + \frac{\partial}{\partial \phi}(r\sin\theta A_\phi) \right} \tag{8}

\nabla \cdot \frac{\hat{r}}{r^2} = \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \cdot \frac{1}{r^2}\right) = 0 \quad (r \neq 0) \tag{9}

= \infty \quad (\text{at } r = 0) \tag{10}