Classical Electrodynamics 1-1

# Chapter 1 Introduction to Electrostatics

# 1.1 Coulomb’s Law (Coulomb’s Inverse-Square Law)

$$\begin{gathered} \vec{F}=\frac 1 {4\pi{\epsilon}_0} \frac {qq_1} {r^2} \vec{e_r} \end{gathered}$$

• $\vec{F} \propto q,q_1$
• $\vec{F} \propto \frac 1 {r^2}$
• $\vec{F} \parallel \vec{e_r}$(central force)
• $qq_1 > 0$ repulsive
• $qq_1 < 0$ attractive

Also, principle of linear superposition

# 1.2 Electric Field

$$\begin{gathered} \lim_{x \to 0} \vec{E}(\vec{x})\equiv\lim_{q \to 0}\frac {\vec{F}} q \end{gathered}$$ $$\begin{gathered} \Rightarrow\vec{E}(\vec{x})=\frac {q_1} {4\pi\epsilon_0 }\frac {\vec{e_r}} {r^2}= \frac {q_1}{4\pi\epsilon_0} \frac {\vec{x}-\vec{x_1}}{|\vec{x}-\vec{x_1}|^3} \end{gathered}$$

principle of linear superposition

$$\begin{gathered} \Rightarrow\vec{E}(\vec{x})= \frac 1 {4\pi\epsilon_0} \sum_{i=1}^{n}q_i \frac {\vec{x}-\vec{x_i}}{|\vec{x}-\vec{x_i}|^3} \end{gathered}$$

continuous charge distribution

$$\begin{gathered} \Rightarrow\vec{E}(\vec{x})= \frac 1 {4\pi\epsilon_0} \int \rho (\vec{x'}) \frac {\vec{x}-\vec{x'}}{|\vec{x}-\vec{x'}|^3} d^3x' \end{gathered}$$

discrete set of point charges

$$\begin{gathered} \rho (\vec{x'})= \sum_{i=1}^{n}q_i\delta(\vec{x'}-\vec{x_i}) \end{gathered}$$ $$\begin{gathered} \vec{E}(\vec{x})= \frac 1 {4\pi\epsilon_0} \int \sum_i q_i \delta(\vec{x'}-\vec{x_i}) \frac {\vec{x}-\vec{x'}}{|\vec{x}-\vec{x'}|^3} d^3x' \end{gathered}$$

Dirac Delta Function
Paul Dirac 1927 “The Physical Interpretation of the Quantum Mechanics”

$$\begin{gathered} *|\psi>=\int f(x)|x>dx \end{gathered}$$ $$\begin{gathered} f(a)=<a|\psi>=\int f(x)\underbrace{<a|x>}_ {\delta (x-a)}dx \end{gathered}$$

Kronecker Delta

$$\sum_{i=m}^n f_i \delta_{ia} = \left\{ \begin{array}{ll} f_a & \text{if } m \le a \le n,\ a \in \mathbb{Z} \\ 0 & \text{otherwise} \end{array} \right.$$ $$\begin{gathered} \textcolor{blue}{\downarrow \text{continuous analogy }} \end{gathered}$$

Dirac Delta Function $\delta (x-a)$

$$\int_{a_1}^{a_2} f(x) \delta (x-a) dx= \left\{ \begin{array}{ll} f(a) & \text{if } a_1 < a < a_2 \in \mathbb{R} \\ 0 & \text{otherwise} \end{array} \right.$$

• $\displaystyle\delta (x-a) = 0 \text{ , for }x \ne a$
• $\displaystyle\int_{a_1}^{a_2} \delta (x-a)dx = 1 \text{ , if } a \in (a_1,a_2)$
• $\displaystyle\int_{a_1}^{a_2} \delta'(x-a) dx = f(x) \delta (x-a) |_{a_1}^{a_2}- \int_{a_1}^{a_2}f'(x) \delta (x-a)dx =-f'(a)$
• $\displaystyle\delta (f(x))= \sum_{i} \frac{1}{|\frac{df}{dx}(x_i)|} \delta (x-x_i)$

( $f(x)$ has simple zeros at $x=x_i$ )

# 1.3 Gauss’s Law (Gauss’s Flux Theorem)

Electric Flux 電通量 $\phi$

$$d \phi = \vec{E} \cdot \vec{n} \, da = \frac{q}{4 \pi \epsilon_0} \frac{cos \theta }{r^2}\,da = \frac{q}{4 \pi \epsilon_0}\,d\Omega$$ $$\phi = \oint_S \vec{E} \cdot \vec{n} \, da = \left\{ \begin{array}{ll} \frac{q}{\epsilon_0} & q \text{ inside } S \\ 0 & q \text{ outside } S \\ \end{array} \right.$$

Integral Form of Gauss’s Law

$$\textcolor{red}{\xrightarrow{\text{superpositon}}} \oint_S \vec{E} \cdot \vec{n} \, da = \sum_i \frac{q_i}{\epsilon_0}$$ $$\textcolor{blue}{\xrightarrow{\text{continuous limit}}} \oint_S \vec{E} \cdot \vec{n} \, da = \frac{1}{\epsilon_0} \int_{\mathbb{V}} \rho (\vec{x} )$$

# 1.4 Differential Form of Gauss’s Law

高斯散度定理 (Gauss’s theorem)

$$\oint_S \vec{E} \cdot \vec{n} \, da = \frac{1}{\epsilon_0} \int_{\mathbb{V}} \rho (\vec{x}) \, d^3 x = \int_{\mathbb{V}} \underbrace{\vec{\nabla} \cdot \vec{E}}_{\textcolor{blue}{\text{for arbitrary volume}}} \, d^3 x$$

(Using the divergence theorem)

$$\oint_S \vec{A} \cdot \vec{n} \, da = \int_{\mathbb{V}} \vec{\nabla} \cdot \vec{A} \, d^3 x$$ $$\Leftrightarrow \underset{\text{Mathematically Equivalent}}{ \boxed{ \vec{\nabla} \cdot \vec{E}= \frac{\rho}{\epsilon_0} }} \textcolor{red}{\text{Differential Form of Gauss's Law}}$$

Note

1. Gauss’s law also applies to moving charges(while Coulomb’s law doesn’t)
2. Newton’s gravitational force field : similar form(inverse square, central force, superposition)
3. $\displaystyle\frac{1}{r^2} \rightarrow \vec{\nabla} \cdot \vec{E}$ does not depend on the shape of the surface, only charge density

Accuracy of the Inverse Square Law

1. $\displaystyle\frac{1}{r^{2+\epsilon}}$
2. $\displaystyle \Phi(r) = \frac{1}{4\pi\epsilon_0} \frac{q}{r} \, e^{-\mu r}$ where $\displaystyle \mu = \frac{m_r/c}{\hbar}$

Concentric Shell Experiment
Inverse square law => Shell Theorems :

1. No $\vec{E}$ inside an uniform spherical shell of charges
2. Uniform spherical shell of charges exerts $\vec{E}$ as if its charges were concentrated at the center point

Q : Gauss’s Law $\Rightarrow$ Coulomb’s Law ?

No ! Need special assumptions : electrostatic -> E-field symmetric with point charge
Mathematically : Need $\vec{\nabla} \cdot \vec{E} \text{ , } \vec{\nabla} \times \vec{E}$ + Boundary condition to find $\vec{E}$

# Delta Function in 3D

1. Cartesian Coordinates

$$\vec{x}=(x_1,x_2,x_3)= (x,y,z)$$ $$\delta(\vec{x}-\vec{x_0})\equiv \delta(x-x_0)\delta(y-y_0)\delta(z-z_0)$$ $$d^3x=dxdydz$$ $$\int_{\text{free space}} \delta (\vec{x}-\vec{x_0})d^3x=\int_{-\infty}^{\infty} \delta(x-x_0)dx\int_{-\infty}^{\infty} \delta(y-y_0)dy\int_{-\infty}^{\infty} \delta(z-z_0)dz$$

2. Cylindrical Coordinates

$$\vec{x}= (\rho,\phi,z)$$ $$\delta(\vec{x}-\vec{x_0}) \equiv \frac{1}{\rho} \delta(\rho- \rho_0)\delta(\phi- \phi_0)\delta(z-z_0)$$ $$d^3x=\rho d\rho d\phi dz$$ $$\int_{\text{free space}} \delta (\vec{x}-\vec{x_0})d^3x=\int_{0}^{\infty} \delta(\rho- \rho_0)d \rho \int_{0}^{2\pi} \delta(\phi-\phi_0)d\phi\int_{-\infty}^{\infty} \delta(z-z_0)dz$$

3. Spherical Coordinates

$$\vec{x}= (r,\theta,\phi)$$ $$\delta(\vec{x}-\vec{x_0}) \equiv \left\{ \begin{array}{ll} \displaystyle\frac{1}{r^2\sin{\theta}}\delta (r-r_0) \delta(\theta-\theta_0) \delta(\phi-\phi_0) \\ \displaystyle\frac{1}{r^2}\delta (r-r_0) \delta(\cos{\theta}-cos{\theta_0}) \delta(\phi-\phi_0) \end{array} \right.$$ $$d^3x= \left\{ \begin{array}{ll} \displaystyle r^2\sin{\theta}drd\theta d\phi \\ \displaystyle r^2drd(\cos{\theta})d\phi \end{array} \right.$$ $$\int_{\text{free space}} \delta (\vec{x}-\vec{x_0})d^3x= \left\{ \begin{array}{ll} \displaystyle \int_{0}^{\infty} \delta(r-r_0)dr \int_{0}^{\pi} \delta(\theta- \theta_0)d \theta \int_{0}^{2\pi} \delta(\phi-\phi_0)d \phi \\ \displaystyle \int_{0}^{\infty} \delta(r-r_0)dr \int_{-1}^{1} \delta(\cos{\theta}- cos{\theta_0})d cos{\theta} \int_{0}^{2\pi} \delta(\phi- \phi_0)d \phi \end{array} \right.$$

# Volume Charge Density with Delta Functions

1. point charge at $(x_0,y_0,z_0)$

$$\rho (\vec{x'}) = q \delta (x'-x_0) \delta (y'-y_0) \delta (z'-z_0)$$ $$\vec{E}(\vec{x}) = \frac{1}{4\pi \epsilon_0} \int \frac{\rho (\vec{x'})(\vec{x}-\vec{x'})}{|\vec{x}-\vec{x'}|^3 }d^3x' = \frac{q}{4\pi \epsilon_{0}}\frac{(\vec{x}-\vec{x_0})}{|\vec{x}-\vec{x_0}|^3} \textcolor{red}{\propto \frac{1}{r^2} }$$

2. line charge (infinite wire along the z-axis)

$$\rho (\vec{x'}) = \lambda \delta (x') \delta (y')$$ $$\begin{align*} \displaystyle E(\vec{x}) &=\int_{-\infty}^{\infty} \frac{\lambda}{4\pi \epsilon_0} \frac{c \vec{e_x} - z' \vec{e_z}}{|c \vec{e_x} - z' \vec{e_z}|^3}dz' \\ &=\frac{\lambda}{4\pi \epsilon_0} \int_{-\infty}^{\infty} \frac{\vec{e_x}c \, dz'}{(c^2+z'^2)^{\frac{3}{2}}} \\ &=\frac{\lambda}{4\pi \epsilon_0}\vec{e_x}\left[\frac{z'/c}{(c^2+z'^2)^{\frac{1}{2}}}\right]_{z'=-\infty}^{\infty} \\ &=\frac{1}{4\pi \epsilon_0} \frac{2\lambda}{c}\vec{e_x}\quad \textcolor{red}{\propto \frac{1}{r}} \end{align*}$$

3. surface charge (infinite plate in the x-y plane)

$$\rho (\vec{x'}) = \sigma \delta (z)$$ $$\begin{align*} \displaystyle \vec{E}(\vec{x})&=\frac{\sigma}{4\pi \epsilon_0} \int \frac{(-\vec{x'})\vec{e_x}+(-\vec{y'})\vec{e_y}+c\vec{e_z}}{|(-\vec{x'})\vec{e_x}+(-\vec{y'})\vec{e_y}+c\vec{e_z}|^3} \, dx' \, dy'\\ &= \frac{\sigma c \vec{e_z}}{4\pi \epsilon_0} \int \frac{r' \, dr \, d\theta}{(r'^2+c^2)^{\frac{3}{2}}} \\ &= \frac{\sigma}{4 \pi \epsilon_0} \int_{0}^{2\pi}d\theta ' \left[ \frac{-1}{\sqrt{r'^2+c^2}} \right]_{r'=0}^{\infty} \\ &= \left\{ \begin{align*} \displaystyle &\frac{\sigma}{2 \epsilon_0} \vec{e_z} \quad c>0 \\ \displaystyle -&\frac{\sigma}{2 \epsilon_0} \vec{e_z} \quad c<0 \end{align*} \right. \quad \textcolor{red}{\propto \frac{1}{r_0} \quad \text{non-divergent}} \end{align*}$$

ref: 古典電力學一 王喬萱教授 上課講義