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Chapter 1 Introduction to Electrostatics

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

F⃗=14πϵ0qq1r2er⃗ \begin{gathered} \vec{F}=\frac 1 {4\pi{\epsilon}_0} \frac {qq_1} {r^2} \vec{e_r} \end{gathered}
  • F⃗∝q,q1 \vec{F} \propto q,q_1
  • F⃗∝1r2 \vec{F} \propto \frac 1 {r^2}
  • F⃗∥er⃗ \vec{F} \parallel \vec{e_r} (central force)
  • qq1>0 qq_1 > 0 repulsive
  • qq1<0 qq_1 < 0 attractive

Also, principle of linear superposition

# 1.2 Electric Field

lim⁡x→0E⃗(x⃗)≡lim⁡q→0F⃗q \begin{gathered} \lim_{x \to 0} \vec{E}(\vec{x})\equiv\lim_{q \to 0}\frac {\vec{F}} q \end{gathered} ⇒E⃗(x⃗)=q14πϵ0er⃗r2=q14πϵ0x⃗−x1⃗∣x⃗−x1⃗∣3 \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

⇒E⃗(x⃗)=14πϵ0∑i=1nqix⃗−xi⃗∣x⃗−xi⃗∣3 \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

⇒E⃗(x⃗)=14πϵ0∫ρ(x′⃗)x⃗−x′⃗∣x⃗−x′⃗∣3d3x′ \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

ρ(x′⃗)=∑i=1nqiδ(x′⃗−xi⃗) \begin{gathered} \rho (\vec{x'})= \sum_{i=1}^{n}q_i\delta(\vec{x'}-\vec{x_i}) \end{gathered} E⃗(x⃗)=14πϵ0∫∑iqiδ(x′⃗−xi⃗)x⃗−x′⃗∣x⃗−x′⃗∣3d3x′ \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”

∗∣ψ>=∫f(x)∣x>dx \begin{gathered} *|\psi>=\int f(x)|x>dx \end{gathered} f(a)=<a∣ψ>=∫f(x)<a∣x>⏟δ(x−a)dx \begin{gathered} f(a)=<a|\psi>=\int f(x)\underbrace{<a|x>}_ {\delta (x-a)}dx \end{gathered}

Kronecker Delta

∑i=mnfiδia={faif m≤a≤n, a∈Z0otherwise \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. ↓continuous analogy  \begin{gathered} \textcolor{blue}{\downarrow \text{continuous analogy }} \end{gathered}

Dirac Delta Function δ(x−a) \delta (x-a)

∫a1a2f(x)δ(x−a)dx={f(a)if a1<a<a2∈R0otherwise \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.
  • δ(x−a)=0 , for x≠a \displaystyle\delta (x-a) = 0 \text{ , for }x \ne a
  • ∫a1a2δ(x−a)dx=1 , if a∈(a1,a2) \displaystyle\int_{a_1}^{a_2} \delta (x-a)dx = 1 \text{ , if } a \in (a_1,a_2)
  • ∫a1a2δ′(x−a)dx=f(x)δ(x−a)∣a1a2−∫a1a2f′(x)δ(x−a)dx=−f′(a) \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)
  • δ(f(x))=∑i1∣dfdx(xi)∣δ(x−xi) \displaystyle\delta (f(x))= \sum_{i} \frac{1}{|\frac{df}{dx}(x_i)|} \delta (x-x_i)

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

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

Electric Flux 電通量 ϕ \phi

dϕ=E⃗⋅n⃗ da=q4πϵ0cosθr2 da=q4πϵ0 dΩ 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 ϕ=∮SE⃗⋅n⃗ da={qϵ0q inside S0q outside S \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

→superpositon∮SE⃗⋅n⃗ da=∑iqiϵ0 \textcolor{red}{\xrightarrow{\text{superpositon}}} \oint_S \vec{E} \cdot \vec{n} \, da = \sum_i \frac{q_i}{\epsilon_0} →continuous limit∮SE⃗⋅n⃗ da=1ϵ0∫Vρ(x⃗) \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)

∮SE⃗⋅n⃗ da=1ϵ0∫Vρ(x⃗) d3x=∫V∇⃗⋅E⃗⏟for arbitrary volume d3x \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)

∮SA⃗⋅n⃗ da=∫V∇⃗⋅A⃗ d3x \oint_S \vec{A} \cdot \vec{n} \, da = \int_{\mathbb{V}} \vec{\nabla} \cdot \vec{A} \, d^3 x ⇔∇⃗⋅E⃗=ρϵ0Mathematically EquivalentDifferential Form of Gauss’s Law \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. 1r2→∇⃗⋅E⃗ \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. 1r2+ϵ \displaystyle\frac{1}{r^{2+\epsilon}}
  2. Φ(r)=14πϵ0qr e−μr \displaystyle \Phi(r) = \frac{1}{4\pi\epsilon_0} \frac{q}{r} \, e^{-\mu r} where μ=mr/cℏ \displaystyle \mu = \frac{m_r/c}{\hbar}

Concentric Shell Experiment
Inverse square law => Shell Theorems :

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

ref: 古典電力學一 王喬萱教授 上課講義