Physics 1409 – Electrostatics, Course Policies & Gauss’s Law

Chapter 16 — Electric Forces & Fields

• Everyday phenomena governed by electromagnetism

  • Friction, springs, fluid dynamics, thermodynamics, chemistry, light, TV/radio

Historical Origin of “Electron”

• Greek amber (ήλεκτρο) — rubbing flint produced charge ⇒ “electric”

• Word “electron” derives from amber

Charge Transfer Examples

• Rubbing two balloons ≈ $10^{12}$ electrons migrate to balloon

• Shuffling feet on carpet (low humidity) yields similar electron movement

Two Kinds of Charge

• Glass-like (vitreous) and amber-like (resinous) — identified by early experimenters

• Benjamin Franklin assigned conventional signs: + / –

• Rule: opposite charges attract, like charges repel

Conservation & Quantization of Charge

• Net charge produced in any process is zero (Franklin’s argument)

• Atomic picture

  • $q_e = -1.602\times10^{-19}\,\text{C}$ (Robert Millikan oil-drop, 1906)

  • Proton charge: $+1e$

  • Charge comes in integer multiples of $e$ (quantized)

Electrical Materials & Resistivity

• Conductors: electrons mobile — metals (Ag, Cu, Al, Au, …)

• Insulators: electrons bound — plastics, glass, rubber, Teflon

• Semiconductors: intermediate — Si, Ge, C (graphite)

• Example resistivities (Ω·m)

  • $\rho<em>{\text{Ag}} = 1.6\times10^{-8}$, $\rho</em>{\text{Teflon}} \approx 10^{14}$, etc.

Coulomb’s Law

• Formulated by Charles Coulomb (1780)

• Magnitude: F = k\dfrac{|Q1Q2|}{r^2},\; k = 8.988\times10^{9}\,\text{N·m}^2!/\text{C}^2

• Vector direction: attractive for unlike, repulsive for like

• In SI, 1 C is huge: two 1 C charges 1 m apart ⇒ $F \approx 9\times10^{9}\,\text{N}$ (≈ ${10}^5$ tons)

Worked Micro-Scale Example

• Electron in hydrogen (Bohr radius $r = 0.53\times10^{-10}\,\text{m}$)

  • $Q<em>p = +e$, $Q</em>e = -e$

  • $F = k \dfrac{e^2}{r^2} \approx 8.2\times10^{-8}\,\text{N}$ (binds electron!)

Vector Superposition Strategy

1. Compute individual forces via magnitudes (use $|Q|$)

2. Assign directions using charge signs

3. Add as vectors (component method, exploit symmetry)

4. Always report direction in words or diagram

3-Charge Problem Recap

• Given $Q<em>1=-8.0\,\mu\text{C},\; Q</em>2=+3.0\,\mu\text{C},\; Q_3=-4.0\,\mu\text{C}$ arranged linearly

  • Calculated pairwise forces $F<em>{12}=2.4\,\text{N},\; F</em>{13}=1.2\,\text{N},\; F_{23}=2.7\,\text{N}$

  • Net forces: $F<em>{Q1}=+1.2\,\text{N},\; F</em>{Q2}=+0.3\,\text{N},\; F_{Q3}=-1.5\,\text{N}$ (signs indicate direction)

Conceptual Checkpoints (ConcepTests)

• Force-zero placement between $+Q$ and $+4Q$ ⇒ point exists; must be closer to smaller charge

  • For equal charges $+Q,+Q$ the null point is midway outside segment, not between

• Mixed sign charges $+Q, -4Q$ ⇒ null point lies beyond the smaller-magnitude charge (on +Q side)

• Vector-direction questions: combining forces in 2-D, scaling of $E$ with $Q$ and $r$

• Field scaling: If $Q \to 2Q$ and $r \to 2r$, $E \to \tfrac12 E_0$

Polarization & Induction Phenomena

• Charged balloon sticks to hair/paper via induced polarization of neutral material

• Two mechanisms

  1. Insulator polarization — electron clouds in atoms shift slightly; surface charge appears

  2. Conductor induction — free electrons redistribute creating surface charge regions

• Polar molecules (e.g., $\text{H}_2\text{O}$) have permanent dipole ⇒ align in external field

Electric Field Concept

• Michael Faraday: replace “action at a distance” with local field vector $\vec{E}$

• Definition: $\vec{E} = \dfrac{\vec{F}}{q_{\text{test}}}$ (N/C) for infinitesimal positive test charge

• For point charge $Q$: $\vec{E} = k\dfrac{Q}{r^2}\hat{r}$

• Superposition: $\vec{E}<em>{\text{net}} = \sum</em>i \vec{E}_i$

• Direction conventions

  • Positive $q$ experiences force along $\vec{E}$

  • Negative $q$ experiences force opposite $\vec{E}$

Worked Field Example (Line Case)

• Charges $Q<em>A=-10\,\mu\text{C}$ at $x=0\,\text{cm}$, $Q</em>B=+40\,\mu\text{C}$ at $x=20\,\text{cm}$

• Field at point $P$ located $x=80\,\text{cm}$

  • $E = E<em>A + E</em>B = -k\dfrac{Q<em>A}{r</em>A^2} - k\dfrac{Q<em>B}{r</em>B^2}$ (both negative direction)

Field Line Visualization

• Lines originate on + charges, terminate on – charges

• Density of lines ∝ $|E|$ magnitude

• For uneven charges, draw proportionally more lines from larger |Q|

• Uniform field between parallel plates: lines parallel & equally spaced

Conductors in Electrostatics

• Inside a good conductor at equilibrium: $\vec{E}_{\text{inside}} = 0$

  • Otherwise electrons would move → non-static

• Excess charge resides on outer surface

• Surface field is perpendicular to surface (no tangential component)

• Demonstrations: Faraday cage (video link provided) prevents internal field penetration

• Induction example: external field polarizes sphere; internal field cancels

Gauss’s Law

• Electric flux: $\Phi_E = \oint \vec{E}\cdot d\vec{A} = EA\cos\phi$ for uniform field region

• Statement: \ointS \vec{E}\cdot d\vec{A} = \dfrac{Q{\text{enc}}}{\varepsilon0},\; \varepsilon0 = 8.85\times10^{-12}\,\text{C}^2!/\text{N·m}^2

• Preferred over Coulomb for high-symmetry charge distributions

Application 1: Charged Spherical Shell

• Outside (r>R): behaves as point charge $E = k\dfrac{Q}{r^2}$

• Inside (r<R): $Q_{\text{enc}}=0 \Rightarrow E=0$ (hollow conductor result)

Application 2: Infinite Line of Charge (linear density $\lambda$)

• Gaussian cylinder (pillbox) radius $r$, length $L$

  • Flux through curved surface: $\Phi=E(2\pi r L)$

  • $Q_{\text{enc}}=\lambda L$

  • $E = \dfrac{\lambda}{2\pi\varepsilon_0 r}$ (radial)

Application 3: Infinite Plane (surface density $\sigma$)

• Gaussian pillbox with faces parallel to plane

  • Flux: $2EA$ (two faces)

  • $Q_{\text{enc}}=\sigma A$

  • $E = \dfrac{\sigma}{2\varepsilon_0}$ (constant, independent of distance)

• Two parallel planes ±$\sigma$ ⇒ uniform field $E = \dfrac{\sigma}{\varepsilon_0}$ between them (capacitor model)

Field vs. Force Concept Questions

• Field at equal distance depends solely on source charge (independent of test charge)

• Force depends on product $qQ$ ⇒ doubling test charge doubles force while field unchanged

Recurring Themes & Practical Implications

• Unit diligence prevents disasters (fuel-mass story)

• Symmetry simplifies mathematics (Gauss, Coulomb problems)

• Sign conventions and vector thinking are crucial (errors propagate)

• Every electrostatic scenario can be tackled via either

  1. Direct Coulomb integration (preferred for few point charges)

  2. Gauss’s Law (preferred for high symmetry)

Ethical & Philosophical Notes

• Academic integrity emphasized: physics relies on honest data & calculation

• Safety tie-in: Faraday cage demonstrations show engineering applications protecting human life

Summary Checklist for Exam Prep

• Memorize key constants: $k,\; \varepsilon_0,\; e$

• Practice unit conversions (Coulombs ↔ electrons, meters ↔ cm, etc.)

• Work vector addition both graphically and component-wise

• Derive Gauss-Law results for sphere, line, plane from scratch

• Redo ConcepTest questions until intuitive

• Attempt homework before lecture discussion to reveal gaps early