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{\text{Ag}} = 1.6\times10^{-8} , \rho{\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} )

    • Qp = +e , Qe = -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 Q1=-8.0\,\mu\text{C},\; Q2=+3.0\,\mu\text{C},\; Q_3=-4.0\,\mu\text{C} arranged linearly

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

    • Net forces: F{Q1}=+1.2\,\text{N},\; F{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}{\text{net}} = \sumi \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 QA=-10\,\mu\text{C} at x=0\,\text{cm}, QB=+40\,\mu\text{C} at x=20\,\text{cm}

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

    • E = EA + EB = -k\dfrac{QA}{rA^2} - k\dfrac{QB}{rB^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