MACQUARIE University Electric Fields and Gauss' Law

PHYS1520 – Lectures 1 and 2
Lecturer: Dr Tayyaba Zafar, School of Mathematical and Physical Sciences
Contact: Room: 12 WW room 505 | Email: tayyaba.zafar@mq.edu.au | Unit Email: phys1520@mq.edu.au
Research Interests:

• Astronomy (Interstellar medium)

• Technology translation and astronomical instrumentation (Telescopes and fibre positioners)

About PHYS1520

• Physics vs Engineering: Understanding physics boosts device quality and intuition.

• Course Structure:

  • Weeks 1-3: Electricity (using "Fundamentals of Physics" textbook)

    • Revises PHYS1510, but includes substantial new material.

  • Weeks 4-8: Electrical circuits (using "Introduction to Electrical Circuits" textbook)

    • Introduces methods for circuit analysis and foundational skills requisite for higher units (e.g., ELEC2070).

  • Weeks 9-10: Magnetism and induction.

  • Weeks 11-12: Electromagnetic radiation, photons, and matter waves (again using "Fundamentals of Physics").

  • Week 13: Revision.

Numeracy Centre

• Location: Room 188, 14 Sir Christopher Ondaatje Avenue.

• Service: Offers a drop-in math clinic for personalized assistance; privacy assured (no names recorded).

• Physics Tutors: Availability roster will be released approximately in week 2.

Concept Questions on Electric Fields and Gauss' Law

1. Coulomb’s Law: Meaning and Units

   • Write down Coulomb’s law for the electrostatic force between two particles. Define each term and state their units.

2. Electric Field vs Electrostatic Force

   • Describe the relationship between the electric field at a point and the electrostatic force on a particle at that point.

3. Positive Test Charge

   • What is the charge of a “positive test charge” in electrostatics?

4. Electric Field Lines Sketch

   • Sketch electric field lines for:

     • (a) A positive point charge of Q1.

     • (b) A negative point charge of twice Q1.

5. Electric Dipole Definition and Sketch

   • Describe and sketch the electric field due to an electric dipole.

6. Law of Superposition

   • Explain how it relates to electric fields.

7. Finding Electric Field Strength due to a Dipole

   • Describe the approach with an accompanying sketch.

8. Electric Field of a Ring of Charge

   • Describe how to find the electric field along the axis of a ring of charge.

9. Electric Flux Definition

   • Define electric flux in words and with a formula.

10. Gauss's Law and its Equation

    • Explain Gauss’s Law and provide the relevant equation, defining each term and stating units.

11. Gaussian Surface

    • What is meant by “a Gaussian surface” in the context of Gauss’ Law?

12. When to use Gauss’ Law

    • Discuss situations for the application of Gauss’ Law versus Coulomb’s law.

Maxwell's Equations

Revision of Key Concepts from PHYS1510

• Electric Charge: Observational experiments (e.g., rubbing materials) demonstrate attraction and repulsion based on charge sign.

  • Charged rods experience forces: same signs repel, opposite signs attract.

• Triboelectric Series: Reference for predicting charge based on interactions.

Electric Charge in Atoms

• Atoms consist of a nucleus with protons and neutrons, surrounded by an electron cloud.

• Rutherford's experiment established the nucleus's mass and positive charge dominance in the atom.

Definition and Conservation of Charge

• Charge is quantized at ${e = 1.602 imes 10^{-19} ext{C}}$ as per Millikan’s experiment.

• Charge is conserved in isolated systems.

• Forces keeping electrons in orbital motion arise from attractive proton-electron interactions.

Classification of Materials Based on Charge Mobility

• Conductors: Free-moving conduction electrons, e.g., metals.

• Insulators (Non-conductors): Limited free charge movement, e.g., rubber and glass.

• Semiconductors: Intermediate charge mobility, e.g., silicon and germanium in electronic applications.

• Superconductors: Offer zero-resistance charge movement at low temperatures, such as copper at cryogenic temperatures.

Inducing Charge and Polarization

• Induction in Conductors: Neutral rods experience charge movement upon proximity to charged rods.

• Polarization in Atoms: Molecules can exhibit dipole moments when subjected to external electric fields.

Coulomb's Law

• Describes the force between charged particles: $F = k rac{|q<em>1 q</em>2|}{r^2}$

  • Where:

    • $F$ = electrostatic force

    • $k = 8.99 imes 10^9 ext{Nm}^2/ ext{C}^2$ = Coulomb's constant

    • $q<em>1$ and $q</em>2$ = charges

    • $r$ = separation distance.

Superposition of Coulomb Forces

• In multiple force scenarios, the total force is the vector sum of individual forces:
  $F<em>{net} = F</em>1 + F<em>2 + F</em>3 + …$

Comparison of Gravitational and Electrostatic Forces

• Analysis for the forces between two protons shows electrostatic forces dominate significantly over gravitational forces.

Electric Field (E) Definition

• The electric field at a point is the ratio of the electrostatic force (F) acting on a positive test charge (q0) to the magnitude of the test charge:
  $E = rac{F}{q_0}$

• Directionality of the electric field:

  • Outward from positive charges, inward toward negative charges.

Electric Field Representation

• Electric field vectors are indicative of strength and orientation, visualized by field lines.

  • Electric field magnitude is inferred from spacing between lines; proximity indicates intensity.

Electric Field from a Dipole

• An electric dipole consists of two charges of equal magnitude but opposite signs, separated by a small distance$d$.

• The electric field at a point due to the dipole is given by $E = rac{1}{4 ext{π} ext{ε}_0} rac{2p}{z^3}$ where

  • $p = qd$ is the dipole moment

  • $z$ is the distance from the center of the dipole to the point of measurement.

Electric Field from a Ring of Charge

• The electric field due to a uniformly charged ring can be calculated at point P on the axis using integrative methods based on charge density$ext{λ}$ (C/m) and geometry.

Gauss' Law

• Gauss’s Law relates electric flux ($ ext{Φ}$) to the charge enclosed ($q{enc}$) within a closed surface (Gaussian surface): $Φ = rac{q</em>{enc}}{ ext{ε}_0}$

• Applicable regardless of the shape of the Gaussian surface and depends on symmetry for simplification in calculations.

Area Vectors and Electric Flux

• The area vector ($dA$) is defined with direction normal to the surface element, facilitating flux calculations via the dot product:
  $dΦ = E \bullet dA$

Applications of Gauss' Law

• Used for spherical, cylindrical, and planar distributions to simplify the computation of electric fields in symmetric scenarios (e.g., spherical shells).

Shell Theorems

• Shell Theorem 1: External charges treat a charged shell as a point charge located at its center.

• Shell Theorem 2: Charged particles inside a shell with uniform charge density experience no net force due to the shell.

Summary

• Electric Field Definition: The electric field is a vector field describing the force per unit charge.

• Gauss’s Law aids calculations, especially in symmetric situations.

• Applications include predicting fields in spherical shells, planars, and from line charges, essential for electrical engineering analyses.