Physics for Engineers II: PHYC10160 Study Notes

Physics for Engineers II: PHYC10160 Ian P. Mercer Th2.1: Electricity

Overview

• Part A: Charge, Force, and Field

  • Electric Charge

  • Coulomb's Law

  • How to calculate the force between charges in various arrangements

  • Electric Field

  • How to calculate an electric field due to point charges

• Part B: Gauss’s Law

  • Electric Flux

  • Using Gauss's law to calculate electric field

  • Charge and field distributions about an electrostatic conductor

• Background Reading: Wiley chapters 21, 22, 23

Atom Structure

• Atoms consist of electrons and a nucleus

• Relative sizes:

  • Atoms: ~$5 imes 10^{-10}$ m (similar size to a peanut)

  • Nuclei: ~$5 imes 10^{-15}$ m (similar size to a football stadium)

• Nucleus consists of two types of particles:

  • Protons (p): Positively charged

  • Neutrons (n): Neutral (zero charge)

  • Electrons (e): Negatively charged

Mass and Charge of Atomic Constituents

• Neutron (n):

  • Mass: $m = 1.675 imes 10^{-27}$ kg

  • Charge: $q = 0$

• Proton (p):

  • Mass: $m = 1.673 imes 10^{-27}$ kg

  • Charge: $q = +1.602 imes 10^{-19}$ C

• Electron (e):

  • Mass: $m = 9.11 imes 10^{-31}$ kg

  • Charge: $q = -1.602 imes 10^{-19}$ C

Notes

1. The symbols “−e” and “+e” are used for the electron and proton charge, known as the elementary charge.

2. Atoms are electrically neutral; the number of electrons = the number of protons (known as the atomic number, symbol: Z).

   • Chemical properties determined exclusively by Z.

3. The sum of protons and neutrons is the mass number (symbol: A).

   • Notation example:

     • Z = 92 (number of protons/electrons)

     • A = 235 (number of protons + neutrons)

     • Atomic number Z = 92 identifies the nucleus as that of a uranium atom: $^{235}_{92}U$

Charge Conservation

• Rubbing does not create charge; it transfers it from one body to another.

• Charge conservation states:

  • No exception to charge conservation has ever been found, even in nuclear reactions.

• Equation:

  • $(Q<em>s + Q</em>g)<em>{before} = (Q</em>s + Q<em>g)</em>{after}$

  • Where $Qg$ and $Qs$ refer to charge before and after.

Electrical Properties of Materials

• Electrical Conductors:

  • Outer shell electrons are freed from atomic cores, leading to freely moving conduction electrons.

  • Examples: Copper, aluminum, silver

• Electrical Insulators:

  • Electric charges do not move freely.

  • Examples: Glass, rubber, ceramics, wood

• Electrical Semiconductors:

  • Display properties between insulators and conductors.

  • Conductivity changes over several orders of magnitude by applying electric fields or controlling impurities (foreign atoms).

  • Examples: Silicon, germanium

Coulomb’s Law

• Discovered by Charles Coulomb (1736 – 1806):

  • The force between two charges is proportional to the amount of each charge and inversely proportional to the square of the distance between them.

  • Formula representation:
    $F = k rac{|q<em>1 q</em>2|}{r^2}$

  • Where $k = rac{1}{4 rac{ ext{pi} imes ext{epsilon}_0}}$

  • Here, $ ext{epsilon}_0$ is the permittivity of free space.

Example Calculation

• Weight a coulomb of charge could lift when separated from an equal and opposite charge by 1m:

  • Given:$k = 8.99 imes 10^9 ext{ N m}^2/ ext{C}^2$

  • Force due to Coulomb's law:
    $F = k rac{(1 ext{ C})^2}{(1 ext{ m})^2}$

• Current definition:

  • $I = rac{dq}{dt}$, where:

  • $I$ = current in amperes (A)

  • $dq$ is the amount of charge that flows through wire's cross-section per unit time.

  • A charge of 1C passing through any cross-section of wire in one second creates a 1A current.

Problem Example

• Scenario with three identical conducting spheres A, B, and C on insulating stands, initially neutral.

• A negatively charged rod contacts A.

• C is removed, then B is removed, and the rod is taken away.

• Final charges:

  • Choices:

  • a) + + −

  • b) + − +

  • c) + 0 −

  • d) − + 0

  • e) − − −

Electric Fields

• Electric fields explain how particles exert forces on each other at distances.

• Defined using a test charge ($q0$): $E = rac{F}{q</em>0}$

  • Units: N/C

Characteristics of Electric Fields

• Symmetry: Objects appear unchanged under specific transformations (e.g., rotations).

  • Example: A featureless sphere displays rotational symmetry.

• Electric Field Generated by Point Charges:

  • For positive charge:

  • Points radially outward.

  • For negative charge:

  • Points radially inward.

  • Follow law:
    $E = rac{q}{4 ext{pi} ext{epsilon_0} r^2}$

• Electric Field Lines:

  • Represent the electric field vector, visually showing the direction of force on a test charge.

  • Field lines are densest in regions of strongest electric fields.

Electric Flux

• Defined as the product of the magnitude of the electric field (E) and surface area (A) perpendicular to the field:
  $ext{Flux} ( ext{Φ}) = E imes A = EA ext{cos}(θ)$

Gauss’s Law

• States that the net electric flux through a closed surface is equal to the net charge inside that surface divided by the permittivity of free space $ ext{epsilon}0$: $ext{Φ} = rac{q</em>{ ext{enc}}}{ ext{epsilon}_0}$

  • Valid for any type of closed surface, chosen arbitrarily.

• Deriving Coulomb's Law from Gauss's Law involves establishing an appropriate Gaussian surface and applying the law to find the electric field generated by a point charge.

Properties of Conductors in Electrostatic Equilibrium

1. Electric field is zero everywhere inside a conductor.

2. Residual charge is located entirely on the surface of an isolated conductor.

3. Electric field just outside a charged conductor is perpendicular to the surface, with a magnitude: $E = \frac{\sigma}{\epsilon_0}$.

4. On irregularly shaped conductors, surface charge density and the electric field are greatest where the radius of curvature is smallest.