Higher Physics Part 1: Particles and Waves - Exhaustive Study Guide

Forces on Charged Particles

• Electrical Charge Overview

  • Electrical charge is a physical property of matter measured in coulombs ($C$).
  • There are two types of electrical charge: positive ($+$) and negative ($−$).
  • Fundamental Interaction Rules: Opposite charges attract each other; like charges repel each other.
  • Charge states: Charge can be static (stationary) or moving.
  • Real-World Applications of Electrostatic Charge:
    • Charging a plastic rod by rubbing it with a cloth to pick up paper or deflect water.
    • Rubbing a balloon on hair or a jumper to stick it to a wall.
    • Van de Graaff generators for creating sparks or making hair stand up.
    • Laser printers and photocopiers (used to make toner stick to paper).
    • Car manufacturing (ensuring spray guns paint cars evenly).

• Electric Field Patterns

  • An electric field is a region of space around a charge where another charge experiences a force.
  • Mapping Fields:
    • Arrows indicate the direction a positive charge would move.
    • Field lines go from positive ($+$) to negative ($−$).
    • Line spacing represents strength: closer lines indicate a stronger field; wider spacing indicates a weaker field.
  • Specific Field Types:
    • Radial Fields: Created by single point charges. Lines radiate outwards from positive charges and inwards towards negative charges.
    • Uniform Fields: Created between two oppositely charged parallel plates. Lines are parallel and equally spaced.

• Charged Particles in Electric Fields

  • Free charged particles in an electric field experience a force and move, which involves work being done.
  • Work and Potential Energy:
    • Moving a positive charge against the direction of the field increases its electrical potential energy.
    • Work done ($W$) is equal to the change in electrical potential energy ($E_{p}$).
    • Relationship: $W = QV$
      • $W$ is work done in joules ($J$).
      • $Q$ is charge in coulombs ($C$).
      • $V$ is potential difference in volts ($V$).
  • Voltage Definition: The voltage of a power supply is the energy given to each coulomb of charge passing through it ($V = \frac{W}{Q}$). One volt equals one joule per coulomb ($1\,V = 1\,J/C$).
  • Conservation of Energy:
    • If a charge moves in the direction of the field, its electrical potential energy converts to kinetic energy ($E_{k}$).
    • For a particle starting from rest: $QV = \frac{1}{2}mv^{2}$.
    • Formula for particles with initial velocity: $W = \Delta E_{k} = E_{k, final} - E_{k, initial}$ or $QV = \frac{1}{2}mv_{final}^{2} - \frac{1}{2}mv_{initial}^{2}$.
  • Deflection: Positive particles follow field lines; negative particles move opposite to field lines.

• Magnetic Fields

  • A magnetic field is a region around a magnet where another magnet experiences a force.
  • Magnets have North ($N$) and South ($S$) poles. Opposite poles attract; like poles repel.
  • Field Lines: Drawn from North to South (direction a North pole would move).
  • Moving Charges and Magnetism:
    • A moving charge (current) produces its own magnetic field.
    • Left Hand Grip Rule: Used for wires. Thumb points in direction of current ($I$); curled fingers show magnetic field direction.
    • Symbols for direction: A dot ($\cdot$) indicates field/current coming out of the page; a cross ($\times$) indicates it going into the page.

• Force on Charges in Magnetic Fields

  • Interaction between a moving charge's field and an external magnetic field creates a force perpendicular to both motion and the external field.
  • Fleming’s Right Hand Rule (for Negative Charges):
    • Thumb: Direction of motion (Force).
    • First Finger: Direction of magnetic field ($N \rightarrow S$).
    • Second Finger: Direction of current (electron flow).
  • Positive Charges: Move in the opposite direction to negative charges (reverse the result of the right hand rule).

Particle Accelerators

• Basic Principles and Components

  • Accelerators increase charged particles (electrons, protons) to high energies/speeds for collisions with targets or other particles to study matter or produce medical radioisotopes.
  • Common Features:
    • Electric fields to accelerate particles.
    • Electromagnets to deflect/steer and confine the beam.
    • Vacuum chambers to prevent obstruction by air/dust.
    • Particle source and beam pipes.

• Types of Accelerators

  • Cathode Ray Tube (Electron Gun):
    • Cathode is heated to produce electrons.
    • Large potential difference ($V$) accelerates electrons between anodes.
    • Deflection coils (magnetic field) steer the beam.
    • Hits a phosphorescent screen to produce light.
  • Linear Accelerators (LINAC):
    • Consist of hollow metal drift tubes in a vacuum.
    • Acceleration occurs only in the gaps between tubes via electric fields.
    • High frequency alternating supply changes tube polarity to keep particles accelerating in one direction.
    • Tubes increase in length as particles speed up so they spend the same amount of time in each tube.
  • Cyclotrons:
    • Two "dees" (D-shaped hollow structures) in a vacuum.
    • Magnetic field causes particles to move in a spiral.
    • Electric field in the gap between dees accelerates particles.
    • Alternating polarity changes the field direction for every gap crossing.
  • Synchrotrons:
    • Linear accelerator bent into a ring.
    • Electromagnets increase field strength as particle speed increases to maintain a circular path.
    • Relativistic effects: As particles reach relativistic speeds, their mass increases, requiring larger forces to maintain the path.

• The Large Hadron Collider (LHC)

  • World’s largest scientific instrument at CERN.
  • circular accelerator $27\,km$ long, situated $100\,m$ underground.
  • Accelerates hadrons (protons or lead ions) to over $10000$ laps per second.
  • Environment:
    • Coldest place: Superconducting magnets cooled to $1.9\,K$ (colder than space at $2.7\,K$).
    • Emptiest place: High vacuum.
    • Hottest place: Collisions create temperatures $> 100000$ times hotter than the sun.
  • Processes $600 \times 10^{6}$ collisions every second to study mass, dimensions, and the composition of the universe (the "96% unknown").

The Standard Model

• Orders of Magnitude

  • Uses powers of 10 to compare sizes.
  • Scale Examples:
    • Quark/Electron: $< 10^{-18}\,m$
    • Proton/Neutron: $10^{-15}\,m$
    • Atomic Nucleus: $10^{-14}\,m$
    • Atom: $10^{-10}\,m$
    • Human: $10^{0}\,m$
    • Earth: $10^{7}\,m$
    • Universe (known): $10^{26}\,m$

• Fundamental Particles (Fermions)

  • Fundamental particles are not made of anything else.
  • Leptons (6 types): Electron ($e^{-}$), Muon ($\mu^{-}$), Tau ($\tau^{-}$), and their neutrinos ($\nu_{e}, \nu_{\mu}, \nu_{\tau}$).
  • Quarks (6 types): Up ($u$), Down ($d$), Charm ($c$), Strange ($s$), Top ($t$), Bottom ($b$).
  • Generations: Quarks and leptons are arranged in three generations.

• Antimatter

  • Every particle has an antiparticle with the same mass but opposite charge.
  • Electron's antiparticle is the Positron ($e^{+}$).
  • Others use a bar: e.g., anti-electron neutrino ($\bar{\nu}_{e}$).
  • Annihilation: Particle meets antiparticle, they destroy each other, converting mass to energy.

• Hadrons (Composite Particles)

  • Made of quarks. Only exist where the overall charge is a whole integer.
  • Baryons: Made of 3 quarks (stable).
    • Proton: $uud$; Charge: $+\frac{2}{3}e + \frac{2}{3}e - \frac{1}{3}e = +1e$.
    • Neutron: $udd$; Charge: $+\frac{2}{3}e - \frac{1}{3}e - \frac{1}{3}e = 0$.
  • Mesons: Made of a quark-antiquark pair (unstable).
    • Positive Pion ($\pi^{+}$): Up quark and anti-down quark ($u\bar{d}$).

• Fundamental Forces and Bosons

  • Strong Nuclear Force: Holds quarks together; carried by gluons. Range: $\approx 10^{-15}\,m$.
  • Weak Nuclear Force: Involved in beta decay; carried by $W^{+}$, $W^{-}$, and $Z$ bosons. Range: $< 10^{-18}\,m$.
  • Electromagnetic Force: Acts on charged particles; holds electrons in orbit; carried by photons. Range: Infinite.
  • Gravitational Force: Acts on mass; carried by the hypothetical graviton. Range: Infinite. Weakest force.

• The Higgs Boson

  • Discovered July 4, 2012, at CERN.
  • Associated with the Higgs field which permeates space and gives particles mass (except photons and gluons).
  • Mass is roughly $133$ times that of a proton ($10^{-25}\,kg$).

Nuclear Reactions

• Atomic Symbols and Isotopes

  • Notation: ${}^{A}_{Z}X$
    • $A$ = mass number (protons + neutrons).
    • $Z$ = atomic number (protons).
  • Isotopes: Atoms of the same element with different numbers of neutrons.

• Radioactive Decay Types

  • Alpha ($\alpha$): Helium nucleus (${}^{4}_{2}He$). Parent mass $-4$, atomic number $-2$.
  • Beta ($\beta$): Fast electron (${}^{0}_{-1}e$) from a neutron decaying to a proton. Mass remains same, atomic number $+1$. Neutrinos ($\bar{\nu}_{e}$) are also emitted (evidence for neutrinos).
  • Gamma ($\gamma$): High-energy wave (${}^{0}_{0}\gamma$). No change in mass or atomic number.

• Mass-Energy Equivalence

  • Total mass after reaction is less than mass before. The "lost mass" ($m$) becomes energy.
  • Formula: $E = mc^{2}$
  • $c = 3 \times 10^{8}\,m/s$ (speed of light); $c^{2} = 9 \times 10^{16}$.

• Fission and Fusion

  • Fission: Heavy nucleus splits into lighter fragments.
    • Spontaneous (random) or Induced (bombarded by neutron).
    • Chain Reaction: Released neutrons cause further fissions.
  • Fusion: Light nuclei join to form a heavier nucleus.
    • Occurs in stars (Hydrogen to Helium).
    • Requires plasma state at high temperature and high pressure to overcome proton repulsion.
    • Containment uses strong magnetic fields.

Inverse Square Law

• Irradiance ($I$)

  • Power per unit area incident on a surface.
  • Formula: $I = \frac{P}{A}$
    • $I$ in $W\,m^{-2}$, $P$ in Watts, $A$ in $m^{2}$.
  • For a point source: Area of sphere $A = 4\pi r^{2}$, so $I = \frac{P}{4\pi r^{2}}$.

• Inverse Square Relationship

  • Irradiance is inversely proportional to the square of the distance ($d$).
  • Formula: $I = \frac{k}{d^{2}}$ or $I_{1}d_{1}^{2} = I_{2}d_{2}^{2}$.
  • Experimental verification involves plotting $I$ vs. $\frac{1}{d^{2}}$ to get a straight line through the origin.

Wave-Particle Duality

• Nature of Light

  • Wave evidence: interference, diffraction, reflection, refraction.
  • Particle evidence: The Photoelectric Effect.

• The Photoelectric Effect

  • Ejection of electrons (photoelectrons) from a metal surface when light (photons) hits it.
  • Key Terms:
    • Photon: Packet of energy ($E = hf$).
    • Threshold Frequency ($f_{0}$): Minimum frequency required for photoemission.
    • Work Function ($hf_{0}$): Minimum energy required for photoemission (property of the specific metal).
  • Relationships:
    • Energy of photon: $E = hf = \frac{hc}{\lambda}$.
    • $h = 6.63 \times 10^{-34}\,Js$ (Planck’s constant).
    • Conservation: $hf = hf_{0} + E_{k}$ or $E_{k} = hf - hf_{0}$.

• Observations using Gold Leaf Electroscope

  • Negatively charged electroscope discharges with Ultraviolet ($UV$) light (frequency $> f_{0}$).
  • Does not discharge with visible light (frequency $< f_{0}$).
  • Positively charged electroscopes do not discharge via this effect.

• Frequency vs. Irradiance Effects

  • Increasing Frequency ($f$): Increases maximum kinetic energy and speed of photoelectrons. No change in current.
  • Increasing Irradiance ($I/brightness$): Increases number of photoelectrons emitted per second (photoelectric current). No change in kinetic energy or speed.