- Foundations of Classical and Statistical Physics
- Oscillations and Waves
- Fields, Relativity and Quantum Physics
- Practicals

FOUNDATIONS OF CLASSICAL AND STATISTICAL PHYSICS

M S Longair (A) and J R Waldram (B)

**Introduction:** What is Physics? The nature of physical understanding, the role of mathematical models.

**Classical Mechanics:** Newton's laws of motion. Basic definitions: time, Galilean frames of reference, force, acceleration, mass, momentum. Units and dimensions. The power of vectors. Newton's laws in vectorial form. Forces and equilibrium, action and reaction, internal forces, shear forces, pressure in a fluid. Work, power, potential energy. Turning moment, centre of mass. Uniform acceleration, motion in a circle. Kinetic energy, conservation of momentum; rockets. Inertial frames of reference; zero momentum frame of reference; impulse, elastic and inelastic collisions.

**Rotational mechanics:** Rotation of a rigid body; angular velocity and acceleration; moment of inertia* I*, angular momentum; rotational kinetic energy; comparison of linear and rotational quantities. Calculation of *I*, parallel and perpendicular axis theorems. Angular velocity, turning moment and angular momentum as vectors; conservation of angular momentum. The gyroscope, precession. Couples. Linear and rotational motion combined. *Boomerangs*.

**Introduction to Statistical Physics — the kinetic theory of gases:** The macroscopic and microscopic descriptions of physical phenomena. The nature of statistical physics; probability; distribution functions, normal distribution of errors. Ideal gas law; gas scale temperature; Boltzmann's constant. Simple kinetic theory; pressure; flux density; partial pressure. Mean free path; *distribution of free paths*. Random walks, Brownian motion. Transport properties at the macroscopic and microscopic levels: thermal conductivity, viscosity and self-diffusion. *Gases at low pressure*.

**Thermal physics, classical and statistical: **Conservation of energy, heat, work, internal energy *U*. Quantum states, quantum jumps and heat flow. The Boltzmann distribution. Calculation of *U* and heat capacity *C*; the two-level system; the oscillator, *derivation of Planck's formula*. Classical form of the Boltzmann distribution and its applications: the isothermal atmosphere, sedimentation. Maxwell-Boltzmann distribution of molecular speeds. Thermal activation and chemical reactions. The equipartition theorem.

**Thermal properties of gases: **Properties of real gases. Degrees of freedom, internal energy of gas, Joule expansion. *C _{v}, C_{p}*, ratio of heat capacities, adiabatic expansion. Breakdown of the classical model.

Dimensional analysis in physics.

**Behaviour of solids, liquids and gases: **Hard-sphere model. Structures of solids and liquids, melting. *Hard-sphere equation of state.* 6-12 potential. Simple properties of solids and liquids: bulk modulus, rigidity; heat capacity, Dulong and Petit's law; *thermal expansion of solids*; latent heat; vapour pressure.

**Empirical equation of state: **Imperfect gases, Boyle temperature. Imperfect gases, critical point, liquefaction. Triple point, sublimation. Phase diagram*. Law of corresponding states - its successes and failures*.

Topics in italics are not for examination and may not be in both courses.

BOOKS

*Foundations of Classical and Statistical Physics Course Companion*. This contains lecturers' problems for use in supervision, and notes on certain important topics not extensively covered in the lectures, such as physical quantities, units and dimensions errors and statistics; and how to tackle problems.

__Useful Mathematical Results for Part 1A Physics.__ This booklet contains a summary of all the mathematical results which will be needed in the Part 1A Physics Course. You should use this as a supplement to the lectures in Part 1A Mathematics for the Natural Sciences.

The following two books cover respectively the first and second parts of the course at about the right level:

*Introduction to Classical Mechanics*, French A P & Ebison M G (Chapman & Hall 1986)

*Gases, Liquids and Solids*, Tabor D (3rd edn CUP 1991)

The following book gives good coverage of much of the material contained in the whole Part IA course at a straightforward level:

*Physics*, Halliday D, Resnik R & Walker J (JohnWiley & Son 6^{th} extended edition 2001).

For a thought-provoking alternative approach to first-year physics (and much more), the* Feynman Lectures on Physics* (Addison-Wesley Publishing Company 1963) are in a class of their own.

J Riley (A) and JR Batley (B)

**Simple Harmonic Motion:** General oscillations. SHM; equation of motion, velocity, acceleration, phase angle. Kinetic and potential energies. Examples. Complex representation *A*e^{iw t}. Superimposed SHM's of the same frequency; phasor diagrams.

**Electronic Components:** Resistors, capacitors, and inductors. Complex impedance. Properties of simple networks; filters.

**Damped Oscillators: **General behaviour. Free motion and damping. Examples. Complex frequencies. Q-values. LCR circuits. Analogies between electronic and mechanical oscillators. Mechanical impedance.

**Driven Oscillators: **Frequency response. Velocity resonance. Power absorption; power bandwidth, its relation to Q. Tuned circuits. Amplitude resonance. *Transients.*

**Feedback:** Linear systems. Feedback; positive and negative. Effective gain, stability, input/output impedances, bandwidth. Operational amplifiers, simple applications. *Sinusoidal oscillator; combining positive and negative feedback.*

**Wave motion:** Harmonic waves, e^{i(kx-w t)}* *notation. Wave equation. Linearity, superposition.

**Standing waves:** Stationary wave pattern, quantisation of *k*. Waves in 1, 2 and 3 dimensions.

Topics in italics are not for examination.

BOOKS

*Vibrations and Waves in Physics*, Main I G (3rd edn CUP 1993)

*The Physics of Vibrations and Waves,* Pain H J (5th edn Wiley 1999)

*Vibrations and Waves, *French A P (Chapman & Hall 1971)

*Analogue and Digital Electronics for Engineers,* Ahmed H & Spreadbury P J (2nd edn CUP 1984) Chapters 5 and 6.

FIELDS, RELATIVITY AND QUANTUM PHYSICS

D A Green (A) and J R Carter (B)

Fields:

**Conservative Fields of Force: **potentials, potential energy, conservation of energy; the gradient operator: _{}, examples.

**Gravity:** Newton's law of gravity, gravity as conservative field of force;* equivalence of gravitational and inertial masses;* Gauss's theorem, flux and solid angle, principle of superposition; examples: field inside and outside uniform sphere.

**Central Orbits: **Kepler's second law; *specific angular momentum*, circular orbits, *elliptical orbits,* Kepler I and III, *central orbits in general relativity.*

**Electrostatics: **Coulomb's law, electrostatic potential,, equipotential; Gauss's law,** **, verification of inverse square law; electric dipole; fields due to charged surfaces, conductors, capacitance; energy in electrostatics.

**Magnetostatics: **absence of magnetic monopoles, magnetic flux density, Gauss's law: , magnetic dipoles, couple on magnet dipole in **B**-field.

**Currents and Magnetic Fields: **equivalence of magnets and small current loops; Ampère's law, Biot-Savart law, examples; force between two currents, definition of the ampere.

**Charged Particles in Electric and Magnetic Fields: **Lorentz force on a charged moving particle, examples: motion in uniform electric field and uniform magnetic field; J.J.Thomson's experiment.

Special Relativity:

**Historical Background: **Revision of Galilean relativity and Newton's laws; stellar aberration, the aether; Michelson-Morley experiment.

**Lorentz Transformations: **Einstein's postulates; derivation of Lorentz transformations; spacetime, relativity of simultaneity, Einstein's train, time dilation, proper time; length contraction; twin paradox, causality; experimental evidence; *geometrical representation of Lorentz transformation.*

**Velocity Transformations: **Velocity transformation and velocity addition; aberration of light, Doppler effect.

**Relativistic Mechanics and Dynamics: **Definition of relativistic momentum, the force equation in relativity; rest mass, kinetic and total energies, conservation of energy and momentum, *E*-** p** invariant;

Quantum Physics:

**The Nature of Light: **Revision of evidence for wave nature of light; Rayleigh-Jeans law and Planck's law for blackbody radiation, ultraviolet catastrophe; photoelectric effect, photons;* Einstein's theory of specific heats; *comparison of wave and particle pictures; Compton scattering.

**Application of Quantisation to Atoms: **Rutherford scattering, Bohr model of the atom,* *** _{}**, problems with Bohr model; proper quantisation of angular momentum:

**The Wave Nature of Particles: **de Broglie waves, Davisson-Germer experiment; wave packets, Heisenberg uncertainty principle, probability interpretation.

**Wave Mechanics: **Schrödinger's equation; 1D infinite potential well, quantisation of energy levels, simple treatment of tunnelling; examples.

**Cosmology: **Cosmological redshift, Hubble's law, microwave background, the Big Bang; expansion rate d*R/*d*t *from Gauss's law; matter- and radiation-dominated universes, density and temperature versus time; history of the Universe: particle-antiparticle annihilation, formation of nuclei.

The core examination syllabus is indicated in Roman type. The items in *italic type* are non-examinable and will be included at the discretion of the lecturer.

BOOKS

For specific treatment of the three major sections of the course, reference can be made to:

*Electricity and Magnetism*, Duffin W J (4th edn McGraw-Hill 1990);

*Dynamics and Relativity, *McComb W D (OUP, 1999)

C A Haniff, G A C Jones, C J B Ford

Michaelmas Term

Four basic experiments are carried out, which are intended to teach experimental skills and error theory as well as reinforcing material developed in lectures. Marks for the first practical do not count towards the final total.

1. **Attenuation of gamma photons.** This aims to introduce the statistical variations of a single variable, via the estimation of the linear attenuation coefficient for gamma photons in lead.

2. **Measuring g by timing the oscillations of a rigid pendulum.** Here an accuracy of 1 in 1000 is required in the result and this is to be achieved through the combinations of uncertainties in measurements.

3. **Gyroscope**. This experiment studies the motion of a spinning bicycle wheel under the influence of an applied couple.

4. **Thermal activation.** This experiment measures the variation of the electrical resistance of a semiconductor with temperature.

Lent Term

The first three sessions consist of an introduction to digital electronics and two experiments which correlate with the lectures in "Oscillations and Waves". Marks for the first practical do not count towards the final total.

5. **Using an Oscilloscope.** This experiment is intended to introduce the oscilloscope as a measuring instrument and demonstrates the action of high- and low-pass filters using a potential divider with a resistance and capacitative impedance.

6. **Electrical resonance and signal filtering.** This is an investigation of resonance in LCR circuits and uses the LCR network to filter a signal from background noise.

7. **The operational amplifier. **An integrated circuit with negative feedback is used to build a simple voltage amplifier, and then, utilising an LCR resonant circuit to build a radio receiver.

**Lent Term** (last session) and **Easter Term**

There is a choice of three experiments from a list of seven.

8. **Mechanical resonance.** This is a study of the resonance of a mechanical system with various degrees of damping.

9. **Spreadsheet physics.** This introduces spreadsheet techniques and uses them to study various topics related to physics courses.

10. **Fourier analysis of vowels.** This experiment studies the harmonic content of your singing voice, or other sounds, or wave forms from a signal generator.

11. **Complex impedance.** The magnitude and phase of impedance is studied as a function of frequency.

12. **Energies and correlation of gamma quanta.** Nuclear decay and positron annihilation are investigated.

13. **Viscosity of polymer solutions.** This experiment studies the dramatic effect of polymer molecules on the viscosity of water. Measurements are used to estimate the size of the polymer molecule.

14. **Camera test.** This experiment shows how to test the performance of a photographic camera.

Head of Class Report

Students are required to write up one of the Lent-term experiments, excepting experiment number 5, as a Head-of-Class report. This is assessed by a Head of Class.

**Projects **(last session of Lent term and Easter term)

Instead of doing three experiments from numbers 8 to 14 and in addition writing a formal Head-of-Class report, a few students may carry out experimental projects for the same exam credit (i.e. the equivalent of three experiments and a formal Head-of-Class report). These projects have to be instigated by the students themselves, but only those students who have done good experimental work in experiments 1 to 7 will be allowed to carry out projects. Contact your Head of Class early in the Lent term for more information.

BOOKS

*Practical Physics* (3rd edition), G. L. Squires, Cambridge University Press (1985). This book introduces the subject in a simple and readable manner.

*Experimental methods*, L. Kirkup, John Wiley (1994). An up to date book at the right level for the course on experimental error analysis, including a section on data gathering using a microcomputer.

*Experimental Physics,* R. A. Dunlap, Oxford University Press (1988).

*An Introduction to Experimental Physics*, Cooke, C., UCL Press (1996).

Copyright © 2001 Cavendish Laboratory, University of Cambridge