Electromagnetism & Optics - PHY00002I

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Module will run

Occurrence	Teaching period
A	Spring Term 2022-23 to Summer Term 2022-23

Module aims

The central aim of this module is to understand how Maxwell unified electricity, magnetism and optics into electromagnetic theory. Knowledge of the basic phenomena of electromagnetism, and a good understanding of the mathematics of vector fields, are essential in achieving the central aim. Maxwell's four equations describe all of electromagnetism, including the propagation of electromagnetic waves. The main aim of the optics part of the course is to understand Fraunhofer and Fresnel diffraction. The subsidiary aim of the course is to provide an account of the electrical and magnetic properties of materials.

Module learning outcomes

Write down Maxwell's equations in differential form, defining all the variables

Show a good grasp of the meaning of the div and curl of a vector field

Understand that the Maxwell equations can be split into two twos for time-independent fields, two for electrostatics, two for magnetostatics

Use the material in Electromagnetism I (lectures 1-9) to solve problems in electrostatics

Derive Maxwell's first equation from Gauss' Flux Law of Electrostatics, using Gauss' theorem

Show a good grasp of the meaning of the gradient of a scalar field, and apply it to obtain electric fields from electric potentials

Recognise Poisson's equation and Laplace's equation of electrostatics

Use the material in EM I (lectures 10-12) to solve problems in magnetostatics, and calculate the motion of charged particles in magnetic fields

Define the Ampere

Relate the differential to the integral form of Ampere's Law, using Stokes' theorem

Use Ampere's Law to calculate the magnetic field of a current-carrying straight wire and a solenoid

Define the vector potential

Use the vector potential to calculate the magnetic field of a current-carrying straight wire

Use the Biot-Savart Law to calculate the magnetic field on the axis of a current-carrying loop

Define electro-motive force (emf)

Write down Faraday's discovery of the three ways in which an emf can be induced in a wire

Relate the differential to the integral form of Faraday's Law, using Stokes' theorem

Apply the integral form of Faraday's Law to calculate the emf induced in a moving circuit, and an AC generator

Calculate mutual inductance, and apply this to transformers

Calculate self-inductance, and apply this to the properties of inductors in AC circuits

Understand that Ampere's Law contravenes the principle of conservation of electric charge

Express charge conservation in differential and integral form, and derive the relationship between them, using Gauss theorem

Understand the new term in the Ampere-Maxwell equation

Show that Maxwell's equations in free space lead to a wave equation

Derive the relationship c^2 =1/u0. (epsilon)0 from the electromagnetic wave equation

Describe all the main features of electromagnetic waves, including energy flow

Define the Poynting vector

Use expressions for the energy density of electric and magnetic fields and the Poynting vector in various simple circumstances

Demonstrate an understanding the basic concepts of polarization, phase, coherence and optical path difference

Describe dispersion by prisms and diffraction gratings

Describe Young's double slit experiment and calculate interference patterns

Describe Fresnel and Fraunhofer diffraction

Determine the Fresnel diffraction pattern from circular apertures.

Understand the connection between Fraunhofer diffraction and the Fourier transform

Derive the laws of reflection and refraction of light from the continuity of the solutions to Maxwell's equations at an interface

Calculate reflection coefficients for light polarized parallel and perpendicular to the plane of incidence

Understand how polarized light is produced by reflection, and by anisotropic materials

Describe plane polarization, circular polarization and elliptical polarization

Calculate the electrostatic potential and field of a dipole

Understand the nature of multipole expansions of the electrostatic potential

Calculate the energy of a dipole in an electric field, and apply understanding of dipoles to the behaviour of dielectric materials in electric fields

Calculate the capacitance of devices with dielectric materials between the charged conductors

Define surface and bulk polarization charges, and the electric polarization vector

Recognise the displacement field

Calculate electrostatic energy for discrete and continuous charge distributions

Calculate the energy density of an electric field, and apply this to the energy stored in a capacitor

Understand the nature of the problem of the infinite energy of a point charge in classical electrodynamics

Calculate the force on a current-carrying wire in a magnetic field

Analyse the forces on a current-carrying loop, to prove that it behaves like a magnetic dipole

Calculate the field of a current-carrying loop, using the vector potential

Calculate the energy of a magnetic dipole in a magnetic field

Show an understanding of the properties of magnetic materials in magnetic fields, including an ability to explain qualitatively the behaviour of diamagnetic, paramagnetic and ferromagnetic materials

Define surface and bulk current densities in magnetic materials, and the magnetization vector

Define magnetic susceptibility and relative permittivity

Recognise typical magnetisation and hysteresis curves for soft iron

Module content

Syllabus

• Maxwell’s equations
• Review of electrostatics covered in Electromagnetism I
• Poisson’s equation and Laplace’s equation
• Differential and integral forms of Ampere’s Law
• Magnetic fields of wires and solenoids
• Vector potential
• Biot-Savart law
• Induced currents; emfs and generators
• Differential and integral forms of Faraday’s Law
• Induction, generators and transformers
• Mutual inductance and self inductance
• Charge conservation
• Ampere-Maxwell equation
• Electromagnetic waves; the speed of light
• Poynting vector
• Energy density of electric and magnetic fields
• Polarization, phase, coherence and optical path difference
• Dispersion by prisms and diffraction gratings
• Young’s double slit experiment and interference patterns
• Fraunhofer diffraction
• Fourier Transform and Fraunhofer diffraction
• Fresnel diffraction from circular and rectangular apertures
• Reflection and refraction of light in terms of Maxwell’s equations at a boundary
• Polarization of electromagnetic waves, Brewster’s angle and polaroids
• Malus’s law. Sequences of polarizing filters
• Plane polarized, circularly polarized and elliptically polarized light, and quarter wave plates
• Electric dipoles
• Properties of dielectric materials
• Capacitance of devices with dielectrics
• Polarization charges, polarization vector and displacement field
• Electrostatic energy
• Magnetic energy
• Magnetic dipoles; analysis of forces on a current-carrying loop
• Field of a current-carrying loop; energy of a magnetic dipole in a magnetic field
• Magnetic materials; diamagnetism, paramagnetism and ferromagnetism
• Surface and volume current densities, magnetization vector and magnetic intensity
• Magnetic susceptibility and relative permeability
• Magnetisation and hysteresis curves for soft iron
• Lasers: radiative processes
• Lasers: population inversion
• Lasers: decay rates
• 3 and 4 level lasers
• Properties of the laser beam
• Lasers: cavity modes

Indicative assessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	80
Essay/coursework	20

Special assessment rules

None

Indicative reassessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Module feedback

Our policy on how you receive feedback for formative and summative purposes is contained in our Department Handbook.

Indicative reading

Feynman: Lectures on Physics volume 2 (Addison-Wesley) ****

Griffiths: Introduction to Electrodynamics (Prentice-Hall) ***

Grant & Philips: Electromagnetism (Wiley) ***

Fleisch: A student's guide to Maxwell's equations (Cambridge University Press) ***

Hecht: Optics (Addison-Wesley) ****

Smith & King: Optics and Photonics (Wiley) ***

The following chapters from the Feynman lectures (vol 2) are particularly useful:

Ch 1 Electromagnetism

Ch 3 Vector Integral Calculus

Ch 8 Electrostatic Energy

Ch 10 Dielectrics

Ch 13 Magnetostatics

Ch 14 The Magnetic Field in Various Circumstances

Ch 16 Induced Currents

Ch 17 The Laws of Induction

Ch 18 The Maxwell Equations

Ch 20 Solutions of Maxwell s Equations in Free Space

Ch 27 Field Energy and Field Momentum

Ch 33 Reflection from Surfaces

Ch 34 The Magnetism of Matter

Ch 36 Ferromagnetism

For the optics part of the course, in Volume 1 of the Feynman lectures:

Ch 28 Electromagnetic Radiation

Ch 29 Interference

Ch 30 Diffraction

In Hecht's book on Optics:

Ch 7 The Superposition of Waves

Ch 8 Polarization

Ch 9 Interference

Ch 10 Diffraction

Ch 11 Fourier Optics