PHY3051 Electromagnetism II 2021-22
Prof. J. Bertolotti

Delivery Weeks: T1:01-11
Level: 6 (NQF)
Credits: 15 NICATS / 7.5 ECTS
Enrolment: 60 students (approx)

### Description

This is the second electromagnetism module taken by Physics students. It builds on PHY2021 (Electromagnetism I) and covers fundamental physics that students are capable of directly observing. The early part of the module provides a brief recap and reinforces the difficult material treated at the end of PHY2021. The Maxwell equations are stated and manipulated to obtain the wave equation, and the form of the solutions discussed. The dielectric and magnetic properties of solids are then introduced, with emphasis on the frequency dependence of their real and imaginary components, and the consequences for wave propagation. Wave propagation at interfaces between dissimilar materials is considered, leading to derivation of Fresnel reflection and transmission coefficients. The need to guide electromagnetic waves of different frequency is discussed, and guiding by transmission lines, waveguides and optical fibers is introduced. Finally the electromagnetic fields generated by charges moving with uniform or oscillatory velocity are discussed. A number of interesting physical phenomena are considered that are important in a wide variety of areas and in many key technologies. This is a core subject for Physics programmes and is supported by Stage 3 tutorials and problems classes.

### Module Aims

The module aims to develop students' understanding of Maxwell's equations and their applications including some advanced topics. Specifically, students will get to the point where they can handle the fundamentals of fields due to moving charges and also to begin to explore the interaction of electromagnetic radiation with matter.

### Intended Learning Outcomes (ILOs)

A student who has passed this module should be able to:

• Module Specific Skills and Knowledge:
1. describe all the fundamental aspects of electromagnetism;
2. explain and solve problems involving the magnetic properties of materials;
3. explain and solve problems involving the dielectric properties of materials;
4. explain many aspects of the interaction of electromagnetic radiation with matter;
5. calculate the effect of such interactions using appropriate vector mathematics;
6. calculate the fields of moving charges;
7. solve problems requiring application of Maxwell's equations to a variety of situations as outlined in the syllabus below;
• Discipline Specific Skills and Knowledge:
1. use vector analysis to solve problems in science and engineering;
• Personal and Key Transferable / Employment Skills and Knowledge:
1. develop and present a coherent solution to a problem;
2. self-evaluate, check and correct solutions to problems.

### Syllabus Plan

1. Maxwell's Equations and Electromagnetic Waves
1. Maxwell's equations for the electromagnetic field.
2. Scalar and vector potentials.
3. The equation of continuity.
4. The wave equation and wave solutions to Maxwell equations.
5. Linear, circular and elliptical polarization states of a wave.
6. Energy of a wave and the Poynting theorem.
7. The electromagnetic stress tensor and the momentum of an electromagnetic field.
8. Gauge invariance and Gauge fixing. The Weyl, Lorenz and Coulomb gauges.
9. Covariance of Maxwell equations and Lorentz transforms
10. Field generated by a moving charge. The LiÃ©nard-Wiechert potentials.
11. Larmor formula
2. Electromagnetic materials
1. Polarization of dielectric materials. Multipole expansion.
2. Electric susceptibility and the displacement field.
3. Clausius-Mossotti relation.
4. Boundary conditions for the electric and the displacement fields.
5. Magnetic dipoles and magnetization.
6. Magnetic susceptibility and the magnetic field.
7. Boundary conditions for the magnetic induction and the magnetic fields.
8. Larmor precession.
9. Paramagnetism and Curie law.
10. Ferromagnetism, spontaneous magnetization, and magnetic hysteresis.
3. Electromagnetism at boundaries and guiding of waves
1. Waves in non-conductive materials.
2. Waves in conductive materials and the skin effect.
3. Dispersive media and the group velocity.
4. Fresnel coefficients and their consequences (Snell's law, Brewster angle, total internal reflection).
5. Reflection and transmission from a conductive material.
6. Transmission lines and impedance.
7. The telegrapher's equation.
8. The rectangular waveguide. TE and TM modes of a waveguide.
9. Optical fibres.
4. Wave propagation
1. Metals as plasmas, and the plasma frequency.
2. Plasma oscillations and plasmons.
3. Surface plasma polaritons.
4. Anisotropic media and the susceptibility tensor.
5. Biaxial and uniaxial media. Waveplates.
6. Double refraction.
7. Nonlinear media and the nonlinear polarization.
8. Nonlinear susceptibility.
9. Phase matching.
10. Wave diffraction. The Fresnel (paraxial) approximation and the Fraunhofer (far field) approximation.

### Learning and Teaching

#### Learning Activities and Teaching Methods

Description Study time KIS type
20×1-hour lectures 20 hours SLT
2×1-hour problems/revision classes 2 hours SLT
3×1-hour tutorials 3 hours SLT
5×6-hour self-study packages 30 hours GIS
4×4-hour problem sets 16 hours GIS
Reading, private study and revision 79 hours GIS

#### Assessment

Weight Form Size When ILOS assessed Feedback
0% Guided self-study 5×6-hour packages Fortnightly 1-10 Discussion in tutorials
0% 4 × Problems sets 4 hours per set Fortnightly 1-10 Solutions discussed in problems classes.
100% Final Examination 2 hours 30 minutes January 1-10 Mark via MyExeter, collective feedback via ELE and solutions.

### Resources

The following list is offered as an indication of the type & level of information that students are expected to consult. Further guidance will be provided by the Module Instructor(s).

Core text:

Supplementary texts:

ELE:

### Further Information

#### Prior Knowledge Requirements

Pre-requisite Modules Electromagnetism I (PHY2021) none

#### Re-assessment

Re-assessment is not available except when required by referral or deferral.

Original form of assessment Form of re-assessment ILOs re-assessed Time scale for re-assessment
Whole module Written examination (100%) 1-10 August/September assessment period

Notes: See Physics Assessment Conventions.

#### KIS Data Summary

Learning activities and teaching methods
SLT - scheduled learning & teaching activities 25 hrs
GIS - guided independent study 125 hrs
PLS - placement/study abroad 0 hrs
Total 150 hrs
Summative assessment
Coursework 0%
Written exams 100%
Practical exams 0%
Total 100%

#### Miscellaneous

 IoP Accreditation Checklist EM-04 Maxwell's equations and plane electromagnetic wave solution; Poynting vector SS-07 Magnetic properties of matter EM-06 Polarisation of waves and behaviour at plane interfaces Availability MPhys only Distance learning YES (see PHY3054) Keywords Physics; Maxwell's equations; Electromagnetic fields; Waves; Radiation; Properties of matter. Created 01-Oct-10 Revised 06-Aug-20