Please make sure you have revised the general results concerning solutions to Laplace's equation by watching Mini-Lecture 3 Video before Monday's lecture in Week 2!

In Week 2 we shall continue our revision of Electrostatics and Magnetostatics.

In addition, you will be encouraged to revise vector calculus, which is an important tool for understanding Electrodynamics.

There are two interactive sessions each week: in the Simon lecture theatre A (Monday) and Stopford lecture theatre 1 (Wednesday).

There are three mini-lectures, with associated small exercises and lecture notes provided.

Solutions to Laplace's equation for a particular physical situation requiring spherical polar coordinates will be discussed in Live-Lecture 4, which will cover approximately the same material as Mini-Lecture 4.

In Wednesday's informal interactive session we'll consider a mathematical detail concerning multipole expansions in electrostatics, as well as some worked examples for vector calculus.

We shall revise Multipole Expansions in Electrostatics (Mini-Lecture 5) and the basics of Magnetostatics (Mini-Lecture 6).

N.B. The bulk of the material presented in Mini-Lectures 5 and 6 will not be covered in a live lecture. Therefore you should follow Mini-Lectures 5 and 6 online.

Please make sure you have revised the general results concerning solutions to Laplace's equation by watching Mini-Lecture 3 Video before Monday's lecture in Week 2!

⚬ Monday 15:00-16:00 - Live Lecture 4: Solutions to Laplace's Equation (example in spherical polar coordinates)

This live lecture covers approximately the same material as Mini-Lecture 4.

Lecture 4 Podcast ............. Lecture notes written to visualiser ............. Exercises from lecture ............. Answers

⚬ Wednesday 9:00-10:00 - Interactive session: a mathematical detail concerning multipole expansions in electrostatics, as well as some worked examples for vector calculus

Podcast ............. Lecture notes written to visualiser

N.B. In preparation for this session please take a look at these Exercises: Revision of vector calculus

In solving these problems you may find the following very brief preliminary remarks on using index notation for vector calculus useful. In addition, here is a somewhat more extensive introduction to using index notation for vector calculus by John Crimaldi of University of Colorado, Boulder.

By clicking on the links given below you will be able to access the video of each mini-lecture, together with the associated small exercises and lecture notes.

Mini-Lecture 4: Solutions to Laplace's Equation (example in spherical polar coordinates)

N.B. The material presented in Mini-Lecture 4 will be covered also in a live lecture.

Video ............. Lecture notes written to visualiser ............. Exercise from mini-lecture ............. Answers

Mini-Lecture 5: Multipole Expansions in Electrostatics

N.B. The bulk of the revision material presented in Mini-Lecture 5 will not be covered in a live lecture. Therefore you should follow Mini-Lecture 5 online.

Video ............. Lecture notes written to visualiser ............. Exercises from mini-lecture

From Wikipedia, here is an alterative (more relevant) picture of the equipotentials for a dipole. In this picture the separation, a, of the two charges is very small and so the equipotentials correspond to the formula we derived in the lecture, which correspond to the case R >> a.

N.B. The revision material presented in Mini-Lecture 6 will not be covered in a live lecture. Therefore you should follow Mini-Lecture 6 online.

IMPORTANT. Please follow Mini-Lecture 6 online BEFORE attending the Monday live lecture in Week 3!

Video ............. Lecture notes written to visualiser

Mini-Lecture 4: Solutions to Laplace's Equation (example in spherical polar coordinates)

Mini-Lecture 5: Multipole Expansions in Electrostatics

Mini-Lecture 6: Magnetostatics

By way of getting an idea of the basic mathematical tools we shall be using it might be useful to take a quick look at:

A summary of useful formulae concerning vector calculus, coordinate systems, etc

Vector Calculus

D.J. Griffiths, Introduction to Electrodynamics: Chapter 1.

Solutions to Laplace's equation

It would be very useful to revise the sections on Laplace's equation, separation of variables, and Legendre polynonmials from your second year course `Maths of Fields and Waves (PHYS20701)'

D.J. Griffiths, Introduction to Electrodynamics: Chapter 3.1 and 3.3.

Multipole Expansions in Electrostatics

D.J. Griffiths, Introduction to Electrodynamics: Chapter 3.4.

Magnetostatics

D.J. Griffiths, Introduction to Electrodynamics: Chapter 5.1, 5.2, 5.3.

Topics in Magnetostatics

D.J. Griffiths, Introduction to Electrodynamics: Chapter 5.

In Mini-Lecture 7a, a general derivation of the curl of B is given starting from the vector potential. For the more "traditional" alternative (but rather long) general derivation of the curl of B starting from the Biot-Savart Law see, e.g.,:

- D.J. Griffiths, Introduction to Electrodynamics: Section 5.3.2.

- J.D. Jackson: Section 5.3.

Revision of the Basics in Electrodynamics

D.J. Griffiths, Introduction to Electrodynamics: Chapter 7.

The Wave Equations for the Potentials and their Solutions

D.J. Griffiths, Introduction to Electrodynamics: Section 10.1.3

Note to try to avoid a potential source of confusion: In the Cambridge University Press (CUP) version of the 4th Edition of D.J. Griffiths, Introduction to Electrodynamics, the chapter entitled "Potentials and Fields" is Chapter 10. Unfortunately, in an older (Pearson) version of the 4th Edition it is chapter 12! I shall refer to the CUP chapter numbers, but if in your edition of Griffiths, chapter 10 is entitled "Radiation" then try looking in chapter 12!

M.A. Heald and J.B. Marion, Classical Electromagnetic Radiation (3rd edition): Section 4.5

Solutions to the Wave Equations for the Potentials (Mini-Lecture 9b)

M.A. Heald and J.B. Marion, Classical Electromagnetic Radiation (3rd edition): Section 8.1 gives a proof along the lines discussed in lecture 9(b)

D.J. Griffiths, Introduction to Electrodynamics: Section 10.2.1, gives a proof that is more "straight-forward", but is more "long-winded".

Special Relativity and 4-vectors: Revision of Physics and Notation

Jeff Forshaw and Gavin Smith, Dynamics and Relativity: Chapters 5, 6, 7, 11 and 12.

D.J. Griffiths, Introduction to Electrodynamics: Chapter 12. (N.B. if in your edition of Griffiths, chapter 12 in entitled "Potentials and Fields" then try looking in chapter 11!)

N.B. The book by Griffiths also has many problems that would form useful revision exercises.