In the Dark

I was tidying up some old files earlier today and came across some old examination papers, including those I took for my final examinations in Part II of the Natural Sciences Tripos in 1985. There were six of these, in the space of three consecutive days…

I picked one of the questions to share here because it covers similar ground to my current (!) Advanced Electromagnetism module for final-year students in Maynooth. Sorry it’s a bit grubby!

It’s been a long time since I took my finals and I’d largely forgotten what the format was. The question above was taken from Paper II which consisted of nine questions altogether in three Sections, A (Solid State Physics), B (Statistical Physics) and C (Electromagnetism, from which Q9 above was taken; I think the course was actually called Electrodynamics & Relativity). The examination was 3 hours in duration and students were asked to answer four questions, including one from each Section. That means each question would be expected to take about 45 minutes.

Looking at the paper in general and the above question in particular, a number of things sprang to mind about differences between then in Cambridge and now in Maynooth:

1. Our theoretical physics papers in Maynooth are 2 hours in duration in which time students are to answer four questions, so that the questions are a bit shorter – 30 minutes each rather than 45.
2. Our papers are also on a single subject rather than a composite of several; we typically don’t offer the students choice; my Advanced Electromagnetism paper has four questions and students have to answer all four for full marks.
3. The questions on the old Tripos papers are less structured. There is no indication of the marks allocated to each part of the question in the question above.
4. As far as I can recall there was no formula booklet back in 1985, though there was a sheet of physical constants. My Advanced Electromagnetism examination this year comes with a couple of pages of useful formulae from vector calculus and key equations in EM theory. One might argue that the old Cambridge papers relied rather more on memory (especially when you take into account that everything was in the space of three days).
5. Back to Question 9, it is true that this along with the other Electromagnetism questions is at a similar level to what I have been teaching this Semester. If I recall correctly the relevant course in Cambridge was of 24 lectures, the same length as the course I’m teaching this year.
6. Students taking my course should know how to do both parts of Question 9 without too much difficulty.

On the final point, the easiest way to tackle this sort of problem is to do what the question says: determine the electric and magnetic potentials, derive the electric and magnetic fields from them, then work out the Poynting vector quantifying the energy flux. The part of this that survives in the far-field limit gives you the radiated power then – Bob’s your Uncle – the answer is basically the Larmor Formula which is ubiquitous in problems of this type. The case of an oscillating dipole is a standard derivation but this method works for any time-varying source, as long as you remember to include the retarded potentials if it’s not periodic.

Had I been writing this question for a modern exam I think I would at very least have ended the first part with “Show that the radiated power is…” and then given the formula, so that it could be used for the second part even if a student could not derive it.