A (Physics) Problem from the Past
I’ve been preparing material for my new 2nd year lecture course module The Physics of Fields and Flows, which starts next week. The idea of this is to put together some material on electromagnetism and fluid mechanics in a way that illustrates the connections between them as well as developing proficiency in the mathematics that underpins them, namely vector calculus. Anyway, in the course of putting together the notes and exercises it occurred to me to have a look at the stuff I was given when I was in the 2nd year at university, way back in 1983-4. When I opened the file I found this problem which caused me a great deal of trouble when I tried to do it all those years ago. It’s from an old Cambridge Part IB Advanced Physics paper. See what you can make of it..
(You can click on the image to make it larger…)
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In the Dark 
September 25, 2012 at 9:21 am
“The idea… is to put together some material on electromagnetism and fluid mechanics in a way that illustrates the connections between them…”
Sounds to me like you have been charged (geddit?) with giving a course on plasma physics by another name.
September 25, 2012 at 9:42 am
it has the potential to go that way…
September 25, 2012 at 11:20 am
I’d forgotten how much I like the word “corrugations”….
September 25, 2012 at 12:33 pm
You couldn’t safely assume that all candidates knew its meaning today.
Thinking further on the question, it just asks for the minimum-energy configuration among the subset specified by the formula, ie corrugations. There is no guarantee that this is a global minimum and that oscillations in both x and y (for example) are not more energetically favourable. A fuller analysis would not be too difficult, but does anybody know if experiments been done?
September 25, 2012 at 1:34 pm
That’s basically what flummoxed me at the time. I found the wording a big vague and wasn’t sure whether a more extensive analysis was required. In the end, though, I just did the easy thing…which I think after all was what was expected.
September 26, 2012 at 12:58 pm
I am completing my PhD in electrohydrodynamics and I can do the first part relatively easily (it’s just maths really) but the wording IS a bit vague, for example it dioesn;t say how many dimensions you’re working in for one thing. The other equation you are asked to derive looks a bit like a dispersion relation to me.
September 26, 2012 at 4:39 pm
… and by ‘the electric field’ they mean the uniform electric field there would be in the absence of corrugations?
September 26, 2012 at 4:49 pm
I think you’re meant to assume that based on the smallness of the corrugations…
October 1, 2012 at 10:35 am
Having eventually got this to come out, it does serve as an effective reminder of how question styles have evolved over the years. If we were to set something like this today, there would have to be a lot more detailed hints for the steps involved – it’s impressive how many distinct things this question asks you to do, and how many different bits of the physics toolkit it exercises. The modern version might be something like:
……………………………………………………………………………
Show that a vacuum electric field periodic in x can have a potential
V = f(z) cos kx + E_0 z,
by writing down Laplace’s equation. Solve the equation to show that f(z) propto exp(-kz), stating your assumed boundary condition at large z.
Now consider a wrinkled conducting liquid surface at z = h cos kx. Argue that this surface should be an equipotential, and hence that f(z) = -E_0 h exp(-kz) in order to satisfy this condition to lowest order in h. Apply perturbation theory to deduce the mean potential on the surface, and show that it is V=E_0 k h^2/2.
By Gauss’s theorem, E_0 is proportional to the surface charge density, sigma. By considering the build-up of the surface charge, show that the electrostatic potential energy of this configuration per unit area is sigma E_0 / 2 = epsilon_0 E_0^2 k h^2/4.
Now consider the potential energy of this configuration caused by doing work against gravity and surface tension, and show that the situation is unstable to the formation of surface corrugations if
epsilon_0 E^2 > rho g / k + gamma k,
where gamma is the surface tension [hint: the work done by the surface tension involves the increase in length of an element of the corrugated surface. This may be approximated by a fractional increase of sqrt(1+(dz/dx)^2) -1 simeq (dz/dx)^2/2].
………………………………………………………………………………
The main effect of writing all that down is to make the question look so complicated that most students would probably want to avoid it. Also, while the hints are desirable to keep students from going off the rails, they make it harder to allow for completely different methods of solution. All in all, I prefer the older terse style, since coming up with the shopping list of key steps for solving the question is probably a more intellectually important skill than being able to do the calculations and not mess them up by a factor 2. But I can’t see us being allowed to go back: if I tried to set such a question in the old style, the external examiner would have a fit.
October 1, 2012 at 11:08 am
So you see the question was actually right…
I hope you haven’t spent all week doing this!
October 1, 2012 at 2:49 pm
I’ve probably spent about an hour – how much was allowed in the actual 1983 paper?