Archive for Physics

In Praise of Natural Sciences

Posted in Biographical, Education with tags , , , , on April 24, 2016 by telescoper

The other day I was chatting with some students in the Department of Physics & Astronomy at the University of Sussex. One thing that came up was the fact that I’m basing the material for my Second Year Theoretical Physics module on the notes I took when I was a second-year undergraduate student at Cambridge over thirty years ago. I mentioned that to counter suggestions that are often made that the physics curriculum has been excessively “dumbed down” over the years. It may have been elsewhere, of course, but not on my watch. In fact, despite the misfortune of having me as a lecturer, many of the students in my class are picking up things far faster than I did when I was their age!

Anyway, that led to a general discussion of the changing nature of university education. One point was that in my day there weren’t any four-year “Integrated Masters” degrees, just plain three-year Bachelors. Teaching was therefore a bit more compressed than it is now, especially at Cambridge with its shorter teaching terms. We teach in two 12-week blocks here at Sussex. Week 11 of the Spring Term is about to start so we’re nearing the finishing line for this academic year and soon the examinations will be upon us.

The other thing that proved an interesting point of discussion was that the degree programme that I took was the Natural Sciences Tripos That meant that I did a very general first year comprising four different elements that could be chosen flexibly. I quickly settled on Physics, Chemistry and  Mathematics for Natural Sciences to reflect my A-level results but was struggling for the fourth. In the end I picked the one that seemed most like Physics, a course called Crystalline Materials. I didn’t like that at all, and wish I’d done some Biology instead – Biology of Cells and Biology of Organisms were both options – or even Geology, but I stuck with it for the first year.

Having to do such a wide range of subjects was very challenging. The timetable was densely packed and the pace was considerable. In the second year, however, I was able to focus on Mathematics and Physics and although it was still intense it was a bit more focussed. I ended up doing Theoretical Physics in my final year, including a theory project.

My best teacher at School, Dr Geoeff Swinden,  was a chemist (he had a doctorate in organic chemistry from Oxford University) and when I went to Cambridge I fully expected to specialise in Chemistry rather tha Physics. I loved the curly arrows and all that. But two things changed. One was that I found the Physics content of the first year far more interesting – and the lecturers and tutors far more inspiring – than Chemistry, and the other was that my considerable ineptitude at practical work made me doubt that I had a future in a chemistry laboratory. And so it came to pass that I switched allegiance to Physics, a decision I am very glad I made. It was only towards the end of my degree that I started to take Astrophysics seriously as a possible specialism, but that’s another story.

As we are now approaching examination season I’ve been dealing with some matters in my role as External Examiner for Natural Sciences (Physics) at Cambridge, a position I have held since last year. It’s certaintly extremely interesting to see things from the other side of the fence, thirty years on since my finals. In particular I was struck last year by how many senior physicists there are at Cambridge who actually came as undergraduates expecting, like I did, to do Chemistry but also then switched. No doubt some moved in the opposite direction too, but the point is that the system not only allowed this but positively encouraged it.

Looking back, I think  there were great educational advantages in delaying  the choice of speciality the way a Natural Sciences degree did. New students usually have very little idea how different the subject is at university compared to A-level, so it seems unfair to lock them into a programme from Year 1. Moreover – and this struck me particularly talking to current students last week – a Natural Sciences programme might well prove a way of addressing the gender imbalance in physics by allowing female students (who might have been put off Physics at school) to gravitate towards it. Only 20% of the students who take Physics A-level are female, and that’s roughly the same mix that we find in the undergraduate population. How many more might opt for Physics after taking a general first year?

Another advantage of this kind of degree is that it gives scientists a good grounding in  a range of subjects. In the long run this could encourage greater levels of interdisciplinary thinking. This is important, since some of the most exciting areas of physics research lie at the interfaces with, e.g. chemistry and biology. Unfortunately, adminstrative structures often create barriers that deter such cross-disciplinary activities.

 

 

“British physics” – A Lesson from History

Posted in History, Politics, Science Politics, The Universe and Stuff with tags , , , , , , , , , , , , , , , , on March 13, 2016 by telescoper

The other day I came across the following tweet

The link is to an excellent piece about the history of European science which I recommend reading; as I do with this one.

I won’t pretend to be a historian but I can’t resist a comment from my perspective as a physicist. I am currently teaching a course module called Theoretical Physics which brings together some fairly advanced mathematical techniques and applies them to (mainly classical) physics problems. It’s not a course on the history of physics, but thenever I mention a new method or theorem I always try to say something about the person who gave it its name. In the course of teaching this module, therefore, I have compiled a set of short biographical notes about the people behind the rise of theoretical physics (mainly in the 19th Century). I won’t include them here – it would take too long – but a list  makes the point well enough: Laplace, Poisson,  Lagrange, Hamilton, Euler, Cauchy, Riemann, Biot, Savart, d’Alembert, Ampère, Einstein, Lorentz, Helmholtz, Gauss, etc etc.

There are a few British names too  including the Englishmen Newton and Faraday and the Scot Maxwell. Hamilton, by the way, was Irish. Another Englishman, George Green, crops up quite prominently too, for reasons which I will expand upon below.

Sir Isaac Newton is undoubtedly one of the great figures in the History of Science, and it is hard to imagine how physics might have developed without him, but the fact of the matter is that for a hundred years after his death in 1727 the vast majority of significant developments in physics took place not in Britain but in Continental Europe. It’s no exaggeration to say that British physics was moribund during this period and it took the remarkable self-taught mathematician George Green to breath new life into it.
I quote from History of the Theories of the Aether and Electricity (Whittaker, 1951) :

The century which elapsed between the death of Newton and the scientific activity of Green was the darkest in the history of (Cambridge) University. It is true that (Henry) Cavendish and (Thomas) Young were educated at Cambridge; but they, after taking their undergraduate courses, removed to London. In the entire period the only natural philosopher of distinction was (John) Michell; and for some reason which at this distance of time it is difficult to understand fully, Michell’s researches seem to have attracted little or no attention among his collegiate contemporaries and successors, who silently acquiesced when his discoveries were attributed to others, and allowed his name to perish entirely from the Cambridge tradition.

I wasn’t aware of this analysis previously, but it re-iterates something I have posted about before. It stresses the enormous historical importance of British mathematician and physicist George Green, who lived from 1793 until 1841, and who left a substantial legacy for modern theoretical physicists, in Green’s theorems and Green’s functions; he is also credited as being the first person to use the word “potential” in electrostatics.

Green was the son of a Nottingham miller who, amazingly, taught himself mathematics and did most of his best work, especially his remarkable Essay on the Application of mathematical Analysis to the theories of Electricity and Magnetism (1828) before starting his studies as an undergraduate at the University of Cambridge ,which he did at the age of 30. Lacking independent finance, Green could not go to University until his father died, whereupon he leased out the mill he inherited to pay for his studies.

Extremely unusually for English mathematicians of his time, Green taught himself from books that were published in France. This gave him a huge advantage over his national contemporaries in that he learned the form of differential calculus that originated with Leibniz, which was far more elegant than that devised by Isaac Newton (which was called the method of fluxions). Whittaker remarks upon this:

Green undoubtedly received his own early inspiration from . . . (the great French analysts), chiefly from Poisson; but in clearness of physical insight and conciseness of exposition he far excelled his masters; and the slight volume of his collected papers has to this day a charm which is wanting in their voluminous writings.

Great scientist though he was, Newton’s influence on the development of physics in Britain was not entirely positive, as the above quote makes clear. Newton was held in such awe, especially in Cambridge, that his inferior mathematical approach was deemed to be the “right” way to do calculus and generations of scholars were forced to use it. This held back British science until the use of fluxions was phased out. Green himself was forced to learn fluxions when he went as an undergraduate to Cambridge despite having already learned the better method.

Unfortunately, Green’s great pre-Cambridge work on mathematical physics didn’t reach wide circulation in the United Kingdom until after his death. William Thomson, later Lord Kelvin, found a copy of Green’s Essay in 1845 and promoted it widely as a work of fundamental importance. This contributed to the eventual emergence of British theoretical physics from the shadow cast by Isaac Newton. This renaissance reached one of its heights just a few years later with the publication a fully unified theory of electricity and magnetism by James Clerk Maxwell.

In a very real sense it was Green’s work that led to the resurgence of British physics during the later stages of the 19th Century, and it was the fact that he taught himself from French books that enabled him to bypass the insular attitudes of British physicists of the time. No physicist who has taken even a casual look at the history of their subject could possibly deny the immense importance of mainland Europe in providing its theoretical foundations.

Of course science has changed in the last two hundred years, but I believe that we can still learn an important lesson from this particular bit of history. Science moves forward when scientists engage with ideas and information from as wide a range of sources as possible, and it stagnates when it retreats into blinkered insularity. The European Union provides all scientific disciplines with a framework within which scientists can move freely and form transnational collaborations for the mutual benefit of all. We need more of this, not less. And not just in science.

 

 

 

Preparing for a PhD Interview in Physics

Posted in Biographical, Education, The Universe and Stuff with tags , , , on February 1, 2016 by telescoper

The other day I was chatting to a group of our 4th-year MPhys students about the process for applying  (and hopefully being interviewed) for a PhD. This is the time when students in the UK have started to apply and are awaiting decisions on whether they have to go for an interview. Final decisions are usually made by the end of March so those with interviews have a busy couple of months coming up.

I actually quite enjoy doing PhD interviews, because that involves giving excellent young scientists their first step on the ladder towards a research career. I’m sure it’s not so pleasant for the candidates though. Nerves sometimes get the better of the students in these interviews, but experienced interviewers can calibrate for that. And if you’re nervous, it means that you care…

Anyone reading this who is nervous about doing a PhD interview (or has experienced nerves in one they’ve already had) might reflect on my experience when I was called to interview for a PhD place in Astronomy at the University of Manchester way back in 1985. I was very nervous before that, and arrived very early for my grilling. I was told to wait in a sort of ante-room as the previous interview had only just started. I started to read a textbook I had brought with me. About five minutes later, the door of the interview room opened and the interviewers, Franz Kahn and John Dyson, both of whom are sadly no longer with us, carried out the unconscious body of the previous candidate. It turned out that, after a couple of friendly preliminary questions, the two Professors had handed the candidate a piece of chalk and told him to go to the blackboard  to work something out, at which point said candidate had fainted. When it was my turn to be handed the chalk I toyed with the idea of staging a mock swoon, but resisted the temptation.

The question, in case you’re interested, was to estimate the angle through which light  is deflected by the Sun’s gravity. I hadn’t done any general relativity in my undergraduate degree, so just did it by dimensional analysis which is easy because an angle is dimensionless. That gets you within a factor of a two of the correct answer which, in those days, was pretty goood going for cosmology. That seemed to go down well and they offered me a place … which I turned down in favour of Sussex.

In those days, before detailed information about research in University departments was available online, the interview generally consisted of a discussion of the various projects available and a few odd questions about Physics (and possible Astronomy) to see if the candidate was able to think on their feet (i.e. without fainting).

Nowadays it’s a bit different. You can still expect a bit of questioning about undergraduate material but that is normally preceded by the chance to talk about your final-year project. One reason for that is that selectors are interested in project work because it can provide evidence of an aptitude for research. The other is simply that it gives the candidate a chance to get over any initial nerves by talking about something that they hopefully know well, as they will have been working on it for some time.

My first piece advice for students who have been offered an interview, therefore, is to prepare a short (~10 minute) verbal summary of your project work so you’re not wrong-footed if asked to talk about it.

Students nowadays are also expected to know a bit more about the thesis topic in advance, so my second tip is to  read up a bit of background so you can talk reasonably intelligently about the proposed research. If, for example, you have decided to work on Dark Energy (as many seem to these days), you won’t come across very well if you don’t know what the main issues are. What’s the observational evidence? What kind of theories are there? What are the open questions? Same goes for other fields. It also will do no harm if you read a couple of recent papers by your prospective supervisor, for reasons of flattery if nothing else.

Anyway, I think those are the two main things. If anyone has other advice to offer prospective PhD students, please feel free to add via the comments box.

 

 

 

Why is General Relativity so difficult?

Posted in The Universe and Stuff with tags , , on November 26, 2015 by telescoper

Just a brief post following yesterday’s centenary of General Relativity, after which somebody asked me what is so difficult about the theory. I had two answers to that, one mathematical and one conceptual.

einstein-equation1

The Field Equations of General Relativity are written above. In the notation used they don’t look all that scary, but they are more complicated than they look. For a start it looks like there is only one equation, but the subscripts μ and ν can each take four values (usually 0, 1, 2 or 3), each value standing for one of the dimensions of four-dimensional space time. It therefore looks likes there are actually 16 equations. However, the equations are the same if you swap μ  and ν around. This means that there are “only” ten independent equations. The terms on the left hand side are the components of the Einstein Tensor which expresses the effect of gravity through the curvature of space time and the right hand side describes the energy and momentum of “stuff”, prefaced by some familiar constants.

The Einstein Tensor is made up of lots of partial derivatives of another tensor called the metric tensor (which describes the geometry of space time), which relates, through the Field Equations, to how matter and energy are distributed and how these components move and interact. The ten equations that need to be solved simultaneously are second-order non-linear partial different equations. This is to be compared with the case of Newtonian gravity in which only ordinary different equations are involved.

Problems in Newtonian mechanics can be difficult enough to solve but the much greater mathematical complexity in General Relativity means that problems in GR can only be solved in cases of very special symmetry, in which the number of independent equations can be reduced dramatically.

So that’s why it’s difficult mathematically. As for the conceptual problem it’s that most people (I think) consider “space” to be “what’s in between the matter” which seems like it must be “nothing”. But how can “nothing” possess an attribute like curvature? This leads you to conclude that space is much more than nothing. But it’s not a form of matter. So what is it? This chain of thought often leads people to think of space as being like the Ether, but that’s not right either. Hmm.

I tend to avoid this problem by not trying to think about space or space-time at all, and instead think only in terms of particle trajectories or ligh rays and how matter and energy affect them. But that’s because I’m lazy and only have a small brain…

 

 

Power from Wind Problems

Posted in Cute Problems with tags , , on November 21, 2015 by telescoper

It has been a little while since I posted anything in the Cute Problems category so since today is quite a windy day I thought I’d give you this one, which leads to an extimate of the maximum power that can, in theory, be extracted from wind a windmill.

Assume that the wind far upstream and down-stream of the windmill has speed V and αV respectively, with 0≤α≤1, and let the wind speed at the sails of the windmill, which sweep out an area A, be v.

Now for the problems

(i) By equating the power absorbed by the mill to the rate of loss of kinetic energy of the wind, show that v/V=½(1+α).

(ii) Show that the power obtainable is proportional to AρV3 where ρ is the density of air.

(iii)  Show that the maximum power that can be extracted is 16/27 of the power available initially in the wind.

The final result is known as Betz’s Law and it works for any form of turbine, not just a windmill.

 

 

 

An Open Letter to the Times Higher World University Rankers

Posted in Education, The Universe and Stuff with tags , , , , , , , , on October 5, 2015 by telescoper

Dear Rankers,

Having perused your latest set of league tables along with the published methodology, a couple of things puzzle me.

First, I note that you have made significant changes to your methodology for combining metrics this year. How, then, can you justify making statements such as

US continues to lose its grip as institutions in Europe up their game

when it appears that any changes could well be explained not by changes in performance, as gauged by the metrics you use,  but in the way they are combined?

I assume, as intelligent and responsible people, that you did the obvious test for this effect, i.e. to construct a parallel set of league tables, with this year’s input data but last year’s methodology, which would make it easy to isolate changes in methodology from changes in the performance indicators.  Your failure to publish such a set, to illustrate how seriously your readers should take statements such as that quoted above, must then simply have been an oversight. Had you deliberately witheld evidence of the unreliability of your conclusions you would have left yourselves open to an accusation of gross dishonesty, which I am sure would be unfair.

Happily, however, there is a very easy way to allay the fears of the global university community that the world rankings are being manipulated: all you need to do is publish a set of league tables using the 2014 methodology and the 2015 data. Any difference between this table and the one you published would then simply be an artefact and the new ranking can be ignored. I’m sure you are as anxious as anyone else to prove that the changes this year are not simply artificially-induced “churn”, and I look forward to seeing the results of this straightforward calculation published in the Times Higher as soon as possible.

Second, I notice that one of the changes to your methodology is explained thus

This year we have removed the very small number of papers (649) with more than 1,000 authors from the citations indicator.

You are presumably aware that this primarily affects papers relating to experimental particle physics, which is mostly conducted through large international collaborations (chiefly, but not exclusively, based at CERN). This change at a stroke renders such fundamental scientific breakthroughs as the discovery of the Higgs Boson completely worthless. This is a strange thing to do because this is exactly the type of research that inspires  prospective students to study physics, as well as being direct measures in themselves of the global standing of a University.

My current institution, the University of Sussex, is heavily involved in experiments at CERN. For example, Dr Iacopo Vivarelli has just been appointed coordinator of all supersymmetry searches using the ATLAS experiment on the Large Hadron Collider. This involvement demonstrates the international standing of our excellent Experimental Particle Physics group, but if evidence of supersymmetry is found at the LHC your methodology will simply ignore it. A similar fate will also befall any experiment that requires large international collaborations: searches for dark matter, dark energy, and gravitational waves to name but three, all exciting and inspiring scientific adventures that you regard as unworthy of any recognition at all but which draw students in large numbers into participating departments.

Your decision to downgrade collaborative research to zero is not only strange but also extremely dangerous, for it tells university managers that participating in world-leading collaborative research will jeopardise their rankings. How can you justify such a deliberate and premeditated attack on collaborative science? Surely it is exactly the sort of thing you should be rewarding? Physics departments not participating in such research are the ones that should be downgraded!

Your answer might be that excluding “superpapers” only damages the rankings of smaller universities because might owe a larger fraction of their total citation count to collaborative work. Well, so what if this is true? It’s not a reason for excluding them. Perhaps small universities are better anyway, especially when they emphasize small group teaching and provide opportunities for students to engage in learning that’s led by cutting-edge research. Or perhaps you have decided otherwise and have changed your methodology to confirm your prejudice…

I look forward to seeing your answers to the above questions through the comments box or elsewhere – though you have ignored my several attempts to raise these questions via social media. I also look forward to seeing you correct your error of omission by demonstrating – by the means described above – what  changes in league table positions are by your design rather than any change in performance. If it turns out that the former is the case, as I think it will, at least your own journal provides you with a platform from which you can apologize to the global academic community for wasting their time.

Yours sincerely,

Telescoper

How to Solve Physics Problems

Posted in Cute Problems, Education with tags , , , , , , on September 18, 2015 by telescoper

It’s Friday afternoon at the end of Induction Week here at the University of Sussex. By way of preparation for lectures proper – which start next Monday – I gave a lecture today to all the new students in Physics during which I gave some tips about how to tackle physics problems, not only in terms of how to solve them but also how to present the answer in an appropriate way.

Richard-Feynman-cornellI began with Richard Feynman’s formula (the geezer in the above picture) for solving physics problems:

  1. Write down the problem.
  2. Think very hard.
  3. Write down the answer.

That may seem either arrogant or facetious, or just a bit of a joke, but that’s really just the middle bit. Feynman’s advice on points 1 and 3 is absolutely spot on and worth repeating many times to an audience of physics students.

I’m a throwback to an older style of school education when the approach to solving unseen mathematical or scientific problems was emphasized much more than it is now. Nowadays much more detailed instructions are given in School examinations than in my day, often to the extent that students  are only required to fill in blanks in a solution that has already been mapped out.

I find that many, particularly first-year, students struggle when confronted with a problem with nothing but a blank sheet of paper to write the solution on. The biggest problem we face in physics education, in my view, is not the lack of mathematical skill or background scientific knowledge needed to perform calculations, but a lack of experience of how to set the problem up in the first place and a consequent uncertainty about, or even fear of, how to start. I call this “blank paper syndrome”.

In this context, Feynman’s advice is the key to the first step of solving a problem. When I give tips to students I usually make the first step a bit more general, however. It’s important to read the question too. The key point is to write down the information given in the question and then try to think how it might be connected to the answer. To start with, define appropriate symbols and draw relevant diagrams. Also write down what you’re expected to prove or calculate and what physics might relate that to the information given.

The middle step is more difficult and often relies on flair or the ability to engage in lateral thinking, which some people do more easily than others, but that does not mean it can’t be nurtured.  The key part is to look at what you wrote down in the first step, and then apply your little grey cells to teasing out – with the aid of your physics knowledge – things that can lead you to the answer, perhaps via some intermediate quantities not given directly in the question. This is the part where some students get stuck and what one often finds is an impenetrable jumble of mathematical symbols  swirling around randomly on the page. The process of problem solving is not always linear. Sometimes it helps to work back a little from the answer you are expected to prove before you can return to the beginning and find a way forward.

Everyone gets stuck sometimes, but you can do yourself a big favour by at least putting some words in amongst the algebra to explain what it is you were attempting to do. That way, even if you get it wrong, you can be given some credit for having an idea of what direction you were thinking of travelling.

The last of Feynman’s steps  is also important. I lost count of the coursework attempts I marked this week in which the student got almost to the end, but didn’t finish with a clear statement of the answer to the question posed and just left a formula dangling.  Perhaps it’s because the students might have forgotten what they started out trying to do, but it seems very curious to me to get so far into a solution without making absolutely sure you score the points.  IHaving done all the hard work, you should learn to savour the finale in which you write “Therefore the answer is…” or “This proves the required result”. Scripts that don’t do this are like detective stories missing the last few pages in which the name of the murderer is finally revealed.

So, putting all these together, here are the three tips I gave to my undergraduate students this morning.

  1. Read the question! Some students give solutions to problems other than that which is posed. Make sure you read the question carefully. A good habit to get into is first to translate everything given in the question into mathematical form and define any variables you need right at the outset. Also drawing a diagram helps a lot in visualizing the situation, especially helping to elucidate any relevant symmetries.
  2. Remember to explain your reasoning when doing a mathematical solution. Sometimes it is very difficult to understand what students are trying to do from the maths alone, which makes it difficult to give partial credit if they are trying to the right thing but just make, e.g., a sign error.
  3.  Finish your solution appropriately by stating the answer clearly (and, where relevant, in correct units). Do not let your solution fizzle out – make sure the marker knows you have reached the end and that you have done what was requested. In other words, finish with a flourish!

There are other tips I might add – such as checking answers by doing the numerical parts at least twice on your calculator and thinking about whether the order-of-magnitude of the answer is physically reasonable – but these are minor compared to the overall strategy.

And another thing is not to be discouraged if you find physics problems difficult. Never give up without a fight. It’s only by trying difficult things that you can improve your ability by learning from your mistakes. It’s not the job of a physics lecturer to make physics seem easy but to encourage you to believe that you can do things that are difficult.

To illustrate the advice I’ve given I used this problem, which I leave as an exercise to the reader. It is a slightly amended version the first physics problem I was set as tutorial work when I began my undergraduate studies way back in 1982. I think it illustrates very well the points I have made above, and it doesn’t require any complicated mathematics – not even calculus! See how you get on…

problem

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