Back to the day job, teaching Advanced Electromagnetism, I put up these two nice tutorial problems about the Lorentz transformation of Electric and Magnetic fields, as a prelude to doing the fully covariant formulation of the Maxwell equations. You might like to have a go at these exercises:
So today I gave my first lecture of the new Semester, it being a gentle reintroduction to Maxwell’s equations for a 4th-year class on Advanced Electromagnetism. They have seen these equations before but it doesn’t do any harm to spend a bit of time refreshing the memory. In what follows I do some potential theory, applications to electrostatics (method of images, multipole expansions, use of the complex potential, etc), dielectric materials and polarization, magnetostatics, relativistic formalism of electromagnetism, gauge invariance, electromagnetic radiation and energy transport, and (if time) plasma physics (if time).
When I taught this module last year I did it all remotely from home – using the blackboard shown above – but this year until further notice I’ll be doing it in person in an actual lecture theatre, though I will be recording the lectures in case any students wish to look at them again for revision, etc, and webcasting them for any students unable to attend on campus. I know there are differences of opinion on this, but I think recording of lectures should become routine practice – as it is in all UK universities I’m aware of – but that is difficult here in Maynooth because the equipment available is inadequate (by which I mean virtually non-existent). Let’s hope the necessary investment will be made at some point.
Follow @telescoperI chose a different route for today’s post-prandial exercise around Maynooth University’s South Campus, and thus came across this attractive little tree-lined path:
At the other end lies the historic cemetery of Maynooth College where, among many other things, I found this memorial:
This gives me an excuse to rehash an old one about the person whose memorial it is.
Father Nicholas Joseph Callan (or, more formally, The Reverend Professor Nicholas Joseph Callan) was born in County Louth in 1799 went to the seminary of St Patrick’s College, Maynooth, in 1816 on order to train as a priest. During his time as a seminarian Callan studied ‘Natural Philosophy’ and became interested in experiments involving electricity. In 1823 Callan was ordained as a priest, and went to Rome in 1826 to obtain his doctorate in Divinity. At the time Italy was a centre for research into electricity and here Callan became familiar with the work of the Italian physicist Alessandro Volta who had developed the world’s first battery. Callan returned to Maynooth where he was made chair of Natural Philosophy, a post he would hold until his death in 1864.
Callan is most famous for inventing the induction coil (in 1836). By connecting two copper wire coils to a battery and electromagnet and then interrupting the current he was able to generate much larger voltages than could be obtained from batteries alone. His 1837 version that used a clock mechanism to interrupt the current 20 times a second is estimated to have produced 60,000 volts – the largest artificially generated charge at that time. It is said that his induction coil could produce sparks 15″ long, which must have been fun to watch.
Callan’s biggest induction coil, unfinished at the time of his death, can be found in the National Science Museum of Ireland (which is in Maynooth). This was one of the largest in the world at the time. The iron core is 109 cm long. The secondary windings are 53 cm in diameter and consist of about 50 km of iron wire insulated with beeswax. They were made in three separate rings separated by air gaps, so wires carrying large voltage differences would not lie adjacent to each other, reducing the risk of the insulation breaking down. At the left end is a vibrating mercury ‘contact breaker’ in the primary circuit, actuated by the magnetic field in the primary, which interrupted the primary current to generate potentials of over 200,000 volts.
Sadly Callan’s work was forgotten for quite a period after his death – experimental electromagnetism was not a priority for St Patrick’s College at this time – for which reason the invention of the induction coil has often been attributed to Heinrich Ruhmkorff who made his first device (independently) about 15 years after Callan. More recently, however, Callan’s achievements have been more widely recognized and in 2000 the Irish government issued a stamp in his honour.
The Callan Building
Nicholas Callan was laid to rest in the College Cemetery at Maynooth in 1864. The Callan Building (above) on the North Campus of the present-day Maynooth University is named in his honour, as of course is Callan Hall, on the South Campus.
P.S. The first interment in the Cemetery took place in 1817, 200 years before I started work in Maynooth.
I’ve had a busy day doing various things that took me to various places on the Maynooth University campus, culminating in my first visit to the Department of Mathematics and Statistics (which is based in Logic House) to give a colloquium. On my travels I passed this sign, which I hadn’t noticed before:
Since I’ve been too busy to write a post today I thought I’d rehash an old one about the person named in the plaque, which is above Callan Hall (on the South Campus)
Father Nicholas Callan (or, more formally, The Reverend Professor Nicholas Joseph Callan) was born in County Louth in 1799 went to the seminary of St Patrick’s College, Maynooth, in 1816 to train as a priest. During his time as a seminarian Callan studied ‘Natural Philosophy’ and became interested in experiments involving electricity. In 1823 Callan was ordained as a priest, and went to Rome in 1826 to obtain his doctorate in Divinity. At the time Italy was a centre for research into electricity and here Callan became familiar with the work of the Italian physicist Alessandro Volta who had developed the world’s first battery. Callan returned to Maynooth where he was made chair of Natural Philosophy, a post he would hold until his death in 1864.
Callan is most famous for inventing the induction coil (in 1836). By connecting two copper wire coils to a battery and electromagnet and then interrupting the current he was able to generate much larger voltages than could be obtained from batteries alone. His 1837 version that used a clock mechanism to interrupt the current 20 times a second is estimated to have produced 60,000 volts – the largest artificially generated charge at that time. It is said that his induction coil could produce sparks 15″ long, which must have been fun to watch.
Callan’s biggest induction coil, unfinished at the time of his death, can be found in the National Science Museum of Ireland (which is in Maynooth). This was one of the largest in the world at the time. The iron core is 109 cm long. The secondary windings are 53 cm in diameter and consist of about 50 km of iron wire insulated with beeswax. They were made in three separate rings separated by air gaps, so wires carrying large voltage differences would not lie adjacent to each other, reducing the risk of the insulation breaking down. At the left end is a vibrating mercury ‘contact breaker’ in the primary circuit, actuated by the magnetic field in the primary, which interrupted the primary current to generate potentials of over 200,000 volts.
Sadly Callan’s work was forgotten for quite a period after his death – experimental electromagnetism was not a priority for St Patrick’s College at this time – for which reason the invention of the induction coil has often been attributed to Heinrich Ruhmkorff who made his first device (independently) about 15 years after Callan. More recently, however, Callan’s achievements have been more widely recognized and in 2000 the Irish government issued a stamp in his honour.
The Callan Building
Nicholas Callan was laid to rest in the College Cemetery at Maynooth in 1864. The Callan Building (above) on the North Campus of the present-day Maynooth University is named in his honour, as of course is Callan Hall which I mentioned at the start.
Historically speaking, Maynooth is more strongly associated with theology than with science but I thought I’d mention here one famous pioneering physicist, who happened also to be a Roman Catholic priest, who spent his working life in these parts.
Father Nicholas Callan (or, more formally, The Reverend Professor Nicholas Joseph Callan) was born in County Louth in 1799 went to the seminary of St Patrick’s College, Maynooth, in 1816 to train as a priest. During his time as a seminarian Callan studied ‘Natural Philosophy’ and became interested in experiments involving electricity. In 1823 Callan was ordained as a priest, and went to Rome in 1826 to obtain his doctorate in Divinity. At the time Italy was a centre for research into electricity and here Callan became familiar with the work of the Italian physicist Alessandro Volta who had developed the world’s first battery. Callan returned to Maynooth where he was made chair of Natural Philosophy, a post he would hold until his death in 1864.
Callan is most famous for inventing the induction coil (in 1836). By connecting two copper wire coils to a battery and electromagnet and then interrupting the current he was able to generate much larger voltages than could be obtained from batteries alone. His 1837 version that used a clock mechanism to interrupt the current 20 times a second is estimated to have produced 60,000 volts – the largest artificially generated charge at that time. It is said that his induction coil could produce sparks 15″ long, which must have been fun to watch.
Callan’s biggest induction coil, unfinished at the time of his death, can be found in the National Science Museum of Ireland (which is in Maynooth). This was one of the largest in the world at the time. The iron core is 109 cm long. The secondary windings are 53 cm in diameter and consist of about 50 km of iron wire insulated with beeswax. They were made in three separate rings separated by air gaps, so wires carrying large voltage differences would not lie adjacent to each other, reducing the risk of the insulation breaking down. At the left end is a vibrating mercury ‘contact breaker’ in the primary circuit, actuated by the magnetic field in the primary, which interrupted the primary current to generate potentials of over 200,000 volts.
Sadly Callan’s work was forgotten for quite a period after his death – experimental electromagnetism was not a priority for St Patrick’s College at this time – for which reason the invention of the induction coil has often been attributed to Heinrich Ruhmkorff who made his first device (independently) about 15 years after Callan. More recently, however, Callan’s achievements have been more widely recognized and in 2000 the Irish government issued a stamp in his honour.
The Callan Building
Nicholas Callan was laid to rest in the College Cemetery at Maynooth in 1864. The Callan Building (above) on the North Campus of the present-day Maynooth University is named in his honour.
On Wednesday this week I spent a very enjoyable few hours in London attending the Inaugural Lecture of Professor Alan Heavens at South Kensington Technical College Imperial College, London. It was a very good lecture indeed, not only for its scientific content but also for the plentiful touches of droll humour in which Alan specialises. It was also followed by a nice drinks reception and buffet. The talk was entitled Cosmology in the Dark, so naturally I had to mention it on this blog!
At the end of the lecture, the vote of thanks was delivered in typically effervescent style by the ebullient Prof. Malcolm Longair who actually supervised Alan’s undergraduate project at the Cavendish laboratory way back in 1980, if I recall the date correctly. In his speech, Malcolm referred to the following quote from History of the Theories of the Aether and Electricity (Whittaker, 1951) which he was kind enough to send me when I asked by email:
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 which 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.
But as to the possible reason for the lack of recognition for John Michell who was clearly an important figure in his own right (he was the person who first developed the concept of a black hole, for example) you’ll have to read Malcolm Longair’s forthcoming book on the History of the Cavendish Laboratory!
Yesterday I overheard some Electrodynamics students talking about the fact that all the famous names attached to pioneering laws or theorems in that subject seem to be either French (Biot-Savart, Laplace, Poisson..) or German (Gauss, Helmholtz…). Why are there no British names in this list?
Well, there was Faraday, of course. But Michael Faraday was primarily an experimentalist rather than a theorist, which sets him apart from the others already mentioned. So why is it that British theoretical was behind continental Europe in the early part of the 19th Century when all this important work on electricity and magnetism was being done.
There was also Maxwell, but he came along a bit later; he published his theory of electromagnetism in 1861/2. So why were the British so slow to enter this field?
Well, my theory of this is that it’s all the fault of Isaac Newton. I came to this conclusion when reading about the work 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. George Green 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, after his father died, and he leased out the mill he consequently inherited, to pay for his studies).
Extremely unusually for British 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).
Great scientist though he was, Newton’s influence on the development of physics in Britain was not entirely positive. 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 which 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.
I came across an article in New Scientist recently on the topic of cosmological magnetism. The piece is about an article by Leonardo Campanelli, which is available on the arXiv and which is apparently due to be published in Physical Review Letters. So it must be right.
Here’s the abstract
We calculate, in the free Maxwell theory, the renormalized quantum vacuum expectation value of the two-point magnetic correlation function in de Sitter inflation. We find that quantum magnetic fluctuations remain constant during inflation instead of being washed out adiabatically, as usually assumed in the literature. The quantum-to-classical transition of super-Hubble magnetic modes during inflation, allow us to treat the magnetic field classically after reheating, when it is coupled to the primeval plasma. The actual magnetic field is scale independent and has an intensity of few \times 10^(-12) G if the energy scale of inflation is few \times 10^(16) GeV. Such a field account for galactic and galaxy cluster magnetic fields.
So why is this interesting? Let me explain….
If you’re stuck for a question to ask at the end of an astronomy seminar and don’t want to reveal the fact that you were asleep for most of it, there are some general questions that you can nearly always ask regardless of the topic of the talk without appearing foolish. A few years ago, “how would the presence of dust affect your conclusions?” was quite a good one, but the danger these days is that with the development of far-infrared and submillimetre instrumentation and the proliferation of people using it, this could actually have been the topic of the talk you just dozed through. However, no technological advances have threatened the viability of another old stalwart: “What about magnetic fields?”.
In theory, galaxies condense out of the Big Bang as lumps of dark matter. Seeded by primordial density fluctuations and amplified by the action of gravity, these are supposed to grow in a hierarchical, bottom-up fashion with little blobs forming first and then merging into larger objects. The physics of this process is relatively simple (at least if the dark matter is cold) as it involves only gravity.
But, by definition, the dark matter can’t be seen. At least not directly, though its presence can be inferred indirectly by dynamical measurements and gravitational lensing. What astronomers generally see is starlight, although it often arrives at the telescope in an unfamiliar part of the spectrum owing to the redshifting effect of the expansion of the Universe. The stars in galaxies sit inside the blobs of dark matter, which are usually called “haloes” although blobs is a better name. In art the whole purpose of a halo is that you can see it.
How stars form is a very complicated question to answer even when you’re asking about nearby stellar nurseries like the Orion Nebula. The basic idea is that a gas cloud cools and contracts, radiating away energy until it gets sufficiently hot that nuclear burning switches on and pressure is generated that can oppose further collapse. The early stages of this processs, though, involve very many imponderables. Star formation doesn’t just involve gravity but lots of other processes, including additional volumes of Landau & Lifshitz, such as hydrodynamics, radiative transfer and, yes, magnetic fields. Naively, despite the complicated physics, it might still be imagined that stars form in the little blobs of dark matter first and then gradually get incorporated in larger objects.
Unfortunately, it is becoming increasingly obvious that this naive picture doesn’t quite work. Deep surveys of galaxies suggest that the most massive galaxies formed their stars quite early in the Big Bang and have been relatively quiescent since then, while smaller objects contain younger stars. In other words, pretty much the opposite of what one might have thought. This phenomenon (known as “downsizing”) suggests that something inhibits star formation early on in all but the largest of the largest haloes. It could be that powerful feedback from activity in the nuclear regions associated with a central black hole might do this, or it could be something a little less exotic such as stellar winds. Or it could be that the whole scheme is wrong in a more fundamental way. I personally wouldn’t go so far as to throw out the whole framework, as it has scored many successes, but it is definitely an open question what is going on.
A paper in Nature a few years ago by Art Wolfe and collaborators revealed the presence of an enormously strong magnetic field in a galaxy at the relatively high redshift of 0.692. Actually it’s about 84 microGauss. OK, so this is just one object but the magnetic field in it is remarkably strong. It could be a freak occurrence resulting from some kind of shock or bubble, but it does seem to fit in a pattern in which young galaxies generally seem to have much higher magnetic fields than previously expected. Obviously we need to know how many more such magnetic monsters are lurking out there.
So why are these results so surprising? Didn’t we already know galaxies have magnetic fields in them?
Well, yes we did. The Milky Way has a magnetic field with a strength of about 10 microGauss, much lower than that discovered by Wolfe et al. But the point is that if we understand them properly, galactic magnetic fields are supposed to be have been much lower in the past than they are now. The standard theoretical picture is that a (tiny) initial seed field is amplified by a kind of dynamo operating by virtue of the strong differential rotation in disk galaxies. This makes the field grow exponentially with time so that only a few rotations of the galaxy are needed to make a large field out of a very small one. Eventually this dynamo probably quenches when the field has an energy density comparable to the gas in the galaxy (which is roughly the situation we find in our own Galaxy).
Hopefully you now see the problem. If the field is being wound up quickly then younger galaxies (those whose light comes to us from a long way away) should have much smaller magnetic fields than nearby ones. But they don’t seem to behave in this way.
A few years ago, I wrote a paper about a model in which the galactic fields weren’t produced by a dynamo but were primordial in origin and quite large from the start. If that’s the case then the magnetic field need not evolve as quickly as it needs to if the initial field is very tiny.
The problem is that it has previously been thought very difficult for any cosmological model involving inflation to generate a significant primordial magnetic field without invoking very exotic physics, such as breaking the conformal invariance of electrodynamics (which would mean, among other things, giving the photon a rest mass).
The interesting thing about Campanelli’s paper is that it suggests a straightforwardmechanism for inflation to generate interesting magnetic phenomena. I’m not an expert on the techniques used in this paper, so can’t comment on the accuracy of the calculations. I’d be very grateful for any comments on this, actually. Me, I’m an old fogey who’s very suspicious of anything that relies too heavily on renormalization. I do however agree with Larry Widrow, quoted in the New Scientist piece.
But even if primordial magnetic fields can be generated by inflation, their impact on the origin and evolution of galaxies and other cosmic structures remains unsolved. Although we know magnetism exists, it is notoriously difficult to understand its behaviour when it is coupled to all the other messy things we have to deal with in astrophysics. It’s a kind of polar opposite of dark matter, which we don’t know (for sure) exists but which only acts through gravity, so its behaviour is easier to model. This is the main reason why cosmological theorists prefer to think about dark matter rather than magnetic fields. I’d hazard a guess that this is one problem that won’t be resolved soon either. Things are complicated enough already!
It is also worth considering the possibility that magnetic fields might play a role in moderating the processes by which gas turns into stars within protogalaxies. At the very least, a magnetic field generates stresses that influence the onset of collapse. Although the evidence is mounting that they may be important, it is still by no means obvious that magnetic fields do provide the required missing link between dark matter haloes and stars. On the other hand, we now have fewer reasons for ignoring them.
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…)
In the Dark 
