Points of Inflection

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I wholeheartedly agree with two statements made by RKN:

(1) Everyone should be free to study whatever they want; and should also be free not to be examined at all.

(2) There is nothing like Higher Mathematics...all mathematics is Higher Mathematics.

Long back in my youth I read Karl Marx predicting that after the Proletarian Revolution takes over, the Communist State would eventually 'wither away'.

I see no signs of the Communist States in India, at least, withering away during my lifetime. Not even W B: it is going strong after 40 years, and even 'producing' Maoist 'children'.

They may 'disintegrate' like the Capitalist State of A P; but no, not 'withering away'.

But RKN's wish about Exams withering away has more or less come true. I am told that students of CBSE may not appear for their Class X Public Exam if they wish not to.....a heresy even a couple of years back; moreover, if they do dare appear, they have no right to continue in the same School! Well, small mercies! During my time, every Class, from I to XII had 'public' exams and every student has to pass every Class to be promoted to the next. I have known a reluctant cousin of mine to fail his SSLC Exam a total of 7 times before he gave it up. And another cousin of mine who 'cleared' her Class X at 45. She was already working in that School as a 'Sewing Teacher' and got a 'special consideration'.

Regarding all math being higher math, I am a living and kicking Example. I knew a Classmate of mine who first looked at the series expansion for 'e' and at once fell in love with 'infinite series'.

I, on the other hand, couldn't make out why people should go to the trouble of 'differentiation' several times, till I chanced upon Maxima and Minima of curves; and since then fell in love with Points of Inflection.

People wondered how a non-math chap like me could pull on with the Math Wizard SDM for five long years and survived unscathed.

The reaosn is simple. The PRS Reprint that he gave me to read was titled: 'Cherenkov Radiation in Anisotropic Media'. Now EM Radiation is PHYSICS! Moreover, he straightaway told me that he was jumping frames from the Lab to the Particle and applying Special Relativity. Jumping frames is always 'fun'; and SR was my strong suit; at best it had 'Cartesian Tensors' which any non-math child could handle.

There was a brilliant student in our B Sc at KGP in the late 70s by name SS. He went to Stony Brook right after B Sc and did his Ph D in Particle Physics. He returned to KGP soon after his 'Qualifiers' and met me in our Central Library, and pleaded with me not to recommend anyone to Stony Brook just after B Sc because it was too too tough to cope; which of course he did, and got a wonderful Degree.

Fast forward 25 years: he visited his alma mater for Recruiting IITians for Motorola, Bangalore, where he was VP; and his wife still in Particle Physics (these choices are forced on you if you marry the same 'Physics Gotra' against all Khap!). He offered to take my son in his Firm anytime he chooses (I hope the offer is still on).

Anyway, when he was in his Third Year, he led a delegation of his classmates to me in the Lab, challenging me to 'explain' why a point charge moving uniformly in vacuum does not radiate, 'without math'. I asked him to jump to the Point Charge Frame where it would be at rest, and vacuum still remains the same vacuum (vacuum doesn't MOVE). And, you will agree that a point charge at rest will not radiate, because there is no H field to give the Poynting Vector. And if 10 photons are emitted in one Frame then 10 photons should be emitted in any other Frame, they being particles countable by 'counters'.

The delegation dispersed without mutiny.

SDM was doing the same trick with the Chrenkov Radiation. But now there is a medium and the uniformly moving charge in the medium could radiate. But jumping to the Rest Frame of the point charge, its fields should be static; but the Radiation MUST be going on. Which means that it is the medium that radiates and not the point charge. And it should be so in any Frame including the Lab Frame.

Same with the Smith-Purcell Radiation: it is the grating that radiates and not the charge, because it would come to rest if you jump to its own Frame.

All in all, Relativity and Electrodynamics won me over and I carried on with SDM despite his formidable reputation as a math wizard.

And did get to predict results that could be verified in High Energy Labs.

Coming back to Points of Inflection, I was first charmed by the Magnetic Field produced by the Tangent Galvanometer (now you see the 'connection') Coil carrying steady current. Our hopeless book by Starling asked us to differentiate it twice and set it equal to zero. And, lo and behold, I did that with lots of enthu and did find that it happens at x = r/2. And, my joy knew no bounds, because that was the first time I found an example in my Student Physics Life where the trouble of differentiating TWICE led to a physical result that could be verified in the Lab. This graph used to be drawn and the Point of Inflection determined. and I could get some confidence that Math could be at times useful in Physics.

And, I went on to use this result later at KGP by designing an Experiment in the Second Year Lab with TWO coils in series (Helmholtz Coils) to get a very nice 'uniform' field that could be plotted over a range. Indeed this is the best way of generating uniform weak magnetic fields useful in NQR etc.

Then again in 1980 or so there was an unusually brilliant student by name Tanmoy Bhattachraya (last seen at Los Alomos) in my First Year Optics Course (Later I had him in his 4th year; and he did his Project with DB, if I remember right).

For the first time I decided to teach Fermat's Principle to his batch in my very first Lecture. I wasn't too happy with the treatment in Jenkins & White where they do say that the Optical Path (Time Traveled) could be a minimum, maximum or stationary. But the example given of 'bending' a mirror continuously didn't satisfy me.

Rest assured that if the teacher is not satisfied with a topic he has to teach, there would at least be one student in his Class (at IIT) who is even more dissatisfied.

And after the first Lecture, Tanmoy met me and asked me to find a better example.

That was a challenge, which took me 6 hours to invent. And in the Next Class both me and Tanmoy had a great time.

I never liked to teach Fermat's Principle the way Feynman did it: the 'life-guard' and the 'drowning beauty'. Because I knew that 'least time' is not always followed by the actual ray. And in any case, it would be giving too much of a 'physical' beauty to an essentially 'formal' beauty.

And Jenkins & White's example is misleading as if the shift from minimum to maximum via the Point of Inflection is a feature of the shape of the 'mirror', which was suspect.

The example I cooked up is s 'beauty'; because I still remember it fondly after 30 years. And is not found in any ext book:

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"Take a hemispherical concave mirror with its pole at P. Take an actual ray incident along the axis of the mirror and getting reflected back on itself. Fix the end point of the path at the center C of the mirror. Take any other point A along the axis of the mirror. Take a 'possible' ray ABC where B is the point of incidence of the light on the mirror. You will find that the actual path APC is the minimum, or stationary, or maximum compared to the alternate paths ABC depending on whether A is towards the mirror, at its center C or away from the mirror".

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All it takes is to differentiate twice the 'possible' Optical Path.

Here you have a clear example of the SAME mirror, SAME ray and the SAME final point giving all possible varieties depending on just your CHOICE of the other point!

So much for any silly 'philosophical' implications of Fermat's Principle of LEAST time!

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