In my academic life, things that have troubled me the most are not some really tough mathematical proofs (perhaps I haven’t reached that point yet), but things that are really obvious. Trivial questions have always left me perplexed. Here I am listing some of these questions, without giving any satisfactory answers, in the hope that they will trigger some thinking in the reader’s mind.
The earliest I can recall is dynamic equilibrium from class 9th. Our physics teacher used to ask one of the students to read the book and he would comment on it, usually restating the same thing in different words. In that class, I never understood the dynamic equilibrium problem, specifically that of a parachute. How can a body move if all the forces on it are zero? As trivial as it sounds, it was a mystery to me and I never understood why would something move if the forces sum to zero? Yes, we did study Newton’s laws but those were in some other chapter of the book.
It was class 11th, when I was sitting in the physics class and the teacher was explaining equilibrium on the black board, that I found the answer to my parachute equilibrium question and understood the fact that parachutes need to be opened someway down the flight so that there is enough inertia to ground it.
Another thing that puzzled me was why a dot product of force and distance results in work but a cross product results in torque? This question was triggered by a question asked of me by an examiner in the final viva exam of physics in class 9th. The question was probably 'what is torque?' and I answered ‘force into distance’, the examiner wanted to know what did I mean by that, so he wrote it on the paper $F \cdot d$ or $F \times d$, hinting at $F \times d$ and I chose the cross product. My transcript says I got 11/15 and that is the average you get when you don’t impress the examiner who is usually from another school or city.
I don’t know when I found the answer but I am currently content as far as this question is concerned. I eventually found that these are just mathematical constructs that help us explain different phenomena and that one gets used to it over time.
Since my childhood I was interested in electronics and I am really lucky to become an electronic engineer. However, my high school physics textbook (or perhaps my own dumbness) was not very helpful in igniting my interest in electronics. In the chapter on electronics, there was this topic 'transistor as a switch' and as I perceived it, the crux of the topic was something like if you press the button, the current will start flowing through the transistor. I was deeply troubled by this, if we press the button then we are acting as a switch operator, the transistor has nothing to do with controlling the current, its me, I pressed the button!
Later, I found that in a transistor one current (the base current supplied by us through the button push) controls another current passing through the transistor (the collector-emitter current).
A variable, as the name itself suggests, is variable, you can assign any value to it, then what is the difference between a variable and a random variable? Painful! The question rouse in my mind because I took a class in probability and I wrote the final exam with this question still lurking in my mind.
In the final year, my undergrad advisor Dr. Jawad taught me that a random variable is something whose values can't be determined by an analytical formula, for example, if a variable $w$ represents tomorrow’s weather, you can't write a formula for $w$.
For a long long time I was puzzled by the idea of doping and that there is a hole current and an electron current. I got started with this problem when I was in school but I didn’t find an answer through out my four year electronics engineering degree. I could simply not get the idea how would holes flow, no matter how you explain it to me. Unlike the other problems, this problem was of a very serious nature to me. For example, in term 3 of my undergrad, I wasn’t able to get over Chapter 1 of the textbook of the electronic circuit design course that discussed the characteristics of pn junction in terms of holes and electrons. In term 7 (the penultimate term), I took power electronics, there I found that BJTs are better than MOSFETs because it had something to do with these holes and electrons at the junctions. Well, finally in term 8, I took VLSI design, we learned the high level physics and chemistry but after all it was a final year engineering class and we were expected to know holes and electrons and pn junctions. The class was more of how to design logic circuits in silicon rather than how silicon works.
I started reading Floyd, Sedra and Razavi but the more I read, the more I got confused. I was not able to get how this law of mass action can be true? I discovered that there are Fermi levels and density of states and wave functions and stuff but none of those directly answered my question. At some point I got content with the answer that the flow of electrons in the conduction band is called electron current and the flow of electrons in the valence band is called hole current. I have not worked much with electronics after that, so I don’t know if this answer will raise any further questions but I was content with it at that time.
Apart from these, I struggled quite a bit with imagery numbers, negative frequencies, how an EM wave emancipates from an antenna, where does the Laplace transform comes from, what on earth is convolution etc.
In the beginning, my questions were more of a fundamental nature and even if I discussed with someone, I seldom got a satisfactory answer. It is not that the answers were not there in the books or that the people I asked from were not able to answer but probabily it had to do with looking at things from different abstraction levels and assuming different background knowledge.
Recently, however, my problems are more of managing complexity, directing resources and sorting priorities, for example, I came across Hamiltonian mechanics and I couldn’t get the essence of it, but I know that it is quite well written in books and if I wade through any mechanics book, I can get a good grasp of Hamiltonian, so in a sense things have got easier now than before. It has probably to do with a gradual maturity in thinking and access to help from people who specialize in the particular domains.
Hope you enjoyed the read! If you want to get in touch, shoot me an email at najeeb dot khan at usask.ca