Snell's Law for Light and Students

Optics, the study of visible light, is one of the earliest branches of science.  An introductory physics class will usually highlight what happens when a light ray encounters a new medium.  That is, a light ray may be travelling in a straight line unperturbed through medium 1, air, when it suddenly encounters medium 2, a thick plane of glass.  Before the light ray strikes the boundary, it is referred to as an incident ray.

The incident light ray will divide itself upon striking the new medium: some of the ray will reflect, and some of it will refract.  The reflection of a light ray is simple - it is no different than what happens to a billiard ball when it strikes a band.  Just as a billiard ball rebounds off of the flat border with an angle equal to that which it struck with, the angle of the reflected ray is equal to the angle of the incident one.

However, light can also permeate the new medium (an ability that no billiard ball possesses).  The portion of the light that passes through the boundary, continuing its travels into medium 2, is referred to as the refracted ray.  As shown in the figure below, the angle of each ray is measured with respect to an invisible 'normal' line, which is drawn perpendicular to the surface.

Water and Electricity Do Mix

*** Disclaimer: Do not wet your finger and plunge it into a live electrical outlet.  The title of this article was intended to be a play on words.  You know what?  If that was not already obvious to you, a severe electrical shock might do you some good.

Some science students are initially baffled when studying electricity.  This is not so surprising, as it is hard to relate to the concept of current, the rate of flow of charge.  The charge that is transported through an electrical circuit is composed of electrons - many of them.  I have never seen an electron, and while I have felt a charge by means of electric shock, I have trouble sensing exactly how many Coulombs are being transferred to me (or are they being taken from me?  I can never tell.).  I have an easier time visualizing, say, mass than I do visualizing charge, and I imagine this to be true for most people.

This is why it is appealing to explore the analogy between electrical current through a circuit and fluid flow rate through a system of pipes.  As a student, I could never remember whether current was constant through resistors connected in series or parallel.  The answer becomes obvious when one considers the flow rate of water through pipes rather than the current through wires.

Entropy in Autumn

There are two things that I associate with the Fall season: Halloween and raking leaves.  While the former brings out my inner child who yearns for costumes and candy, the latter is the focus of this article, as it provides a wonderful illustration of the second law of thermodynamics.

Ah, the second law... Personally, I have never been particularly drawn to the study of thermodynamics, but I have always had an affinity for the second law and all of its fascinating philosophical implications.

The zeroth law of thermodynamics discusses thermal equilibrium, and in so doing, gives meaning to the term temperature.  The first law describes the conservation of energy, and dictates that energy cannot be created nor destroyed.  The second law says, in a variety ways, that no system can be 100% efficient, and as such, every system has losses. 

If you are curious, the reason that there is a zeroth law is that it was developed decades after the first two, but precedes them in terms of scientific principle, in that it lays a foundation for further study.  Rather than shift laws one and two up to two and three, it was decided that they be left alone, and that the new law be referred to as law zero.  This was a great choice, because it gave scientists the opportunity to use the term "zeroth", and as a result, sound very intelligent.

In summary, law zero says there will be lunch, law one says that it will not be free, and law two says that actually, you have to tip.

Quantum Discomfort

Quantum physics has always been this mystical thing to me.  Throughout my years of academic studies in engineering, the word quantum may have been uttered by a professor, but it was never explored in any kind of depth.  Engineers are practical, and while quantum physics is fascinating, it is of little practical importance to most engineers at present.

A physics professor might say, "Classical physics is not theoretically sound, as it is only an approximation, and may only be accurately applied to things that do not move too fast and are not too small."  An engineering professor might then respond, "Most things are reasonably large and move reasonable slowly... In any case, any error less than 5% is good enough for me."

For nearly all engineers, the entire field of study may be ignored, and is.  For theoretical physicists on the other hand, it appears that there is not much outside quantum physics to ponder anymore.

In reality, for an object of reasonable size, the amount of error introduced by simply ignoring quantum aspects is negligible.  Still, this field of study is so fundamentally different from all of the science that came before it, that it is always categorized separately; that is, there is quantum physics, which centers around wave functions and probabilities, and then there is classical physics, which is everything else.

Quantum physics as a field of study was established in the mid-1920s with physicists such as Max Planck, Albert Einstein, Neils Bohr, and Werner Heisenberg.  The fundamental principle of quantum physics was discovered by Heisenberg, who showed that one could not be 100% certain of both the momentum and position of a given particle.  The law is known as the 'uncertainty principle', not to be confused with the 'what the heck is going on principle' commonly exemplified by certain physics students.  

Wavelengths and Personality Types

Light is a funny thing. 

It behaves like an electromagnetic wave (it has a phase - it can undergo interference), yet it also behaves like a stream of particles (it consists of photons, as proven by Einstein's photoelectric effect).  The split personality exhibited by light has bewildered scientists for one hundred years.  And, while we will never see a photon with our naked eye, our eyes would see nothing at all if not for light.  These photons reveal the nature of the matter around us, but keep their own true nature under wraps.

The light that reaches our pupils may come directly from a source or it may arrive after a series of reflections off of other surfaces; either way, the light emanates from a source, like the sun or a light bulb.  Both of these particular sources produce what is known as white light. 

All electromagnetic waves, whether they are radio waves or x-rays, have a wavelength associated with them.  Visible light fills a very narrow band within the electromagnetic spectrum: 400 nm to 700 nm.  All wavelengths in this range correspond to a particular colour.  Violet light has a wavelength of 400 nm, while that of red is 700 nm.  A rainbow will have these colours on either side (violet on the bottom, red on top), and all other colours in between ordered by wavelength.  White light is what you get if you superimpose all colours in the visible spectrum on top of one another.  Black, on the other hand, represents the absence of colour.

With this background information on light, we can properly appreciate why a red sports car looks like a red sports car.  It is less obvious than you might think...

The sun undergoes fusion, a nuclear reaction, which produces, among other things, white light.  These white light rays travel in all directions at about 300,000 km/s.  After travelling a little over eight minutes, some of these white light rays hit the red sports car that sits before you.  Then, something critical happens: some of the white light is reflected by the car's surface, but most of it is absorbed.  In fact, the surface absorbs every wavelength of light with the exception of its own, red, 700 nm.  The 700 nm wavelength of the light is reflected by the surface.  These reflected light rays head off in many directions, including yours.  Your brain interprets this wavelength to be red.

So, while the car may be red, we could not make this deduction if not for a source of light that contains, at a minimum, this particular wavelength.  It is kind of like that falling tree... If an object is not lit up by a source, does it have a colour?

The car's red colour gives it a certain character.  And, while one's personality is more complex than a single data point along a spectrum, colour codes for personality classification are often surprisingly telling.

High Pressure Situations

A relatively small number of people deal with intense situations on a regular basis.  Overseas military, first responders and ER doctors are part of a select group that deal with high stress on a daily basis.  The rest of us can probably count on one hand the number of traumatic experiences that we have encountered in our modern, sheltered lives.  Personally, I can count this number on two fingers.

The first intense situation that I recall occurred when I was eighteen and living at home.  I started the backyard barbecue, and then went back inside the house while it warmed up.  When I looked back at the barbecue a few minutes later, it was a massive fireball.  I jumped to my feet, and ran towards the fire extinguisher.  As I approached the fire, I was fearful, realizing that the propane tank could have blown up at any moment.  Nevertheless, I sprayed out the blazing flames.

My one and only firefighting experience ended well.    Thinking back now, my actions that evening were instinctive; they involved no thought.  Intense situations do not leave one with much time to think.

Intense situation number two made yesterday a day I would like to forget.

Differential Equations Govern the Future

Mathematics is the language used to quantify anything in science.  Still, some aspect of the tool we call math is utilized in nearly every field from business, to sales, and of course, engineering.

The field of Mathematics may be divided into several branches like calculus, linear algebra, and statistics.  While all of these subjects are fascinating, let us focus on calculus, which can be summarized as "the study of change."  If no facet of the universe ever changed, we would have no use for calculus, but then again, such a universe could not support life altogether (life cannot exist without chemical reactions).

At its heart, calculus focuses on functions, which are equations describing how variables are related to one another.  The simplest kind of function consists of two variables; one variable is dependent, the other is independent.  If a taxi driver charges a customer $1.50 per minute, then the cost function for riding in the taxi would be C = 1.5t, where C is in dollars and t is in minutes.  Here, time is an independent variable, and cost is a dependent variable (as the cost depends on the duration of the cab ride).  In other words, cost is a function of time, or C = f (t).

The simple taxi function given above relates cost to time, but functions can describe the relationships between other dependent variables and time.  We could, for example, consider the temperature of a hot cup of coffee.  One can imagine that the coffee's temperature value would decrease as time goes on until it reaches the room's air temperature.  The key difference between the taxi function and the coffee function, is that the function for the ride in the taxi was specified by the taxi driver.  No one specified the temperature function of the coffee.  The coffee's temperature function is the result of a differential equation, which is also known as a governing equation - a law of nature, and in the case of the coffee, thermodynamics.

You can Count on Asimov

Isaac Asimov is on a short list of my favourite science authors.  The list has two names on it: Arthur C. Clarke and Isaac Asimov.  Both write excellent science fiction (Asimov's "I Robot" and Clarke's "Fountain's of Paradise" are my personal favourites) and both write excellent topical science articles and essays.  While Orson Scott Card writes some compelling sci-fi (Ender's Game is probably my favourite novel of all time), he is not a "popularizer" of science - it is rare to find an author that is skilled in both fiction and non-fiction.

Asimov may have been the most prolific author ever, having published upwards of four hundred pieces.  His direct style in story-telling and, at times, redundant style of communicating in non-fiction does not appeal to everyone, but it does appeal to me.

Asimov's largest volume of work involves the communication of science, but physics in particular.  I just finished reading "Asimov on Numbers," which involves mathematics, but really focuses on, well, numbers.  It is a collection of essays written over a period of many years, beginning in 1959.  Although all articles are between four and five decades old, they have aged well.  The only instances where the book feels old is when population and financial figures are discussed, as the absolute values of both have inflated significantly in the ensuing years.

I learned a lot in reading these essays, mostly about the history of mathematics (very fascinating) and the Earth's geography (which, as it turns out, can be described so well numerically).

A Big Day for Science Geeks

There are two interesting developments to discuss, so let's get right to them:


1. Today, several chunks of a decommissioned satellite will strike the surface of the Earth (hundreds of kilograms at hundreds of km/h)

For the past week, earthlings have been warned that a defunct satellite will enter the Earth's atmosphere on Friday, Sept 23, 2011.  Unlike small satellites, this one, the size of a bus, will not entirely burn up upon re-entry.  Although the simulation is hard to predict with high precision, it is certain that a significant portion of the satellite will survive the re-entry, and will find a terminal velocity in the region of 300 km/hr.  Like the weather, aerodynamic problems involving extreme heat transfer are difficult to solve with much accuracy long in advance.  The location of where on Earth these satellite chunks will land is still somewhat up in the air.

NASA explained last week that this should not worry us, as there is a 75% chance that the said pieces will hit water (gee, thanks for the geography lesson).  As of this morning, we now know that the satellite pieces will strike the Earth in its Southern Hemisphere.

This sort of thing happens roughly once per year...

Guitar String Theory

I learned how to play the guitar about thirteen years ago, towards the end of high school, because I hypothesized that it would attract the ladies.  Fortunately for me, it attracted one lady in particular, and I ended up marrying her.  Although this case study was limited to one specimen, I think further investigation would lead to a strong correlation between guitars and ladies.

Over time, the guitar has become much more for me than a tool for garnering attention around a campfire.  The six-stringed instrument has sat on my lap for countless hours, as I figure out songs that I enjoy, perform them live, or just sit alone strumming nothing in particular.  It’s amazing what a wonderful diversion those twelve notes can provide.

I have written previously about how a pulse can travel along a string that is in tension.  What distinguishes the behaviour of a stringed instrument is that both ends of the strings are fixed.  The phenomenon that occurs when a taught string is plucked is known as a standing wave.  The note that we hear, its frequency, is dependent on four parameters for the given string: its length (L), tension (T), mass per unit length (µ), and mode number (n).  The frequency in Hz is given by the following equation:

f = [n/(2L)](T/µ)^0.5