Wednesday, November 16, 2011

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.

The law of reflection says that Ɵ1’ = Ɵ1, which is a fairly intuitive result.  The law of refraction, commonly referred to as Snell's Law, is not nearly as trivial; it says that light bends.  

To fully appreciate the reason for which light bends when it crosses a boundary into a new medium, we need to assign a parameter to each medium.  This parameter, the index of refraction, n, is essentially a measure of how much of a deterrent a given medium is for a light ray.  If a given medium has a high index of refraction, then light will travel significantly slower in that medium that it would in a vacuum.

More specifically, v = c/n, where v is the speed of light in a given medium and c is the speed of light in a vacuum (approximately 300,000 km/s).  The index of refraction in a vacuum is exactly 1, whereas for air, n is just slightly higher than 1 (about 1.0003).  However, for water, n = 1.333, and so light travels at a ratio of 1/1.333 the regular vacuum speed (about 25% slower) in water: just 225,000 km/s.  A medium like glass represents an even greater hindrance for the passage of light, with an index of refraction around 1.5 (it slows down light to the lethargic speed of 200,000 km/s).

Returning then to the figure above, when a light ray crosses the boundary between air and glass, it slows down.  Although the frequency of the light goes unchanged, its wavelength decreases in proportion to the ratio of the refractive indices.  As the wavelength reduces, the curvature of the wave-fronts become less pronounced (the radii grow), and as a result, the ray has no choice but to bend.

The simple equation describing this bending is given by Snell's Law:

n1sin(Ɵ1) = n2sin(Ɵ2)

This equation leads to a few rules regarding refraction:

1.  If n2 > n1Ɵ2 < Ɵ1
2.  If n2 < n1Ɵ2 > Ɵ1
3.  If Ɵ1 = 0, Ɵ2 = 0

The first rule says that transferring to a higher index of refraction (like air to glass) causes a light ray to bend towards the normal line.  This is illustrated in the figure above.  Since the angle shrinks, it is always possible to refract into a medium with a larger refractive index.

Rule #2 says just the opposite.  When a light ray strikes a boundary as it attempts to leave a medium that has a higher refractive index, it bends away from the normal line.  Since the angle grows, there is a spectrum of incident rays that may not refract out into a medium of lower refractive index.  For any two media, there exists a certain critical angle above which there shall be no refraction from high n to low n.  In such a circumstance, the light ray does not come to a stop - it just reflects exclusively; this is known as total internal reflection.

The third and final rule listed above represents the only way that a light ray may cross from one medium to another without altering its course.  For the particular case where light is on a course that is exactly perpendicular to the boundary dividing the two media, the ray will not bend as it crosses over (it will still, however, travel at the new speed prescribed by the new refractive index).

With this introductory information regarding optics, let us now attempt to answer the following question: Do people behave like rays of light when they encounter crossroads in their lives?

I find that in a philosophical sense, Snell's Law is illustrated by people when they attempt to cross a boundary into a new environment.  Let us use the example of a High School graduate who is entering University.

There is no question that the demands on a student increase significantly upon entering University.  As this environment, or medium, requires significantly more attention, and poses a much greater challenge, the refractive index of a University is significantly higher than that of a High School.  As a result, the ray of light that represents this transitioning student will need to slow down and focus, and yes, even study, in order to succeed in this new medium.

Furthermore, this maturing student will undoubtedly shift his or her perspective on life in this new environment.  This can be thought of as a modification of one's direction, or, better yet, as a refraction of oneself.

Eventually, the student will see a new boundary emerge in the distance: the finish line.  To complete university successfully, a student cannot carry him or herself in any which way.  In fact, there is a whole spectrum of students that will not graduate from university for one reason or another.  The line in the sand separating the graduates from the non-graduates is defined by the critical angle.

It is sadly not all that uncommon for a student to bounce around within the prism (which sounds eerily close to "prison") of University.  A student experiencing total internal reflection within the college campus walls can become discouraged.  Perhaps the problem for some students is that they do not slow down enough to learn in this information-dense new environment.  By not adjusting to the new demands, they refuse to refract, and in turn, break Snell's law.

It is exceptionally rare to find a student that can transition from High School to University without making some degree of personal changes.  This is not surprising, as only one ray of light in one million will strike a boundary perfectly perpendicular to it.

In summary, it is easier for light to enter a prism than it is for it to leave.  Correspondingly, it is easier to have a goal than it is to fullfill the obligations it entails.  A very high refractive index correlates to a goal with extremely high demands.  There are media with higher indices of refraction than glass: for diamond, n = 2.42.  The critical angle for light to leave diamond and re-emerge into the air is 24.4 degrees.  Light that hits the interior boundary of a diamond ring at an angle greater than this will merely reflect.  One reason why a diamond ring sparkles so brightly is because light rays have so much trouble escaping.

Before each semester, students ought to plan out their schedules carefully.  To abide by Snell's Law, students need to slow down, and dare I say, devote themselves to learning.  If they choose an appropriate path to pass through the prism of school, they will exit successfully into the real world, refracted.


J Plante said...

Excellent parallel with students' life entering College...!

The Engineer said...

Thanks J.P.,

It is so common to see students that cannot find their way in higher education. Of course, the issue is more complicated than a single angle, but socio/psychological issues are always more complex than the physical sciences (my students HATE when I say that).

Martin said...

Would you address the point that the speed of the light exiting the refracting media will once again increase in speed. How does it do that? What is the accelerating energy and where does it come from?

The Engineer said...


I really like your question. But I think the first point to address is that light is made up of photons, which have no mass, and so to think of the speed at which they travel in terms of kinetic energy is not appropriate.

Photons do have energy... the tiniest quanta of energy, which is associated with the frequency of the light wave. When light refracts at a boundary between two media, its velocity changes, but remains given by v = wavelength * frequency. The frequency of the light (and therefore the energy) do not change due to refraction, but the wavelength and velocity do change as they are linearly dependent on one another.

In short, the bending AND the velocity change may be attributed to the light's change in wavelength (and the wavelength adjusts according to the optical density of the media in which the light travels). Light ray acceleration leaving a dense medium does not imply a gain in energy just as its deceleration upon entering does not represent an energy loss.

If some of the intensity of the light is lost as it passes through an optically dense medium, this is because the medium has been excited by the light (heat transfer by radiation).

The intersection of light absorption and light ray velocity is best described by "complex refractive index", but I think this is outside the scope of your question.

Thanks for reading.