Snell's Law
Quantum and Nuclear | Light, Sound and Waves

Why these rays?

Physics Narrative for 14-16 Supporting Physics Teaching

Rules about rays: where do they come from?

Why does the angle of incidence = the angle of reflection? One answer explored earlier is an appeal to the rules about rays. That's just how you draw them, and once you have the ray, it predicts the beam.

That says how, but does not really even attempt to answer the why? question. That requires another level of enquiry: why are the rays drawn in just that way, and no other? One way of approaching this is to try the alternatives and see what happens. Perhaps playing out what does not happen in our imagination will give an insight that'll suggest why it does not happen when played out in the physical world.

In refraction, just why is this true?  Again, just following the rule tells us how but not why.

Light travels in straight lines: that is, it takes the shortest path from A to B. How can we identify a straight line? Just picking up a ruler may not be reliable enough. One way is to try out a few paths along which the light could travel, and then choose the one with the shortest time – after all, that builds on the rather secure knowledge that the speed of light is constant.

Maybe it's worth imagining doing just this.

Here we're suggesting a series of theoretical explorations to find out what's special about the rays from source to detector, involving propagation, a reflection or a refraction. So we'll draw many possible paths, seeing what varies as we do so. You already know that the trip time is important for radiating, because it gives the delay between what the vibrations at the source are up to and the time when the detector will undergo the same vibration. The trip time is, of course, set by the speed at which the radiations propagate and the distance that they travel, so it is set by the geometry of the situation. The geometry defines the paths by linking the source, a single moveable waypoint and the detector. You could, of course, define more complex paths to try by adding as many waypoints as you like, but the principle would not change.

It turns out that the rays are always drawn along a path that corresponds to the shortest trip time. You could, of course, ask in turn why that is the case and so why the path of exactly least time is selected from amongst the many possible paths. That's indeed a good question. Perhaps it's just because nature is lazy, or maybe there is a deeper unifying principle at work. That's how it goes in physics – every good answer breeds further questions. We'll show you a bit more of the answer to this one in episode 03.

Predicting rays for reflections: why are there certain rules about rays?

Now model a mirror. Will light still follow the path of least time and, if it does, will this account for the rule that you've used without subjecting it to analytical enquiry? One way to answer the question is simply by drawing: a mathematical experiment. Draw a source, a detector and a mirror. Draw in as many paths to try as you like.

From source to mirror will be a straight line, as will from mirror to detector, as these are both straightforward propagation, and you've already established that light travels in straight lines under these circumstances – you don't need to try any of the many curves that link these pairs of points. But the point on the mirror is not pre-ordained, so there are still many pairs of lines to try. For each, measure a distance, and then convert this distance to a trip time, using the universal converter of the speed of light. Based on what we found out about propagation, the best guess would be that there is a pair of lines showing the rays where the angle of incidence is equal to the angle of refraction, and that this pair has a minimum trip time for the journey between source and detector. Then you can repeat for a variety of different locations for source, detector and mirror, so producing an overwhelming quantity of empirical evidence that minimising the trip time in these circumstances is indeed the underpinning principle that accounts for the brute fact. You might reasonably enlist the help of a computer for this kind of repetitive calculation. For proof, you'd need to enlist a mathematician who could do differential calculus and show that the minimum trip time for any situation would always predict the angles correctly. Maybe you have a colleague or clever A-level mathematician who needs a challenge…

Rules about rays used to predict refractions – but where do the rules come from?

Thinking about refraction raised two questions, of which only the first was answered.

  • How, exactly, is the angle of incidence related to the angle of refraction?
  • Why are the angles related in this way?

Let's start with the empirical rule again.  The greater the difference in speed between the materials, the greater the constant. Sines are trigonometric functions, and so connect the angles to distances in triangles. So you have something comparing distances, comparing speeds. Maybe trip times will again provide a unifying principle – they have already provided an account of the brute facts for both propagation and for the law of reflection. Maybe a further unification will be possible. Time for further simple experiments with a pencil and paper, or perhaps for enlisting the assistance of a computer in doing the repetitive computations. Draw up a source and a detector and draw a surface representing a change of medium between them. Set different speeds of propagation in the two media. Then try out many different waypoints on the surface, one at a time. Calculate the trip time from source to detector. The path from source to waypoint can be a straight line, as can the path from waypoint to detector, as these are both propagation, and you already know that the shortest time for these legs will be a straight line. Find the waypoint and corresponding paths that make the trip time a minimum. Ink this pair in, making them rays. These rays will exactly predict the passage of the beam. The principle of minimum trip time has accounted for another fact.

Snell's law is a consequence of least time paths

Snell's law is a simple consequence of the least time principle. Check this by calculating the ratio of the sines of the angles of incidence and refraction for a range of different source and detector locations, so exploring a range of angles of incidence. Try some modelling using a computer. You'll always set a single constant for the chosen pair of speeds for the two different media. Remember that this choice corresponds to the beam traversing the surface between two materials. You could extend this to a situation where a beam traverses a block of a different medium, so there are two possible waypoints and three paths to consider. You'll get exactly the same mathematical prediction – that minimising the trip time can only be done by selecting paths that lie on the rays for which Snell's law is true. If the sides of the block are parallel, the incident beam will simply be offset from its original line but not deviate from this line.

Next, minimise trip times for a beam traversing a block of material where the sides through which the beam enters and exits are not parallel (e.g. a prism). Here the prediction is that the beam deviates. You could have predicted this from applying the brute fact of Snell's law. The new geometry, where the two normals (one for entry and one for exit) are not at the same angle, has the consequence that the beam will deviate. As this is frequency dependent, and the change is being applied in the same sense both times – that is the deviation is rotating the beam in the same direction on entry and on exit (so there is no hint of undoing what was done) – you might also reasonably expect dispersion: the different frequencies present in the single incident beam now travelling out in different directions from the prism.

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