The Prism

I put light rays in prism, but they always escape. (I know, groan…. I can't help it.)

We will deal with three things you might want to know about the prism, a triangular slab of glass or transparent plastic, plus how to do some nifty demos:

a) Calculating the angles between the normals to the surfaces of the prism and the path of a ray through the prism,

II) calculating the overall angle of deviation from the original direction, and

3) explaining the separation of white light into the colors it is composed of.

(That last sentence reminds me about the rule, never end a sentence with a preposition- George Bernard Shaw said something to the effect that it was a rule "up with which I shall not put." Or maybe it was Churchill.)

The left sketch below shows the path of the light and the relevant angles a , b , g , d ; on the right is the overall angle of deviation D.

 

Here is a quickie how-to:

1)Given a and index of refraction n of the prism, find b by Snell's law.
b) Given upper prism angle f , find g = f - b .
iii) Find d by Snell's law.
IV) Find angle of deviation D = a + d - f . Incidentally, the minimum D occurs when a = d .

If you do these things for an assignment or to prepare for a test, you better know the whys, or you may be in trouble. What to know besides Snell's law: The sum of the angles of a triangle is 180^o = p radians. From this it can be deduced that the exterior angle of a triangle is equal to the sum of the remote interior angles, and the sum of the angles of a quadrilateral is 360^o or 2p .

The two dotted normal lines and the sides of the prism up to the top, in the above picture on the left, form a quadrilateral with two angles of p /2 each, so the sum of f and q , the top and bottom angles, must be p . But q + b + g = p also, because they are angles of a triangle, so eliminate q and show that

g = f - b .

D is the exterior angle of a triangle, and the two remote interior angles are q ₁ and q ₂ which are a -b and d -g . Summing these and substituting g = f - b gives us

D = a +d -f .

The minimum D (call it D_m) occurs when a = d , so also b = g . From this and the equations above it is easy to show that
a = (D_m + f )/2 and b = f /2. Then Snell's law says

sin[(D_m + f )/2]=nsin(f /2).

This is a reasonably good way to experimentally find n, because D_m and f are easy to measure.

That last equation tells us that if white light is split into its colors, it must be because different colors have (slightly) different n values. (I hope you know that it is the white folks who are the most colored.) For the usual purposes we say n is a constant for a given material, but if you want high precision, it varies a little as a function of wavelength: longer l means smaller n. I do not know why.

If you want to see a nice spectrum, the rainbow-like display of the prism, cover most of an overhead projector surface with two pieces of cardboard or something so that only a sliver of light remains in the middle. Put the prism in the beam after it emerges from the final lens. Your instructor probably knows that, but here is one that (s)he probably doesn't know about: Put a little strip of cardboard or a pencil on the overhead so that you now have just the opposite as the above situation: light where there was dark and dark where there was light. The regular spectrum was the standard red to green to blue; this one is the complements, cyan to magenta to yellow. This is tricky to explain, and I don't want to put the necessary time into it. Now zip back to the main junk on waves. Or

my other main pages:

mechanics
fluids, heat, electricity and magnetism
quantum
index

Comments, questions: fredrick.gram@tri-c.edu