Video Transcript
The diagram shows the path of a light ray through a triangular prism. Which of the following is a correct formula for determining the angle 𝛼? (A) 𝛼 equals Φ one plus 𝜃 two plus 𝐴. (B) 𝛼 equals Φ one plus 𝜃 two minus 𝐴. (C) 𝛼 equals Φ two plus 𝜃 one plus 𝐴. (D) 𝛼 equals Φ two plus 𝜃 one minus 𝐴. (E) 𝛼 equals Φ one minus 𝜃 two plus 𝐴.
In our diagram, we see the angle 𝛼 indicated here. This is the angle between where the incoming ray of light would have gone if it hadn’t been refracted by the prism and where it actually did go because it was. Along with the angle 𝛼, we have the apex angle of our prism 𝐴 indicated as well as the initial angle of incidence of our ray Φ one, the initial angle of refraction 𝜃 one, the second angle of incidence Φ two, and the second angle of refraction 𝜃 two. As our problem statement tells us, we want to develop an equation for 𝛼 in terms of these other angles.
To start doing that, let’s clear some space on screen. And we’ll focus at first on this orange triangle highlighted. Drawing an expanded version of this triangle, we don’t yet know any of the interior angles, but we can make a start on identifying these angles based on the other information given to us. For example, this angle here, according to our diagram, is 𝛼. That tells us that this interior angle must be equal to 180 degrees minus 𝛼 because this angle plus this angle equals 180 degrees. Along with this, we know that our ray’s initial angle of incidence is Φ one and its initial angle of refraction is 𝜃 one.
Looking at the geometry of these two angles, Φ one and 𝜃 one, we can see that this interior angle of our orange triangle is Φ one minus 𝜃 one. We know that to be the case because this interior angle and this angle here 𝜃 one itself must add up to give Φ one. In a similar way, we can sketch in the second angles of incidence and refraction Φ two and 𝜃 two, respectively, which, by considering these angles, shows us that this final interior angle of our triangle is 𝜃 two, the larger of the two angles, minus Φ two.
Let’s now remember that for any three-sided shape, for any triangle, the sum of that shape’s interior angles is always 180 degrees. Therefore, if we add this interior angle to this interior angle and this one, that gives us Φ one minus 𝜃 one plus 180 degrees minus 𝛼 plus 𝜃 two minus Φ two. All of this must add up to 180 degrees.
Notice that this angle appears on both sides of the equation. If we subtract 180 degrees from both sides, we cancel out this angle. That leaves us with the left side of our equation being equal to zero. And note that if we add the angle 𝛼 to both sides, that angle cancels out on the left, and we find that 𝛼 equals Φ one minus 𝜃 one plus 𝜃 two minus Φ two. The problem with this though is that all of our answer options are given in terms of the apex angle 𝐴, whereas our solution here is not.
Let’s clear away our zoomed-in triangle then and work on finding an expression for this apex angle. To help us do that, let’s focus on this quadrilateral in pink. Considering the interior angles involved, we know that one of those angles is the apex angle 𝐴. Then there’s one interior angle of exactly 90 degrees. This is so because this angle is defined by a line normal to the surface of the prism. And in fact, there’s even a second interior angle, which is also a right angle. This is a 90-degree angle for the same reason as before. It’s defined by the surface of the prism on this side and aligned normal to that surface. The last interior angle of our quadrilateral is this unlabeled one here.
Let’s recall at this point that for any four-sided object, if we add together all four of the interior angles on that shape, we find they always add up to 360 degrees. This means if we take 𝐴 and our two 90-degree angles and our unknown angle that we’ll leave blank for now and we add them all together, their sum will be 360 degrees. Note that 90 degrees plus 90 degrees is 180 degrees. And if we subtract 180 degrees from both sides, that factor cancels on the left, and on the right we have 180 degrees. The angle 𝐴 then plus our unknown angle equals 180 degrees. And finally, if we subtract 𝐴 from both sides canceling 𝐴 out on the left, we find that the unknown or unlabeled angle in our pink quadrilateral is 180 degrees minus 𝐴.
On our diagram, the angle that we’ve just identified is this one right here. That angle, we’re saying, equals 180 degrees minus 𝐴. To see how this helps us, let’s now focus on this green triangle that we’ve highlighted. An expanded view of that triangle looks like this. Here, we have a triangle with interior angles 𝜃 one, Φ two, and 180 degrees minus 𝐴. Our triangle rule from earlier tells us that if we add up all three of these angles, their sum will be 180 degrees. If we then subtract that angle from both sides of the equation, 180 degrees cancels out, giving us this expression. And if we then add 𝐴 to both sides so that 𝐴 minus 𝐴 on the left is equal to zero, we find that our apex angle 𝐴 could be expressed as 𝜃 one plus Φ two.
Notice now that in our expression for 𝛼, we do have a 𝜃 one and Φ two term. In fact, we can group these two terms together so that we get a new expression for 𝛼 that looks like this. Note that in this expression, the value in parentheses, 𝜃 one plus Φ two, is equal to 𝐴. That means we can make this substitution.
And clearing a bit of space, we can now write 𝛼 as Φ one plus 𝜃 two minus 𝐴. This agrees, we see, with option (B) on our answers’ list. The total angle this ray of light was deviated 𝛼 is equal to Φ one, the initial angle of incidence, plus 𝜃 two, the second angle of refraction, minus 𝐴, the apex angle of our prism.
