Video Transcript
Which of the following formulas correctly relates 𝛼 zero, the minimum deviation angle of a triangular prism, to 𝜙 zero, the angle of incidence of light onto the prism that corresponds to the minimum deviation angle, and 𝐴, the apex angle of the prism? (A) 𝜙 zero equals 𝛼 zero plus 𝐴. (B) 𝜙 zero equals two times the quantity 𝛼 zero plus 𝐴. (C) 𝜙 zero equals 𝛼 zero plus 𝐴 over two. (D) 𝜙 zero equals 𝛼 zero divided by two plus 𝐴. (E) 𝜙 zero equals the quantity 𝛼 zero plus 𝐴 all divided by two.
We’re considering here a situation with a triangular prism with apex angle 𝐴, where a ray of light passes through this prism in such a way that the deviation angle of that ray, what we usually call 𝛼, is minimized. So we give it the special label 𝛼 zero. At this deviation angle, the original angle of incidence of our ray is labeled as 𝜙 zero. Our question asks us to identify the correct relationship between these three values: 𝜙 zero, 𝐴, and 𝛼 zero. We’ve seen that two of these values, 𝛼 zero and 𝜙 zero, are labeled as such because we have a special case of the minimum angle of deviation of our ray here.
In general, though, the angle of deviation 𝛼 of any ray passing through a triangular prism can be written this way. Here, 𝜙 one is the original angle of incidence of the ray as it enters the prism. 𝜃 two is the angle of refraction of the ray as it leaves the prism. On our diagram, that will be indicated by this angle in pink. And then 𝐴 is the apex angle of the prism. As we’ve seen, we’re working with a special case where the angle 𝛼 is minimized. When that’s true, we give 𝛼 the label 𝛼 zero. 𝜙 one, the original angle of incidence of the ray, becomes 𝜙 zero. This is the angle of incidence corresponding to the minimum angle of deviation. And then, let’s consider 𝜃 two, the angle of refraction of our ray as it leaves the prism.
Whenever a ray of light passes through a triangular prism and is deviated the least amount, that is, 𝛼 equals 𝛼 zero, that ray as it travels through the prism follows a horizontal line. That fact that this ray is horizontal as it’s inside the prism means that the ray’s original angle of incidence, what we’ve called 𝜙 zero, and its final angle of refraction, what we’ve called 𝜃 two, must be equal.
One way to see that this is the case is to imagine that our ray changes direction and instead of traveling left to right travels right to left. This ray would follow the same path as before. And so we can say the path of the ray is symmetric about the center of the prism. All this means that in our special case, where 𝛼 is equal to 𝛼 zero, we can call 𝜃 two as 𝜙 zero. This angle of refraction is the same as the original angle of incidence. The apex angle 𝐴 stays the same since it’s a constant.
So we now have a formula for the angle of deviation for the special case where that angle is minimized. Simplifying just a bit, we can write that 𝛼 zero equals two times 𝜙 zero minus 𝐴. And notice that in our answer options, 𝜙 zero is the subject in all of these equations. So let’s add the apex angle 𝐴 to both sides, canceling that angle on the right, and then divide both sides of the equation by two, removing that factor from numerator and denominator on the right. We find that 𝜙 zero equals 𝐴 plus 𝛼 zero all divided by two.
This corresponds with answer choice (E) in our list. The angle of incidence 𝜙 zero corresponding to the minimum angle of deviation equals the quantity 𝛼 zero, that minimum angle of deviation, plus 𝐴, the apex angle of the prism, all divided by two.
