A triangular prism produces a minimum angle of deviation of light rays through it when the angle of incidence of light rays onto the prism is 44 degrees. The minimum angle of deviation of the prism is 28 degrees. What is the apex angle of the prism? Answer to the nearest degree.
Let’s say that this is our prism, with unknown apex angle we’ll call 𝐴. That’s the angle we want to solve for. And to help us do it, we’re told that when a ray of light passes through this prism, the smallest possible angle of deviation of that ray, this angle here, is equal to 28 degrees. We’ll call this angle 𝛼 zero.
We’re told further that the angle of incidence of this ray, this angle right here, is equal to 44 degrees. And we’ll call this angle 𝜙 zero. Whenever we have a ray being deviated by a triangular prism with a minimum angle of deviation, like we have here, we can recall a mathematical relationship between these variables 𝜙 zero, 𝐴, and 𝛼 zero. This relationship, which only holds true at the minimum angle of deviation 𝛼 zero, states that the angle of incidence for that angle of deviation 𝜙 zero equals the minimum angle of deviation of a ray 𝛼 zero plus the apex angle of the prism 𝐴 all divided by two.
In our scenario, it’s not 𝜙 zero but rather 𝐴 that we want to solve for. So, to help us do that, let’s multiply both sides of this equation by two, canceling that factor out on the right. And then as a next step, we could subtract the minimum angle of deviation 𝛼 zero from both sides. On the right-hand side, 𝛼 zero minus 𝛼 zero adds up to zero, leaving us with this equation that two times 𝜙 zero minus 𝛼 zero equals the apex angle 𝐴. Note that we know the angles 𝜙 zero and 𝛼 zero. 𝜙 zero is given as 44 degrees, and 𝛼 zero is 28 degrees. 𝐴 equals two times 44 degrees, or 88 degrees, minus 28 degrees, which equals 60 degrees. This is the apex angle of the prism.