Isaac Physics Optical Dipstick Question

1.

What Makes Light Stay Inside? For that "super-efficient mirror" effect (total internal reflection) to kick in, two things need to happen. First, light has to be going from a place where it travels slower (the prism, presumably) to a place where it travels faster (air). Second, the angle at which it hits that boundary has to be pretty steep – steeper than a certain angle called the critical angle.

2.

Tracking the Light's Angle: Look at the prism's shape and that 45° angle. We need to figure out the angle at which the light beam smacks into the bottom surface. When it bounces off, the angle it goes in is the same as the angle it comes out. Then, as that reflected beam heads towards the prism-air edge, its angle changes again based on the prism's geometry. We need to find the final angle of incidence at the prism-air surface.

3.

The Tipping Point: That angle we just found? It has to be at least as big as the critical angle for the light to stay trapped inside and bounce back completely.

4.

Snell's Law to the Rescue: The critical angle is a special case. It's the angle inside the prism that makes the light bend so much when it hits the air that it skims right along the surface (at 90° to the boundary). We can use Snell's Law here: nprism​×sin(θcritical​)=nair​×sin(90°)

5.

We know that nair​ is roughly 1, and sin(90°) is exactly 1. So, our equation simplifies to: nprism​×sin(θcritical​)=1

6.

Putting It All Together: Now, that θcritical​ in the equation is related to the angle of incidence we found in step 2. By connecting these pieces and doing a little math (you've got this!), you can solve for prism​. That value will be the minimum refractive index the prism needs for the total internal reflection. The question mentions it "exceeds a certain value," and that "certain value" is what you'll calculate. Make sure to round your final answer to three significant figures.