[College Physics II: Waves/Optics] Index of Refraction

[AgentRedfield]

Homework Statement

A solid glass cube with edge length of 10.0 mm and index of refraction n=1.75 has a small, dark spot dead center of the cube. Find the minimum radius of black paper circles that could be pasted at the center of each cube face to prevent the center spot from being seen, no matter what the direction of viewing.
Picture (link)

Homework Equations

Snell's Law (maybe?)
n₁/n₂=sin(θ₂)/sin(θ₁)

The Attempt at a Solution

I'm really confused where to even start. A hint in the right direction is all I'm hoping for. Thank you for your time.

 

[Simon Bridge]

Hint: total internal reflection.

 

[AgentRedfield]

Hint: total internal reflection.

I was able to figure it out. Thank you!

 

[Simon Bridge]

Well done and no worries :)

 

[HalfLostAndSearching]

To solve this problem, we need to consider the path of light through the glass cube. The light will enter the cube through one face, travel through the glass, and then exit through another face. The index of refraction of the glass is 1.75, which means that the speed of light in the glass is 1.75 times slower than in vacuum.

To prevent the dark spot from being seen, we need to block the light from reaching the center of the cube. This can be achieved by placing a black paper circle at the center of each face of the cube. The radius of the circle will determine the amount of light that is blocked.

To determine the minimum radius of the black paper circle, we can use Snell's law. This law relates the angles of incidence and refraction for a light ray passing through two different media. In this case, we can use it to find the angle of refraction for a light ray passing through the glass cube.

Using the given information, we can set up the following equation:

n1*sin(θ1) = n2*sin(θ2)

Where n1 is the index of refraction of the first medium (air), n2 is the index of refraction of the second medium (glass), θ1 is the angle of incidence, and θ2 is the angle of refraction.

Since we want to block the light from reaching the center of the cube, we can set θ2 to be 90 degrees, which means that the light will be refracted at a right angle. This will ensure that the light does not reach the center of the cube.

Substituting in the values, we get:

1*sin(θ1) = 1.75*sin(90)

Simplifying, we get:

sin(θ1) = 1.75

Using a calculator, we can find that the angle of incidence is approximately 63.43 degrees. Now, we can use this angle to find the minimum radius of the black paper circle.

We can draw a triangle with one side being the edge length of the cube (10.0 mm) and the other side being the radius of the circle (r). The angle between these two sides is θ1, which we just found to be 63.43 degrees.

Using basic trigonometry, we can set up the following equation:

sin(θ1) = r/10.0

Solving for