How Much of a Glass Cube's Surface Must Be Covered to Hide a Central Spot?

[kreil]

Homework Statement

A glass cube has a small spot at its center. What parts of the cube face must be covered to prevent the spot from being seen, no matter what the direction of viewing? What fraction of the cube face must be covered? Assume a cube edge of 1 cm and a refractive index of 1.50. (Neglect subsequent behaviour of an internally reflected ray.)

Homework Equations

I think the only relevant equations are snells law and the critical angle equation:

$$n_1 sin( \theta_1)=n_2 sin( \theta_2)$$

$$\theta_c=\frac{n_2}{n_1},,,,(n_2<n_1)$$

The Attempt at a Solution

I have no clue how to even start this. I need some hints first.

-Josh

 

[andrevdh]

The rays from the spot that do not suffer total internal reflection need to be blocked.

Draw a little square with a dot in the middle (a top view of the situation). All rays from the one straight towards the front of the cube up to first ray that will be refracted along the face of the cube (the "last ray") need to blocked off. This region will then form a circle on the side face. Notice that the incident angle of the "last ray" will be the critical incident angle.

 

[m2003]

The first step in solving this problem is to understand the concept of refraction. Refraction is the bending of light as it passes through a medium with a different refractive index. In this case, the glass cube has a refractive index of 1.50, which means that light will bend as it passes through the cube.

To prevent the spot from being seen, we need to cover the parts of the cube face where light enters and exits the cube. This means that we need to cover two opposite faces of the cube. This is because when light enters the cube, it will bend towards the normal (an imaginary line perpendicular to the surface of the cube) and when it exits the cube, it will bend away from the normal. By covering two opposite faces, we ensure that no light enters or exits the cube, thus preventing the spot from being seen.

As for the fraction of the cube face that needs to be covered, we can use the critical angle equation to determine this. The critical angle is the angle of incidence at which the refracted ray will travel along the surface of the cube. In this case, the critical angle is given by:

θc = sin^-1 (1/1.50) = 41.81°

This means that any incident angle greater than 41.81° will result in total internal reflection, where the light will not enter or exit the cube. Therefore, the fraction of the cube face that needs to be covered is the area of the face that corresponds to an angle greater than 41.81°. This can be calculated by dividing the area of the cube face by the total surface area of the cube, which is 6 cm^2. This gives us a fraction of 2/6 or 1/3 of the cube face that needs to be covered.

In conclusion, to prevent the spot from being seen, we need to cover two opposite faces of the cube and the fraction of the cube face that needs to be covered is 1/3. I hope this helps you to understand the problem better. Good luck with your homework!