Question

Consider scenarios A to F in which a ray of light traveling in material 1 is incident onto the interface with material 2.Part BFor the scenarios in which total internal reflection is possible, rank the scenarios on the basis of the critical angle, the angle above which total internal reflection occurs. At this angle, the refracted ray is at 90 degrees from the normal.Rank from largest to smallest. To rank items as equivalent, overlap them.n1water=1.33, n2air=1.00n1diamond=2.42, n2air=1.00n1diamond=2.42, n2water=1.33n1benzene=1.50, n2water=1.33

Answer 1

Scenarios where total internal reflection is possible are ranked by the critical angle, which is largest when the difference in indices of refraction is greatest. This happens for a diamond-air boundary and is smallest for a diamond-water boundary. Critical angles are determined using Snell's Law and the indices of refraction for the two materials.

Step-by-step explanation:

Total internal reflection occurs when a ray of light strikes the boundary between two materials with different indices of refraction and the incident angle is greater than the critical angle for that pair of materials. The critical angle is the incident angle that produces a refracted ray at 90° to the normal at the boundary. When ranking scenarios of total internal reflection by critical angle, we consider the indices of refraction for the materials involved.

Using Snell's Law, we find the critical angle (θc) where θc is the incident angle for which the refracted angle (θ2) is 90°. The formula derived from Snell's Law for critical angle is:

θc = arcsin(n2/n1)

Ranking the critical angles from largest to smallest for the given scenarios:

1. n1diamond=2.42, n2air=1.00 (Largest θc)
2. n1water=1.33, n2air=1.00
3. n1benzene=1.50, n2water=1.33
4. n1diamond=2.42, n2water=1.33 (Smallest θc)

The critical angle is largest when the difference in indices of refraction is greatest (diamond to air) and smallest when the difference is the least (diamond to water).

Answer 2

D>A>C>B

Step-by-step explanation:

When a ray of light, traveling in a material with a given index of refraction, incides onto other material with a index of refraction, as it changes its speed when crossing the boundary between them, its trajectory binds (only at the boundary) , which is explained saying that the ray refracts.

The relationship between the angle that the incident ray makes with the normal, and the one that the refracted ray makes with it, is explained by Snell's Law:

n₁* sin i = n₂* sin r

where n₁ is the index of refraction of the incident material, i is the incident angle (from the normal to the interface between the two materials), n₂ is the index of refraction of the 2nd material, and r is the angle that the refracted ray makes with normal.

We can arrange this equation as follows:

`(sin i)/(sin r) =(n2)/(n1)`

While sin r remains < 1, there will be a refraction. When sin r = 1, the refracted ray makes a 90º with the normal, so it is parallel to the interfase between materials.

The incident angle that meets this condition, is called the critical angle. Beyond this angle, no energy will be refracted, and all it will be reflected back to the first material (this is the principle in which fiber optics communications is based on).

The critical angle, then, can be expressed as follows:

θ = sin⁻1 (n₂/n₁)

For the different scenarios, from A to D, we have:

A) n₂ =1.00 n₁ = 1.33 ⇒ sin θ = 0.752 ⇒ θ = 48.7º

B) n₂ =1.00 n₁ = 2.42 ⇒ sin θ = 0.413 ⇒ θ = 24.4º

C) n₂ =1.33 n₁ = 2.42 ⇒ sin θ = 0.550 ⇒ θ = 33.4º

D) n₂ =1.33 n₁ = 1.50 ⇒ sin θ = 0.887 ⇒ θ = 62.5º

Ranking the different scenarios based on the value of the critical angle, we have the following progression:

D>A>C>B