Question

Using Snell’s Law, derive an equation for the critical angle θc. Write your final equation in the answer textbox below.

n1 sinθ , = n2sinθ2
where n and n2 are the indices of refraction for the two materials, θ1 is the angle of incidence, and θ2 is the angle of refraction. In cases where ni > n2, there exists a range of angles at which all light incident on the second material can be trapped inside the first material, with none of it being transmitted to the second material. The minimum angle at which this occurs is the critical angle c, which is defined as the incident angle that would result in a refraction angle θ2 = 90°

Answer 1

The critical angle, represented as θc, can be derived using Snell's Law by setting the angle of refraction to 90° and solving for θ1, which results in θc = sin-1(n2/n1).

Step-by-step explanation:

Snell's Law describes the relationship between the angles of incidence and refraction when a light ray passes from one transparent medium to another. The law is articulated as n1 sin θ1 = n2 sin θ2, where n1 and n2 are the indices of refraction for the two materials, and θ1 and θ2 are the angles of incidence and refraction, respectively. When dealing with the critical angle, θc, this condition occurs when the angle of refraction, θ2, equals 90°, which makes sin θ2 equal to 1.

Here's how we can derive the equation for the critical angle from Snell's Law:

1. For total internal reflection to occur, set θ2 to 90°, where sin θ2 = 1.
2. Snell's Law can be rearranged to solve for the critical angle, θ1 (or θc): n1 sin θc = n2 × 1.
3. Isolate sin θc on one side of the equation to yield sin θc = n2/n1.
4. Take the inverse sine of both sides to get θc = sin-1(n2/n1).

Therefore, the formula for the critical angle when the light is moving from a denser medium to a less dense medium is θc = sin-1(n2/n1).