The equilateral glass prism shown has an index of refraction of 1.56. The prism is surrounded by air. A ray of light is incident on the middle of the left face of the prism. What is the smallest angle of incidence for which the ray of light will emerge from the right face of the prism.
Incident ray hits the left side of the triangle going up and then refract down toward normanl inside the prism.
To calculate the smallest angle of incidence for which the ray of light will emerge from the right face of the prism, we can use Snell's law and the concept of critical angle.
Snell's law states that the ratio of the sine of the angle of incidence (θ1) to the sine of the angle of refraction (θ2) is equal to the ratio of the indices of refraction of the two mediums:
n1(sinθ1) = n2(sinθ2)
In this case, the light is coming from air (n1 = 1) and entering the prism (n2 = 1.56).
Since the prism is equilateral, all its angles are 60 degrees. The ray of light hits the left face of the prism, which is perpendicular to the incident ray. Therefore, the angle of incidence (θ1) is 90 degrees minus the angle of the prism (60 degrees). So, θ1 = 30 degrees.
To find the angle of refraction (θ2), we need to solve Snell's law for θ2 in terms of θ1 and the refractive indices:
sinθ2 = (n1/n2) * sinθ1
Plugging in the given values, we get:
sinθ2 = (1/1.56) * sin(30 degrees)
Using a calculator, we find sinθ2 ≈ 0.367
To determine the smallest angle of incidence, we need to find the critical angle (θc) at which the ray of light will no longer emerge but will be totally internally reflected instead. The critical angle occurs when the angle of refraction (θ2) is 90 degrees.
Therefore, sinθc = (n2/n1) = 1.56/1
Using a calculator, we find sinθc ≈ 0.577
Since sinθc is larger than sinθ2, we conclude that the ray of light will never reach the critical angle and will always emerge from the right face of the prism. Therefore, the smallest angle of incidence for which the ray of light will emerge from the right face of the prism is 30 degrees.
Note: The critical angle represents the angle at which the ray of light travels parallel to the interface between the prism and air, and any angle larger than that will result in total internal reflection. In this case, the angle of incidence does not reach the critical angle, so the light will always emerge from the prism.