Question

A ray of light is incident onto the interface between material 1 and material 2.

Given the indices of refraction n_1 and n_2 of material 1 and material 2, respectively, rank these scenarios on the basis of the phase shift in the refracted ray.

1) Rank from largest to smallest. To rank items as equivalent, overlap them.

a) n1,water=1.33
n2,air=1.00

b) n1,water=1.33
n2,quartz=1.46
c) n1,water=1.33
n2,diamond=2.42
d) n1,air=1.00
n2,quartz=1.46
e) n1,benzene=1.50
n2,water=1.33
f) n1,air=1.00
n2,water=1.33

2) Rank these scenarios on the basis of the phase shift in the reflected ray.
a) n1,benzene=1.50
n2,water=1.33
b) n1,air=1.00
n2,quartz=1.46
c) n1,water=1.33
n2,quartz=1.46
d) n1,water=1.33
n2,air=1.00
e) n1,water=1.33
n2,diamond=2.42
f) n1,air=1.00
n2,water=1.33

Answer 1

The concepts used to solve the given problem are total internal reflection and critical angle.

First, determine the expression of the critical angle from the expression of Snell's law. Next, use the expression of critical angle obtained previously to calculate the critical angle for each of the four cases and arrange the values thus obtained in decreasing order.

Critical angle: When the incident ray of light traveling through a denser medium, progresses towards the interface with the rarer medium, the light ray gets refracted at an angle, with the normal at the interface. However, when the incidence angle is such that the refraction angle is equivalent to 90°, then the incidence angle is referred to as the critical angle of incidence.

Total internal reflection: At any angle greater than the critical angle of incidence the incident ray of light gets totally internally reflected and this phenomenon is known as the total internal reflection.

Expression for Snell's law is written as,

Here, θ_i represents the incidence angle, θ_r represents the refraction angle, n₂ represents the refractive index of the refraction medium, and n₁ represents the refractive index of the incident medium.

(B)

Consider, θ_c is the critical angle of incidence

From Snell's law,

Here, n₂ is the refractive index of the rarer medium, and n₁ is the refractive index of the denser medium.

Substitute θ_c for θ_i, and 90° for θ_r.

Thus,

                                                                                                       

The total internal reflection of a light ray is possible only when the refractive index of the medium in which the light ray travels initially, is greater than the other medium at the interface, that is, the light ray travels from a denser medium to a rarer medium.

Use the expression for Snell'law and solve for the critical angle of incidence defined as the angle of incidence for which the refraction angle is 90° .

Do not consider the refraction angle to be greater than 90°  for the incidence angle to be the critical angle of incidence; it is incorrect. The critical angle is referred for an incidence angle at which the refraction angle is equivalent to 90° .

Use the expression of critical angle obtained in step 1 and calculate the critical angle for each of the four cases and arrange the values thus obtained in decreasing order.

Calculate the critical angle.

Case 1:

The expression for critical angle is,

Substitute 1.00 for n₂ and 2.42 for n₁.

Case 2:

The expression for critical angle is,

Substitute 1.33 for n₂ and 2.42 for n₁.

Case 3:

The expression for critical angle is,

Substitute 1.33 for n₂ and 1.50 for n₁.

Case 4:

The expression for critical angle is,

Substitute 1.33 for n₁ and 1.00 for n₂.

Part B

For the given scenarios, the critical angles are ranked in decreasing order as.

Use the expression of the critical angle of incidence obtained in step (1) and calculate the critical angle for the given four conditions and arrange the critical angles thus obtained in decreasing order.

Do not write the expression for critical angle as, , because the correct expression for critical angle is .