Question

Red light is incident in air on a 30^o - 60^o - 90 ^o prism as shown. The incident beam is directed at an angle of φ₁ = 47.6^o with respect to the horizontal and enters the prism at a height h = 15 cm above the base. The beam leaves the prism to the air at a distance d = 37.6 cm along the base as shown.

1)

What is φ₂, the angle the beam in the prism makes with the horizontal axis?

degrees

2)

What is n, the index of refraction of the prism for red light?

3)

What is φ₃, the angle the transmitted beam makes with the horizontal axis?

degrees

4)

What is φ_1,max, the maximum value of φ₁ for which the incident beam experiences total internal reflection at the horizontal face of the prism?

degrees

5)

The red beam is now replaced by a violet beam that is incident at the same angle φ₁ and same height h. The prism has an index of refraction n_violet = 1.7 for violet light. Compare d_violet, the exit distance for violet light, to d, the exit distance for red light.

d_violet < d

d_violet = d

d_violet > d

6)

Suppose now that the violet beam is incident at height h, but makes an angle φ_1,v = 60^o with the horizontal. What is φ_3,v, the angle the transmitted beam makes with the horizontal axis?

degrees

Answer 1

1 )

h / d-x = tan30

x = d - ( h / tan30 )

= 0.376 - ( 0.15 / tan30 )

x = 0.1161 m

tan₂ = h / x

₂ = tan^-1 ( 0.15 / 0.1161 )

₂ = 52.26^o

2 )

r = 90 - 30 - 52.26

r = 7.73^o  

i = 90 - 30 - 47.6

i = 12.4^o

n₂ = n₁ ( sini / sinr )

= 1 x ( sin12.4 / sin7.73 )

n₂ = 1.596

3 )

i₂ = 90 - ₂

= 90 - 52.26

i₂ = 37.74^o

n₂ sini₂ = n₁ sinr₂

r₂ = sin^-1 ( 1.596 x sin37.74 / 1 )

r₂ = 77.655^o

₃ = 90 - r₂

= 90 - 77.655

₃ = 12.344^o  

4 )

sin_c = n₁ / n₂

_c = sin^-1 ( n₁ / n₂ )

= sin^-1 ( 1 / 1.596 )

_c = 38.797^o

₂^' = 90 - _c

= 90 - 38.797

₂^' = 51.2^o

r ' = 90 - 30 - ₂^'

r ' = 8.8^o

n₁ sini' = n₂ sinr'

i' = sin^-1 ( 1.596 x sin8.8 / n₁ )

= 14.13^o

_1max^' = 90 - 30 - i '

= 45.87^o