The index of refraction for red light in water is 1.331 and that for blue light

Question

The index of refraction for red light in water is 1.331 and that for blue light is 1.340. A ray of white light enters the water at an angle of incidence of 46.8 degree. What is the underwater angle of refraction for the red component of the light? What is the underwater angle of refraction for the blue component of the light? (Give your answer to three significant figures.) The index of refraction for violet light in silica flint glass is 1.66, and that for red light is 1.62. What is the angular spread of visible light passing through a prism of apex angle 60.0 degree if the angle of incidence is 52.0 degree? See figure below.

Explanation / Answer

A. Using snell's law

n1*sin A1 = n2*sin A2

n1 = 1 for Air

A1 = angle of incidence = 46.8 deg

n2)red = 1.331

n2)blue = 1.340

1.

A2)red = arcsin (n1*sin A1/n2)red)

A2)red = arcsin (1*sin 46.8 deg/1.331)

A2)red = 33.208 deg

2.

A2)blue = arcsin (n1*sin A1/n2)blue)

A2)blue = arcsin (1*sin 46.8 deg/1.340)

A2)blue = 32.957 deg

B. A = theta

using snell's law

n1*sin A1 = n2*sin A2

Ans, for the incoming ray

sin A2 = n1*sin A1/n2

A2)violet = arcsin (1*sin 52/1.66) = 28.3402 deg

A2)red = arcsin (1*sin 52/1.62) = 29.1059 deg

for outgoing ray

A3 = 60 - A2

A3)violet = 60 - 28.3402 = 31.6598 deg

A3)rec = 60 - 29.1059 = 30.8941 deg

Now for A4

n3*sin A3 = n4*sin A4

n4 = 1

sin A4 = n3*sin A3

A4 = arcsin (n3*sin A3)

A4)violet = arcsin (1.66*sin 31.6598 deg) = 60.6091 deg

A4)red = arcsin (1.62*sin 30.8941 deg) = 56.2834 deg

The angular dispersion will be

dA = A4)violet - A4)red = 60.6091 - 56.2834 = 4.3257 deg

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