The indices of refraction for violet light (λ = 400 nm) and red light (λ = 700 nm) in diamond are 2.46 and 2.41, respectively. A ray of light traveling through air strikes the diamond surface at an angle of 51.0 ∘ to the normal.

Calculate the angular separation between these two colors of light in the refracted ray.

The angular separation between the two colors of light in the refracted ray is of 0.42°.

Given data:

The wavelength of violet light is, .

The wavelength of red light is, .

The index of refraction of violet light is, n = 2.46.

The index of refraction of red light is, n' = 2.41.

The angle of incidence is, i = 51.0°.

We can apply the Snell's law of refraction which states that the ratio of the angle of sine of incidence to angle of sine of refraction is equal to a constant for a given pair of media. Mathematically,

sin i/sin r = n

Here,

r is the angle of refraction.

Solving as,

sin(51.0)/sin(r) = 2.46

sin r = sin51°/2.46

sin r = 0.316

r = sin^-1(0.316)

r = 18.42°

Now similarly for the red light we have,

sin i/sin r' = n'

Here,

r' is the angle of refraction of red light

Solving as,

sin(51.0)/sin(r') = 2.41

sin r' = sin51°/2.41

sin r' = 0.323

r' = sin^-1(0.323)

r' = 18.84°

The angular separation between these two colors of light in the refracted ray will be the difference between there angle of refraction.

Angular separation = r' - r

                                 = 18.84° - 18.42°  

                                = 0.42°

Thus, we can conclude that the angular separation between the two colors of light in the refracted ray is of 0.42°.

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0.42°

Explanation:

Using Snell's law of refraction which states that the ratio of the angle of sin of incidence to angle of sine of refraction is equal to a constant for a given pair of media. Mathematically,

Sin(i)/sin(r) = n

n is the refractive index of the medium

FOR VIOLET LIGHT:

n = 2.46

i = 51°

r = ?

To get r, we will use the Snell's law formula.

2.46 = sin51°/sinr

Sinr = sin51°/2.46

Sinr = 0.316

r = sin^-1(0.316)

rv = 18.42°

FOR RED LIGHT:

n = 2.41

i = 51°

r = ?

To get r, we will use the Snell's law formula.

2.41 = sin51°/sinr

Sinr = sin51°/2.41

Sinr = 0.323

r = sin^-1(0.323)

rd = 18.84°

The angular separation between these two colors of light in the refracted ray will be the difference between there angle of refraction.

Angular separation = rd - rv

= 18.84° - 18.42°

= 0.42°

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