Question

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.

Answer 1

Answer:

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°

Answer 2

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, $\lambda_{v}=400 \;\rm nm =400 \times 10^{-9}\;\rm m$.

The wavelength of red light is, $\lambda_{r}=700 \;\rm nm =700 \times 10^{-9} \;\rm m$.

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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