Calcite and Refraction of Light

Calcite is a birefringent material, which means it has two different refractive indices for different polarizations of light. When unpolarized light enters a calcite plate at an angle of incidence, it splits into two rays with different refractive indices. The angular separation between these two emerging rays can be determined using the laws of refraction.

Refractive Indices of Calcite

The refractive index of calcite depends on the polarization of light. The ordinary refractive index (no) corresponds to the refractive index for light polarized perpendicular to the optic axis, while the extraordinary refractive index (ne) corresponds to the refractive index for light polarized parallel to the optic axis.

For calcite, no = 1.6584 and ne = 1.4864.

Angle of Incidence and Angle of Refraction

When unpolarized light enters the calcite plate at an angle of incidence, it splits into two rays with different angles of refraction. Let's consider the angle of incidence as θi and the angles of refraction for the ordinary and extraordinary rays as θo and θe, respectively.

Snell's Law for Refraction

Snell's law relates the angles of incidence and refraction for a given medium. It is given by:

n1 * sin(θ1) = n2 * sin(θ2)

where n1 and n2 are the refractive indices of the initial and final mediums, and θ1 and θ2 are the angles of incidence and refraction, respectively.

Calculation of Angle of Refraction

In the case of calcite, the two rays experience different refractive indices. Therefore, we can apply Snell's law separately for each ray.

For the ordinary ray:
n0 * sin(θi) = no * sin(θo)

For the extraordinary ray:
n0 * sin(θi) = ne * sin(θe)

Angular Separation between the Emerging Rays

The angular separation between the two emerging rays can be determined by subtracting their angles of refraction. In other words, the angular separation is given by:

Angular separation = θe - θo

Calculation and Solution

Given that the angle of incidence is 50 degrees, we can calculate the respective angles of refraction for the ordinary and extraordinary rays using Snell's law.

For the ordinary ray:
1.6584 * sin(50) = 1.6584 * sin(θo)
sin(θo) = sin(50) / 1.6584

Using the inverse sine function, we find that θo = 29.41 degrees.

For the extraordinary ray:
1.6584 * sin(50) = 1.4864 * sin(θe)
sin(θe) = sin(50) / 1.4864

Again, using the inverse sine function, we find that θe = 32.55 degrees.

Finally, we can calculate the angular separation between the two emerging rays:
Angular separation = θe - θo = 32.55 - 29.41 = 3.14 degrees.

Thus, the angular separation between the two emerging rays