To find the critical angle $\theta_{ce}$ for e-waves in calcite, we’ll use Snell's law and the concept of the critical angle:

1. Snell's Law: $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$

2. Critical Angle Condition: The critical angle $\theta_{ce}$ is defined as the angle of incidence in calcite for which the angle of refraction in air ($\theta_2$) is 90°.

Since the refractive index of air ($n_2$) is approximately $1.000$, and the refractive index for e-waves in calcite ($n_1$) is $1.4864$, we set $\theta_2 = 90^\circ$ and rearrange Snell's law as follows:

$\sin(\theta_{ce}) = \frac{n_2}{n_1}$

3. Calculate $\theta_{ce}$: $\sin(\theta_{ce}) = \frac{1.000}{1.4864}$

   Solving for $\theta_{ce}$:

   $\theta_{ce} = \arcsin\left(\frac{1.000}{1.4864}\right)$

Let's compute this.import math

Given values

n_air = 1.000 # Refractive index of air n_calcite_e_wave = 1.4864 # Refractive index for e-waves in calcite

Calculating the critical angle in degrees

theta_ce_radians = math.asin(n_air / n_calcite_e_wave) theta_ce_degrees = math.degrees(theta_ce_radians) theta_ce_degrees42.2809388169988442.28093881699884successThe critical angle $\theta_{ce}$ for e-waves in calcite is approximately $42.28^\circ$ (to four significant figures).

Would you like further details on this calculation, or do you have any other questions?

Here are five related questions for deeper exploration:

1. How would the critical angle change if the refractive index for e-waves were slightly higher or lower?
2. How does birefringence in calcite affect the behavior of o-waves compared to e-waves?
3. What is the critical angle for o-waves in calcite, given the refractive index for o-waves?
4. How does total internal reflection depend on the critical angle in materials with high birefringence?
5. In what practical applications is the birefringence of calcite used, especially concerning polarized light?

Tip: For highly birefringent materials like calcite, critical angles will differ for ordinary and extraordinary waves, making them useful in polarizing optics.