Dispersion and polarization of light

Solution:

Decomposition of Light

White light is a mixture of seven monochromatic colors (red, orange, yellow, green, blue, indigo, violet), i.e., a mixture of electromagnetic waves with different frequencies.

Individual colored lights propagate in a medium at different speeds $v < c$,
therefore they have different refractive indices and refract at different angles.

$$n = \frac{\lambda_0}{\lambda} = \frac{c}{\frac{c}{\lambda f}} = \frac{c}{\lambda f}$$ $$v = \frac{c}{n} = \frac{c}{\frac{c}{\lambda f}} = \lambda f$$

Polarization of Light

Natural light is unpolarized. It is a transverse electromagnetic wave, characterized by oscillations of mutually perpendicular vectors
$\vec{E}\perp \vec{B}$. On the retina of the eye, only the oscillations of the vector of electric field intensity $\vec{E}$ act.
In polarized light, the oscillation of vector $\vec{E}$ occurs in one plane.

Complete polarization occurs when the angle of incidence is the so-called Brewster’s angle $\alpha_p$.

$$\tan \alpha_p = \frac{n_2}{n_1}$$

At the polarization angle $\alpha_p$, the reflected ray is perpendicular to the refracted ray.

Solution:

Analysis:

The wavelength of the hydrogen line H₂ in water is λ = 493.5nm.

Solution:

a) In vacuum:

$$n = \frac{c}{v}$$$$$$

$$v = \frac{c}{n} = \frac{c}{1} = c$$$$$$

$$v = c \;-\; \text{does not depend on frequenc$$

b) In a medium:

$$n = \frac{c}{v}$$$$$$

$$v = \frac{c}{n} = \frac{c}{f\lambda} = f\lambda$$$$$$

$$v = f\lambda \;-\; \text{depends on frequenc$$ 

In vacuum, the speed of light does not depend on frequency, in a medium it does.

Solution:

At the polarization angle α_p, the reflected beam is perpendicular to the refracted beam.
 

The formula for calculating Brewster's angle is tgα_p = n_1,2

Solution:

Analysis:

The refractive index of enamel is n₂ = 1.6

Solution:

Water:      $\mathrm{ tan}{\alpha }_p=n=1.33\Rightarrow {\alpha }_p={53}^{\circ }$
Glass:        $\tan \alpha_p = n = 1.5 \Rightarrow \alpha_p = 56.3^\circ$
Diamond: $\tan \alpha_p = n = 2.465 \Rightarrow \alpha_p = 68^\circ$

Brewster's angle of complete polarization is for water 53⁰, glass 56.3⁰ and diamond 68⁰.

Solution:

Solution:

The red ray deviates the least from the original direction, the violet ray deviates the most.

Solution:

The wavelength of the violet component of light in vacuum is λ₀ = 3.84·10^–7 m, and in the diamond λ_D = 1.55·10^–7 m.

Solution:

The refractive index of diamond is n_D = 2.475. The angle of refraction is β = 22^o.

Solution:

Complete polarization occurs when the angle of incidence equals the polarization angle α_P = 45.5^o.