The relation between the polarizing angle and the refractive index of a material is defined by Brewster's Law.
Brewster's Law Explained
Brewster's Law establishes a direct connection between the polarizing angle (often called the Brewster angle, denoted by $\theta_p$ or $\theta_B$) and the refractive indices of the two media involved when light is incident upon an interface.
According to the reference provided, Brewster's Law describes the relationship between the index of refraction and the polarizing angle as $(\theta) = \frac{n_2}{n_1}$ where $\theta$ is the polarizing angle, $n_1$ is the index of refraction for the medium where the reflected light travels (usually $n_1 = 1.0$ referring to the medium of air), and $n_2$ is the index of the second medium.
In this formula:
- $\theta$ (or $\theta_p$, $\theta_B$) represents the polarizing angle. This is the specific angle of incidence at which light reflecting off a non-metallic surface is completely polarized parallel to the surface.
- $n_1$ is the refractive index of the medium through which the incident light travels (e.g., air). As the reference states, $n_1$ is usually approximated as $1.0$ for air.
- $n_2$ is the refractive index of the second medium (the material the light is reflecting or refracting from, e.g., glass, water).
Essentially, the tangent of the polarizing angle is equal to the ratio of the refractive index of the second medium to the refractive index of the first medium.
Key Relationship Insight
This formula shows that the polarizing angle is dependent only on the relative refractive indices of the two materials forming the interface.
- Higher refractive index (n₂): For a given $n_1$, a higher refractive index $n_2$ of the second medium results in a larger polarizing angle $\theta$.
- Lower refractive index (n₂): A lower $n_2$ results in a smaller polarizing angle $\theta$.
Practical Applications and Examples
Brewster's Law has several practical applications:
- Polarizing Sunglasses: These often use filters oriented to block horizontally polarized light, which is the orientation of light reflected off horizontal surfaces like water or roads at or near the Brewster angle. This significantly reduces glare.
- Optical Instruments: Used in the design of polarizers and other optical components.
- Determining Refractive Index: By measuring the polarizing angle for a material, its refractive index can be calculated if the refractive index of the surrounding medium is known.
Example:
Imagine light in air ($n_1 \approx 1.0$) is incident on a type of glass with a refractive index ($n_2$) of 1.5.
Using Brewster's Law:
$(\theta) = \frac{n_2}{n_1} = \frac{1.5}{1.0} = 1.5$
To find the polarizing angle $\theta$, you take the inverse tangent (arctan or $^{-1}$) of 1.5:
$\theta = ^{-1}(1.5) \approx 56.3^\circ$
This means that when light hits this glass at an angle of incidence of about 56.3 degrees, the reflected light will be almost completely polarized parallel to the glass surface.
Summary Table
| Term | Symbol | Description | Relation in Brewster's Law |
|---|---|---|---|
| Polarizing Angle | $\theta$ | Angle of incidence for maximum reflection polarization | $(\theta)$ |
| Refractive Index | $n_1, n_2$ | Measures how much a medium slows down light | Ratio $n_2/n_1$ |
In essence, Brewster's Law provides a specific angle at which reflection yields maximum polarization, and this angle is solely determined by the refractive properties of the materials involved.
