Does the polarizing angle e, for a transparent medium depend upon the wavelength of light?

Answer

To determine whether the polarizing angle (denoted as \( e \)) for a transparent medium depends upon the wavelength of light, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Polarizing Angle**:
The polarizing angle is defined as the angle at which the reflected light is completely polarized. At this angle, the incident ray, reflected ray, and refracted ray are related such that the reflected and refracted rays are perpendicular to each other.

2. **Using Snell's Law**:
According to Snell's Law, the relationship between the angles and the refractive indices of the two media can be expressed as:
\[ n_1 \sin(i) = n_2 \sin(r) \]
where \( n_1 \) is the refractive index of the first medium (air, usually taken as 1), \( n_2 \) is the refractive index of the second medium, \( i \) is the angle of incidence, and \( r \) is the angle of refraction.

3. **Relationship Between Angles**:
At the polarizing angle \( e \), the reflected and refracted rays are perpendicular:
\[ r + r' = 90^\circ \]
This implies that \( r' = 90^\circ - r \).

4. **Deriving the Polarizing Angle**:
From the geometry of the situation, we can express:
\[ i + r' = 90^\circ \implies i + (90^\circ - r) = 90^\circ \implies i = r \]
Hence, at the polarizing angle, we have:
\[ \tan(e) = n_2 \]
This means that:
\[ e = \tan^{-1}(n_2) \]

5. **Refractive Index and Wavelength**:
The refractive index \( n_2 \) is defined as:
\[ n_2 = \frac{c}{v} \]
where \( c \) is the speed of light in vacuum and \( v \) is the speed of light in the medium. The speed of light in a medium is related to the wavelength and frequency:
\[ v = f \cdot \lambda \]
where \( \lambda \) is the wavelength and \( f \) is the frequency of the light.

6. **Dependence on Wavelength**:
Since the refractive index can also be expressed in terms of the wavelengths in different media:
\[ n_2 = \frac{\lambda_0}{\lambda} \]
where \( \lambda_0 \) is the wavelength in vacuum and \( \lambda \) is the wavelength in the medium. Thus, the polarizing angle \( e \) does depend on the wavelength of light because \( n_2 \) changes with \( \lambda \).

### Conclusion:
Yes, the polarizing angle \( e \) for a transparent medium does depend upon the wavelength of light.