Question

What is optical path? A monochromatic light ray of 6000 Å is incident on a glass slab. Refractive index of the slab is 1.5. Find the velocity and wavelength of reflected and refracted rays from the slab.

Hint:

When light enters a medium with a refractive index greater than 1, its velocity decreases, and its wavelength shortens, while the frequency remains constant.

Answer 1

Step 1: Definition of Optical Path.
The optical path is the path length traveled by light in a medium, which is modified by the refractive index of the medium. It is given by the formula: \[ \text{Optical Path} = \frac{\text{Physical Path}}{\text{Refractive Index}} \]
Step 2: Given Data.
- Wavelength of the incident light \( \lambda_0 = 6000 \, \text{Å} = 6.0 \times 10^{-7} \, \text{m} \), - Refractive index of the slab \( n = 1.5 \), - Speed of light in vacuum \( c = 3.0 \times 10^8 \, \text{m/s} \).
Step 3: Wavelength of Light in the Slab.
The wavelength of light inside a medium is related to the wavelength in vacuum by the formula: \[ \lambda_{\text{medium}} = \frac{\lambda_0}{n} \] Substituting the given values: \[ \lambda_{\text{medium}} = \frac{6.0 \times 10^{-7}}{1.5} = 4.0 \times 10^{-7} \, \text{m} = 4000 \, \text{Å} \] So, the wavelength of the refracted light inside the glass slab is \( 4000 \, \text{Å} \).
Step 4: Velocity of Light in the Slab.
The velocity of light in a medium is given by: \[ v = \frac{c}{n} \] Substituting the given values: \[ v = \frac{3.0 \times 10^8}{1.5} = 2.0 \times 10^8 \, \text{m/s} \] So, the velocity of light inside the glass slab is \( 2.0 \times 10^8 \, \text{m/s} \).
Step 5: Reflected Ray.
The reflected ray will have the same wavelength and speed as the incident ray because it does not enter the medium. The wavelength of the reflected ray remains \( 6000 \, \text{Å} \), and its speed is \( c = 3.0 \times 10^8 \, \text{m/s} \).
Step 6: Conclusion.
- The wavelength of the refracted ray in the glass slab is \( 4000 \, \text{Å} \), - The velocity of the refracted ray in the slab is \( 2.0 \times 10^8 \, \text{m/s} \), - The reflected ray has the same wavelength and speed as the incident ray: \( 6000 \, \text{Å} \) and \( 3.0 \times 10^8 \, \text{m/s} \).