Question

If a ray of light passes through a medium, its frequency and wavelength are \( 4 \times 10^{14} \) Hz and 450 nm respectively. Then the refractive index of the medium is:

• A. 1.67
• B. 1.5
• C. 1.414
• D. 1.33
• E. 1.2

Hint:

The refractive index is the ratio of the speed of light in vacuum to the speed of light in a medium. It can also be determined using the ratio of the wavelength in vacuum to the wavelength in the medium.

Answer 1

The Correct Option is A

The refractive index \( n \) of a medium is related to the speed of light in vacuum \( c \), the speed of light in the medium \( v \), and the wavelength of light in the medium \( \lambda \) and in vacuum \( \lambda_0 \) as: \[ n = \frac{c}{v} = \frac{\lambda_0}{\lambda} \] Where: - \( c \) is the speed of light in vacuum (\( 3 \times 10^8 \, {m/s} \)),
- \( \lambda_0 \) is the wavelength in vacuum,
- \( \lambda \) is the wavelength in the medium.
From the problem, we are given:
- The frequency \( f = 4 \times 10^{14} \, {Hz} \),
- The wavelength in the medium \( \lambda = 450 \, {nm} = 450 \times 10^{-9} \, {m} \).
Now, we know that the speed of light in a medium is related to the frequency and wavelength by the equation: \[ v = f \lambda \] Substitute the values for frequency and wavelength in the medium: \[ v = (4 \times 10^{14}) \times (450 \times 10^{-9}) = 1.8 \times 10^8 \, {m/s} \] Next, using the equation for refractive index: \[ n = \frac{c}{v} = \frac{3 \times 10^8}{1.8 \times 10^8} = 1.67 \] Thus, the refractive index of the medium is \( \boxed{1.67} \).