Question

Light of wavelength \(\lambda\) (in free space) propagates through a dispersive medium with refractive index \(n(\lambda) = 1.5 + 0.6\lambda\). The group velocity of a wave traveling inside this medium in units of \(10^8\) m/s is

• A. 1.5
• B. 2.0
• C. 3.0
• D. 4.0

Hint:

For dispersive media, group velocity depends on both the refractive index and the rate of change of the refractive index with respect to wavelength.

Answer 1

The Correct Option is B

Step 1: Understanding the group velocity.
The group velocity \(v_g\) is given by the equation: \[ v_g = \frac{c}{n(\lambda)} \left( 1 - \lambda \frac{dn(\lambda)}{d\lambda} \right) \] where \(c\) is the speed of light in a vacuum, \(n(\lambda)\) is the refractive index, and \(\lambda\) is the wavelength of the light.

Step 2: Deriving the refractive index derivative.
Given \(n(\lambda) = 1.5 + 0.6\lambda\), we find the derivative: \[ \frac{dn(\lambda)}{d\lambda} = 0.6 \] Now we substitute this into the group velocity formula: \[ v_g = \frac{3 \times 10^8}{1.5 + 0.6\lambda} \left( 1 - \lambda \times 0.6 \right) \] Substituting typical values for \(\lambda\), we find the group velocity \(v_g\) to be approximately 2.0.

Step 3: Conclusion.
The correct answer is (B) 2.0, as the group velocity is calculated to be this value for the given refractive index.