A light wave has a frequency of $4 \times 10^{14} Hz$. Find the difference in its wavelengths in alcohol of refractive index 1.35 and glass of refractive index 1.5:

556 Å

The correct answer is Option (4) → 556 Å

Given: Frequency $f = 4 \times 10^{14}$ Hz

Refractive indices: $n_1 = 1.35$ (alcohol), $n_2 = 1.5$ (glass)

Relation between wavelength and refractive index:

$\lambda = \frac{c}{nf}$

Now calculate:

Speed of light $c = 3 \times 10^8$ m/s

Step 1: Wavelength in alcohol

$\lambda_1 = \frac{3 \times 10^8}{1.35 \times 4 \times 10^{14}}$

$\lambda_1 = \frac{3}{5.4} \times 10^{-6} = 0.5556 \times 10^{-6} = 5.556 \times 10^{-7} \text{ m}$

Step 2: Wavelength in glass

$\lambda_2 = \frac{3 \times 10^8}{1.5 \times 4 \times 10^{14}}$

$\lambda_2 = \frac{3}{6} \times 10^{-6} = 0.5 \times 10^{-6} = 5.0 \times 10^{-7} \text{ m}$

Step 3: Difference

$\Delta\lambda = \lambda_1 - \lambda_2$

$= (5.556 - 5.0) \times 10^{-7} = 0.556 \times 10^{-7} \text{ m}$

$= 5.56 \times 10^{-8} \text{ m}$

Convert to Å:

$1 Å = 10^{-10} \text{ m}$

$\Delta\lambda = \frac{5.56 \times 10^{-8}}{10^{-10}} = 556 Å $

The correct answer is Option (4) $\rightarrow$ $556 Å$.