Question

Two waves of light in air have the same wavelength and are initially in the same phase. They then travel through plastic layers with thicknesses of $L_1=3.5\text{mm}$ and $L_2=5.0\text{mm}$ and indices of refraction $n_1=1.7$ and $n_2=1.25$ as shown in the figure. The rays later arrive at a common point. The longest wavelength of light for which constructive interference occurs at the point is

• A. $0.8\text{mm}$
• B. $1.2\text{mm}$
• C. $1.7\text{mm}$
• D. $2.9\text{mm}$

Answer 1

The correct option is B $1.2\text{mm}$

Let $\Delta x$ be the path difference between the two light waves

$\therefore \Delta x=[n_1{L}_1-L_1]-[n_2{L}_2-L_2]$

$\Delta \varphi$= phase difference between two waves

For longest wavelengths, $\Delta \varphi$ is the smallest.

$\Rightarrow \Delta {\varphi }_{min}=\frac{2\pi }{{\lambda }_{max}}\Delta x$

For constructive interference, $\Delta {\varphi }_{min}=2\pi$

$\therefore \Delta x={\lambda }_{max}$

$\therefore {\lambda }_{max}=n_1{L}_1-n_2{L}_2+(L_2-L_1)$

$=\left(1.7\right)(3.5\times {10}^{-3})-\left(1.25\right)(5.0\times {10}^{-3})+(5.0-3.5)\times {10}^{-3}$

$=(5.95-6.25+1.5)\times {10}^{-3}=1.2\times {10}^{-3}\text{m}$

$\Rightarrow {\lambda }_{max}=1.2\text{mm}$

Hence, option $\left(\text{B}\right)$ is correct.