Question

Two transparent slabs have the same thickness as shown. One is made of material A of refractive index 1.5. The other is made of two materials B and C with thickness in the ratio $1:2$. The refractive index of C is 1.6. If a monochromatic parallel beam passing through the slabs has the same number of waves inside both, then

• A. the optical path length travelled in B and C are equal
• B. the optical path length travelled in B is lesser than in C.
• C. refractive index of B is 1.3
• D. None of the above

Answer 1

Answer: (B), (C)

The correct options are
B the optical path length travelled in B is lesser than in C.
C refractive index of B is 1.3

For A : Let the thickness be (as in diagram) $t$
Total number of waves = $\frac{\left(1.5\right)t}{\lambda }$ ---(1)
$\because \left(\begin{array}{c}\text{Total number}\\ \text{of waves}\end{array}\right)=\left(\frac{\text{optical path length}}{\text{wavelength}}\right)$
For B and C: The thickness of B is $\frac{t}{3}$ and C is $\frac{2t}{3}$ as shown in diagram.

Total number of waves = $\frac{n_B\left(\frac{t}{3}\right)}{\lambda }+\frac{\left(1.6\right)\left(\frac{2t}{3}\right)}{\lambda }$ ---(2)
Equating (1) and (2) we get
$n_B=1.3$

Optical path length travelled in B $=\frac{n_B\left(\frac{t}{3}\right)}{\lambda }=0.43\frac{t}{\lambda }$

Optical path length travelled in C $=\frac{\left(1.6\right)\left(\frac{2t}{3}\right)}{\lambda }=1.07\frac{t}{\lambda }$