Question

The figure shows a transparent slab of length 2 m placed in air. The refractive index of the slab varies along x-axis as μ = 2 + x2 (0 ≤ x ≤ 1). The optical path length of ray R will be

Answer 1

The optical path length of the ray R will be

 $\int\limits^1_0 {2+x^2} \, dx$  

= $2x + \frac{x^3}{3} \left \| {{x=1} \atop {x=0}} \right.$

= $2 + \frac{1}{3}$

= $\frac{7}{3} m$

Answer 2

Given :

Length  of the transparent slab = 2m

Refractive index of slab =  = 2 + x² (0≤ x ≤ 1)

To find :

The optical path length of ray R

Solution :

• Optical path length of a material =   $\int\limits^1_0{2+x^{2} } \, dx$

                                                      =$\left \{ {{x=1} \atop {x=0}} \right$ 2x+(x³/3)

                                                      =2+(1/3)

                                                      =7/3m

• The optical path length of the material os 7/3m