A long rectangular slab of transparent medium of thickness d is placed on table with length parallel to x axis and width parallel to the y -axis. A ray of light travelling along y -axis enters at orig

Question

A long rectangular slab of transparent medium of thickness d is placed on table with length parallel to x axis and width parallel to the y -axis. A ray of light travelling along y -axis enters at origin such that angle of incidence θ ( θ → 9 0 ∘ ) . The refractive index μ of the medium varies as μ = 1 + e x / d ​ . The refractive index of the air is 1 . If the x -coordinate of the point A , where the ray intersects the upper surface of the slab-air boundary is ( kd ) ℓ n 2 , then find the value of k .

TestHub · Accepted Answer

Apply Snell's law, μ sin θ = constant From Snell's law 1 = μ sin θ sin θ = μ 1 ​ dx dy ​ = tan θ = c o s θ s i n θ ​ dx dy ​ = μ ( μ 2 − 1 ) 1/2 ​ 1/ μ ​ dx dy ​ = e 2 d − x ​ ∫ dy = ∫ e 2 d − x ​ dx y = − 2 de 2 d − x ​ + C at x = 0 , y = 0 , C = 2 d at y = d y = − 2 d e 2 d − x ​ + 2 d − d = − 2 de 2 d − x ​ 2 1 ​ = e 2 d − x ​ ℓ n 2 = 2 d x ​ 2 d ℓ n 2 = x

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