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A ray of light travelling in air is incident at an angle of incidence of 30 degrees on one surface of a slab in which the refractive index varies with the y-coordinate. The light travels along the curve y = 4x² (y and x in metres) inside the slab. Find the value of 2*mu, where mu is the refractive index of the slab at y = 1/2 m.

1. (A) 1
2. (B) 2
3. (C) 4
4. (D) 8

Correct answer: (B) 2

Solution

For a graded-index medium, Snell's law in integral form gives mu*sin(theta) = constant, where theta is the angle the ray makes with the y-axis (the direction of refractive index gradient). At the entry surface, theta = 30 deg and mu_air = 1, so mu*sin(theta) = 1*sin(30 deg) = 1/2 throughout the slab. At y = 1/2 m, find x: 1/2 = 4x² => x = 1/(2*sqrt(2)). The curve y = 4x² has dy/dx = 8x. At x = 1/(2*sqrt(2)), dy/dx = 8/(2*sqrt(2)) = 4/sqrt(2) = 2*sqrt(2). The ray travels along the curve, so the tangent direction has slope 2*sqrt(2). The angle phi with the y-axis satisfies tan(phi) = 1/(dy/dx) = dx/dy = 1/(2*sqrt(2)). sin(phi) = 1/sqrt(1+(2*sqrt(2))²) = 1/sqrt(1+8) = 1/3. Using mu*sin(phi) = 1/2: mu*(1/3) = 1/2 => mu = 3/2. Therefore 2*mu = 3.

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