# 8 = r, - r, = dsin 0 (36.1) This condition takes into account two factors: (1) the difference in path length for the two rays (the term ma,) and (2) the 180° phase change upon reflection (the term A,). Because A, = A/n, we can write Equation 36.11 as 2nt = (m+ )A m = 0, 1, 2, ... (36.12) If the extra distance 2t traveled by ray 2 corresponds to a multiple of A,, the two waves combine out of phase and the result is destructive interference. The general equation for destructive interference in thin films is 2nt 3D ma т %3D 0, 1, 2, ... (36.13)

Question

The quantity nt as shown is called the optical path length corresponding to the geometrical distance t and is analogous to the quantity δ as shown, the path difference. The optical path length is proportional to n because a larger index of refraction shortens the wavelength, so more cycles of a wave fit into a particular geometrical distance. (a) Assume a mixture of corn syrup and water is prepared in a tank, with its index of refraction n increasing uniformly from 1.33 at y = 20.0 cm at the top to 1.90 at y = 0. Write the index of refraction n(y) as a function of y. (b) Compute the optical path length corresponding to the 20.0-cm height of the tank by calculating

∫20cm n(y)dy

(c) Suppose a narrow beam of light is directed into the mixture at a nonzero angle with respect to the normal to the surface of the mixture. Qualitatively describe its path.

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8 = r, - r, = dsin 0 (36.1) This condition takes into account two factors: (1) the difference in path length for the two rays (the term ma,) and (2) the 180° phase change upon reflection (the term A,). Because A, = A/n, we can write Equation 36.11 as 2nt = (m+ )A m = 0, 1, 2, ... (36.12) If the extra distance 2t traveled by ray 2 corresponds to a multiple of A,, the two waves combine out of phase and the result is destructive interference. The general equation for destructive interference in thin films is 2nt 3D ma т %3D 0, 1, 2, ... (36.13)