Question

# Use Huygens' Principle to show how a plane wave front propagates from a denser to rarer medium. Hence, verify Snell's law of refraction.

Solution

## We assume a plane wave front AB propagating in denser medium incident on the interface PP at angle i as shown in Fig. Let T be the time taken by the wave front to travel a distance BC. If vi is the speed of the light in medium I.So, $$BC = v_1 T$$ In order to find the shape of the refracted wave front, we draw a sphere of radius $$AE = v_2 T$$ , where $$v_2$$ is the speed of light in medium II (rarer medium). The tangent plane CE represents the refracted wave front In $$\Delta ABC = sin \ i = \frac{BC}{AC} = \frac{v_1t}{AC}$$and in $$\Delta ACE = sin \ r = \frac{AE}{AC} = \frac{v_2t}{AC}$$$$\therefore \ \ \frac{sin \ i}{sin \ r} \frac{BC}{AE} = \frac{v_1t}{v_2t} = \frac{v_1}{v_2}$$       ...(1)Let c be the speed of light in vacuum So, $$\mu_1 = \frac{C}{v_1}$$ and $$\mu_2 = \frac{C}{v_2}$$$$\frac{\mu_2}{\mu_1} = \frac{v_1}{v_2}$$                                                                            ...(2)From equations (1) and (2), we have$$\frac{sin \ i}{sin \ r} = \frac{\mu_2}{\mu_1}$$$$\mu_1 \ sin \ i = \mu_2 \ sin \ r$$It is known as Snell's law.Physics

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