Why light bends towards the normal when it travels from rarer to denser medium. Explain with diagram.

When light travels from rarer to denser medium , the one which travels at an angle will slow down its velocity and changes its direction . The wave which is coming at perpendicular direction to the plane normal , changes its velocity but not direction. Now , we know λ (= v/f) , as  speed slows down then  λ  will become small if frequency has to be kept constant as shown in Fig. 1 .  Additionally, All the wave fronts are not reaching at same time  to the  boundary because they are coming at an angle . the one which reaches first will slow down its speed first i.e decreases its wavelength and at the same time others will travel with their own regular velocity. And  when will slow down their velocity when reaches to boundary. As the incoming  wave has the same wavelength through out . suddenly it changes at the boundary , in order to maintain same frequency it has to change its direction. so light bends. Now the rays  which is travelling at perpendicular direction to the boundary  has all the wave fronts reaching at same time, so they will only slows down their velocity but will not bend. 

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This can be explained on the wave theory of light . Yov will understand it by a diagram
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by theorem of light.
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by light theorem
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The light propagation is described, in the simplest case, by Asin(kr2πνt). The quantity ϕ=kr2πνt  is called phase, and a surface on which at a given time t the phase is constant, is called wave-front. The phase-velocity is the ratio of the distance between two neighbor wave-fronts of the same phase ϕ, and the time T needed for the light to travel a distance equal to the distance dbetween two neighbor wave-fronts. This time is given by




Note that the light frequency ν doesn't change from medium to medium, therefore T doesn't change.

For illustration of the situation imagine the following scenario:

enter image description here

Consider a front wave of phase ϕ (red line) touching at the time t0 the point A on the separation surface between the two media, then at a time t1 the point B of the front wave touches a point B1 of the separation surface, and at a time t2 the point C of the front wave touches the point C2 of the separation surface. Let the points A,B,C be chosen so as t1=t0+T, and t2=t1+T. As the velocity of the wave is smaller in M2,




the distance that the wave can travel during the time T is smaller,


(iii)d1=v1T,   d2=v2T==>d2<d1


(hence the distance between two consecutive wave-fronts is smaller.)

In consequence, the angle θ of the wave-front with the separation surface is smaller in M2 than in M1. Finally, note that the angle between the front wave in M2 and the separation surface is exactly equal to the refraction angle (i.e. between the normal to the separation surface and the normal to the wave-front).

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