Reflection On A Plane Mirror And Refraction Of A Wave Front

Reflection On a Plane Mirror:

Consider the light rays 1, 2 and 3 shown by solid lines. The wave fronts which are perpendicular to these light rays are shown by the thin lines. The secondary wave fronts generated are the circular fronts described.

At point a, a wave front is generated due to the secondary source on ray 2. At the same time, other wave fronts are generated at points c and b. Since wave fronts at points a and b are generated at the same time a c = c b. Thus the triangle a c b is isosceles and the angles θ1 = θ2.

Note that θ1 is the angle of incidence and θ2 is the angle of reflection.

Thus, angle of incidence = angle of reflection

Reflection

Reflection of light by a plane mirror

Refraction of a Wave Front:

Reflection

Refraction of a wavefront

Consider the light rays 1, 2 and 3 shown by solid lines refracted to rays 1’, 2 and 3’ respectively. The wave fronts which are perpendicular to these light rays are shown by the thin lines. Consider the wave fronts to be one wavelength apart in their respective media where refractive indices n1 < n2.  The incident angle is θ1 and the refracted angle is θ2.  Consider the wave front at c, the front is bent in the new medium because the speed of light is slower. However, since the frequency of the waves are a constant, the wavelengths change across media to accommodate the change in speed.

i.e.,

\(ϑ_1\) = \(ϑ_2\)

\({v_1}{λ_1}\) = \({v_2}{λ_2}\)

Also from figure the side ac is common to the triangles abc and adc,

\(ac\) = \(ac\)

\(\frac{bc}{sinθ_1}\) = \(\frac{ad}{sinθ_2}\)

But the line segments ac and ad represent the wavelengths in their respective media,

\(\frac{λ_1}{sinθ_1}\) = \(\frac{λ_2}{sinθ_2}\)

\(\frac{v_1}{sinθ_1}\) = \(\frac{v_2}{sinθ_2}\)

\(\frac{sin~ θ_1}{sin~θ_2}\) = \(\frac{v_1}{v_2} \)

\(\frac{sin~θ_1}{sin~θ_2}\) = \(\frac{\frac{c}{v_2}}{\frac{c}{v_1}}\) = \({n_2}{n_1} \)

Where, c is the speed of light in vacuum.

Thus, the Huygens principle can be used to prove the law of refraction. A similar exercise can be conducted for n1 > n2.

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