question_answer

Construct a refracted wavefront using Huygens' principle, when a plane wavefront is incidented on a plane surface from an optically denser medium, further define critical angle and obtain the condition for total internal reflection.

Answer:

When a plane wavefront PQ travelling in an optically denser medium of refractive index \[{{n}_{1}}\] with a speed \[{{v}_{1}}\] is incident on the surface of a rarer medium at an angle of incidence, i, then the wavelets spread in the rarer medium of refractive index, \[{{n}_{2}}\] (where, \[{{n}_{2}}<{{n}_{1}}\]) with a speed, \[{{v}_{2}}\](where \[{{v}_{2}}>{{v}_{1}}\]). Thus, a refracted wavefront RS is formed as shown in the figure. The refracted wavefront subtends an angle of refraction, r. As \[{{v}_{2}}>{{v}_{1}},\] hence, the angle of refraction is greater than the angle of incidence (i.e. r > i). However, Snell's law holds good, according to which             \[\sin i/\sin r={{n}_{2}}/{{n}_{1}}={{v}_{1}}{{v}_{2}}\] Critical angle As angle i increases, value of angle r also increases. If for a certain value of \[i={{i}_{c}},\] the angle of refraction just becomes\[90{}^\circ ,\] then             \[\frac{\sin \,{{i}_{c}}}{\sin 90{}^\circ }=\frac{{{n}_{2}}}{{{n}_{1}}}=\frac{{{v}_{1}}}{{{v}_{2}}}\] or         \[\sin {{i}_{c}}=\frac{{{n}_{2}}}{{{n}_{1}}}=\frac{{{v}_{1}}}{{{v}_{2}}}\] This angle \[{{i}_{c}}\] is called the critical angle. Condition for total internal reflection If angle of incidence \[i>{{i}_{c}},\] then it is not possible to find refracted wavefront. In such a case, no refraction takes place and whole wavefront is totally reflected back into the denser medium. It is known as the phenomenon of total internal reflection.