A ray of monochromatic light is incident on the plane surface of separation between two media X and Y with angle of incidence ‘i’ in medium X and angle of refraction ‘r’ in medium Y. The given graph shows the relation between sin i and sin r. If VX and VY are the velocities of the ray in media X and Y respectively, then which of the following is true ?
VX = √3 VY
Snell's law gives \(n_X\sin i = n_Y\sin r\), and the refractive index of a medium is inversely proportional to the speed of light in it, \(n = \dfrac{c}{V}\). The frequency of light is fixed by the emitter and does not change on refraction.
Step 1 — The graph of sin r against sin i is a straight line through the origin at 30° to the sin i axis, so \(\dfrac{\sin r}{\sin i} = \tan 30^\circ = \dfrac{1}{\sqrt{3}}\).
Step 2 — From Snell's law, \(\dfrac{\sin r}{\sin i} = \dfrac{n_X}{n_Y} = \dfrac{V_Y}{V_X}\), hence \(\dfrac{V_Y}{V_X} = \dfrac{1}{\sqrt{3}}\), i.e. \(V_X = \sqrt{3}\,V_Y\).
Step 3 — Since \(n_Y = \sqrt{3}\,n_X\), medium Y is optically denser than X.
\(V_X = \dfrac{1}{\sqrt{3}}V_Y\) inverts the speed ratio. Total internal reflection needs light to travel from the denser to the rarer medium, so it can occur for light incident in Y, not in X. The frequencies in the two media are equal, so a frequency ratio of √3 is impossible.
VX = √3 VY option is correct.
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