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Physics Guru is an online education and competitive examinations preparation portal founded by Manish Verma, an IIT Madras alumnus, providing learning resources for competitive examinations.

The emphasis is on conceptual understanding, analytical thinking and problem solving rather than simply memorising formulas and standard methods.

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Discontinuity at Critical Angle

The Question

Consider an object inside a pond full of water. At the critical angle C, the refracted ray just grazes the water-air interface. If the angle of incidence i > C, the light undergoes total internal reflection. Is the angle that the ray leaving the water-air interface makes with the upward normal a discontinuous function at i = C?

The Answer

When the light grazes the water-air interface, the angle that the refracted ray makes with the upward normal is 90°.

However, when the angle of incidence i is slightly greater than C, refraction ceases to occur and the light undergoes total internal reflection. The reflected ray then makes an angle of 180° − i with the upward normal.

Thus, as i approaches C from below, the angle approaches 90°, whereas as i approaches C from above, it approaches 180° − C.

Since these two limiting values are unequal, the angle that the ray leaving the water-air interface makes with the upward normal is a discontinuous function at i = C.

Conclusion

The transition from refraction to total internal reflection at the critical angle produces a discontinuity in the angle of the outgoing ray, when measured with respect to the upward normal.