Question

A plane light wave propagating from a rarer into a denser medium, is incident at an angle \( i \) on the surface separating two media. Using Huygen’s principle, draw the refracted wave and hence verify Snell’s law of refraction.

Hint:

Huygen's principle is a useful tool to explain refraction by considering each point on a wavefront as a source of secondary wavelets. Snell's law can be derived from this principle by considering the change in the speed of light when moving between media of different refractive indices.

Answer 1

Step 1: According to Huygen’s principle, each point on a wavefront serves as a source of secondary wavelets that propagate in the forward direction. The position of the new wavefront at any later time is the envelope of these secondary wavelets.
Step 2: When a plane light wave passes from a rarer medium (with refractive index \( n_1 \)) into a denser medium (with refractive index \( n_2 \)), the wavefronts bend towards the normal due to a change in the speed of light. Let the angle of incidence be \( i \) and the angle of refraction be \( r \).
Step 3: To verify Snell's law using Huygen’s principle, consider the following steps:
The wavefronts in the rarer medium will have a velocity \( v_1 \), and the secondary wavelets will move slower in the denser medium with velocity \( v_2 \).
The refracted wavefront is drawn by connecting the positions of the secondary wavelets in the denser medium.
The refracted angle \( r \) is such that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant and equals the ratio of the velocities in the two media:
\[ \frac{\sin i}{\sin r} = \frac{v_1}{v_2} = \frac{n_2}{n_1} \] Thus, Snell's law is verified, which states that the ratio of the sines of the angles of incidence and refraction is equal to the ratio of the velocities (or the inverse of the refractive indices) in the two media.
Conclusion:
The relation derived from Huygen's principle leads to the verification of Snell's law.