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, every point on a wavefront acts as a source for secondary wavelets that travel forward. The new wavefront's position at any subsequent time is the envelope of these secondary wavelets.
Step 2: When plane light waves transition from a less dense medium (refractive index \( n_1 \)) to a denser medium (refractive index \( n_2 \)), the wavefronts bend towards the normal due to the change in light speed. Let the angle of incidence be \( i \) and the angle of refraction be \( r \).
Step 3: To confirm Snell's law using Huygen’s principle, follow these steps:
Wavefronts in the rarer medium travel at velocity \( v_1 \), and secondary wavelets move slower in the denser medium at velocity \( v_2 \).
The refracted wavefront is formed 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 velocities in the two media:
\[ \frac{\sin i}{\sin r} = \frac{v_1}{v_2} = \frac{n_2}{n_1} \] This verifies Snell's law, which states that the ratio of the sines of the angles of incidence and refraction equals the ratio of the velocities (or the inverse of the refractive indices) in the two media.
Conclusion:
The relationship derived from Huygen's principle validates Snell's law.