Question

Show the refraction of light wave at a plane interface using Huygens' principle and prove Snell's law.

Hint:

Huygens' principle explains refraction by treating each point on a wavefront as a source of secondary wavelets. Snell's law can be derived from this principle.

Answer 1

Step 1: Huygens' Principle. Huygens' principle posits that each point on a wavefront serves as a source of secondary wavelets. The subsequent wavefront is formed by the envelope of these wavelets.

Step 2: Refraction at a Plane Interface. Consider light traversing from medium 1 (refractive index \( n_1 \)) to medium 2 (refractive index \( n_2 \)) at a flat boundary. The incident wavefront makes an angle \( \theta_1 \) with the normal. By Huygens' principle, the wavelets at the interface dictate the direction of the refracted ray. 

Step 3: Derivation of Snell's Law. Let \( v_1 \) be the speed of light in medium 1 and \( v_2 \) be the speed in medium 2. The angle of incidence is \( \theta_1 \) and the angle of refraction is \( \theta_2 \). Using the geometry of the wavefronts and the relationship between velocities and refractive indices, we obtain: \[ \frac{\sin \theta_1}{\sin \theta_2} = \frac{v_1}{v_2} = \frac{n_2}{n_1} \] This equation is Snell's law, defining the relationship between the angles of incidence and refraction.