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 states that every point on a wavefront acts as a source of secondary wavelets. The new wavefront is the envelope of these secondary wavelets. 

Step 2: Refraction at a Plane Interface. Consider a light wave traveling from medium 1 (with refractive index \( n_1 \)) to medium 2 (with refractive index \( n_2 \)) at a plane interface. The wavefront is incident at an angle \( \theta_1 \) to the normal. According to Huygens' principle, the wavelets at the interface are in the directions of the refracted ray. 

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