Question

Using Huygens’ principle, explain the refraction of a plane wavefront, propagating in air, at a plane interface between air and glass. Hence verify Snell’s law.

Answer 1

Using Huygens’ Principle, Explain the Refraction of a Plane Wavefront, Propagating in Air, at a Plane Interface Between Air and Glass. Hence Verify Snell’s Law.

Huygens' Principle:

Huygens’ principle states that every point on a wavefront acts as a source of secondary wavelets, and the new position of the wavefront at any later time is the surface tangent to these secondary wavelets.

Consider a plane wavefront propagating from air into glass, which are two different media with different refractive indices \( n_1 \) (for air) and \( n_2 \) (for glass). The wavefront strikes the interface between the two media at an angle \( \theta_1 \) (the angle of incidence), and the refracted wavefront forms an angle \( \theta_2 \) (the angle of refraction) with the normal to the interface.

Step-by-Step Explanation Using Huygens' Principle:

1. **Incident Wavefront:** A plane wavefront is incident on the boundary between air and glass. According to Huygens’ principle, every point on the incident wavefront acts as a source of secondary wavelets. The speed of light is different in the two media, so the wavelets will travel with different speeds in air and glass.

2. **Refraction at the Interface:** As the wavefront encounters the interface, the part of the wavefront in air (which has a higher speed of propagation) reaches the interface before the part in glass. This results in the bending of the wavefront as it enters the glass. The change in speed causes a change in direction, known as refraction.

3. **Secondary Wavelets in Glass:** In the glass, the secondary wavelets propagate slower than in air due to the higher refractive index. The speed of propagation of light in each medium is given by \( v_1 = \frac{c}{n_1} \) for air and \( v_2 = \frac{c}{n_2} \) for glass, where \( c \) is the speed of light in vacuum, and \( n_1 \) and \( n_2 \) are the refractive indices of air and glass, respectively.

4. **Refracted Wavefront:** The refraction of the wavefront occurs due to the difference in the speeds of the secondary wavelets in air and glass. The new wavefront in glass is formed by the tangent to the secondary wavelets, and the refracted angle \( \theta_2 \) is related to the incident angle \( \theta_1 \) through Snell’s law.

Snell's Law Verification:

Snell’s law states that the ratio of the sine of the angle of incidence (\( \theta_1 \)) to the sine of the angle of refraction (\( \theta_2 \)) is constant and is given by the refractive index ratio of the two media:

\[ \frac{\sin \theta_1}{\sin \theta_2} = \frac{n_2}{n_1} \]

This equation can be derived using Huygens’ principle, by considering the time taken for the wavelets to travel in each medium. The change in speed causes the bending of the wavefront at the interface, and the above relation holds true for the refraction process.

Thus, by using Huygens' principle and considering the change in wave speed at the interface, Snell's law is verified, describing the relationship between the angles of incidence and refraction and the refractive indices of the two media.