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 from Air to Glass at a Plane Interface. Verify Snell’s Law.

Huygens' Principle:

Huygens’ principle posits that each point on a wavefront serves as a source of secondary wavelets. The subsequent position of the wavefront is tangent to these wavelets.

Consider a plane wavefront moving from air (refractive index \( n_1 \)) into glass (refractive index \( n_2 \)). The wavefront strikes the interface at an angle of incidence \( \theta_1 \). The refracted wavefront then forms an angle of refraction \( \theta_2 \) with the interface's normal.

Step-by-Step Explanation Using Huygens' Principle:

1. Incident Wavefront: A plane wavefront approaches the air-glass boundary. As per Huygens’ principle, all points on this wavefront are sources of secondary wavelets. These wavelets travel at different speeds in air and glass.

2. Refraction at the Interface: The portion of the wavefront in air, moving faster, reaches the interface before the portion in glass. This causes the wavefront to bend upon entering the glass, a phenomenon termed refraction, due to the change in speed and direction.

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

4. Refracted Wavefront: The speed disparity in secondary wavelets in air and glass leads to wavefront refraction. The new wavefront in glass is tangential to these wavelets, and the refracted angle \( \theta_2 \) relates to the incident angle \( \theta_1 \) via 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 a constant equal to the refractive index ratio of the two media:

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

This law is derivable from Huygens’ principle by analyzing the travel time of wavelets in each medium. The speed change at the interface dictates the wavefront's bending, and the aforementioned relation accurately describes this refraction process.

Therefore, Huygens' principle, when applied to the wave speed variations at the interface, confirms Snell's law, establishing the relationship between the angles of incidence and refraction and the refractive indices of the involved media.