On the basis of Huygens wave theory, prove Snell's law of refraction of light.

Snell's Law of Refraction - Proof Using Huygens' Wave Theory

Huygens' Wave Theory states that every point on a wavefront serves as a source of secondary spherical wavelets. The new wavefront is formed by the tangential surface to these secondary wavelets. This principle can be used to prove Snell's law of refraction.

Let’s consider two media, medium 1 (with refractive index n1) and medium 2 (with refractive index n2), and a light ray passing from medium 1 to medium 2. The angle of incidence is i, and the angle of refraction is r.

Step-by-Step Proof:
Wavefronts:

Let the light wavefront meet the boundary between two media at an angle i with respect to the normal.
In medium 1, the wavefront moves with a speed v1 and in medium 2 with a speed v2. The refractive indices of the two media are n1 and n2, where n = c/v (c is the speed of light in vacuum, and v is the speed of light in the medium).
Huygens' Principle:

According to Huygens' principle, each point on the wavefront in the first medium becomes a source of secondary wavelets. The wavefront will move forward as a tangential surface to these wavelets.
Change in Speed:

As the wavefront crosses the boundary between the two media, the speed of the wave changes due to the difference in the refractive indices. The light slows down in the denser medium (medium 2) and speeds up in the rarer medium (medium 1).
Time Taken for Light to Travel:

Consider the light incident at the boundary with an angle i. For simplicity, consider a small portion of the wavefront. The time taken for the light to travel from the point P to the boundary is the same for both media, since the wavefront moves in sync.
Relating the Angles:

Let’s consider the distance travelled by the secondary wavelets in the two media. In medium 1, the wavelet moves a distance x1 in time t, and in medium 2, the wavelet moves a distance x2 in the same time t.
Since the speed of light is different in the two media, the distances x1 and x2 will be proportional to the speed of light in the respective media.
Proportionality and Snell's Law:

Using trigonometry and the relationship between speed and refractive index, we can show that the angles of incidence (i) and refraction (r) are related by:

n1 * sin(i) = n2 * sin(r)

This is Snell's law of refraction, which describes the relationship between the angles of incidence and refraction in different media.