Question

A parallel beam of light is travelling obliquely from an optically rarer medium to an optically denser medium.

1. Draw a labelled diagram showing incident and refracted wavefronts. Mark angle of incidence as ‘i’ and angle of refraction as ‘r’.
2. Use Huygen’s wave theory to prove Snell’s law.

Answer 1

i.
 

ii. Let SS' stand for the surface that divides media 1 and 2, which have refractive indices of n₁ and n₂, respectively. Let c₁ and c₂ represent the light velocities in the two mediums. Since the second medium has a higher optical density than the first, c₁ > c₂.
APB represents the incident wavefront. When the disturbance at B reaches C, the secondary wavelets from A have traversed a distance AD = c₂t, where t represents the time necessary for the waves to cover the distance BC. Consequently, BC = c₁t and AD = c₂t. A sphere is drawn with center A and radius AD (= c₂t), and a tangent CD is constructed from point C to the sphere. Consequently, CD denotes the refracted plane wavefront. To demonstrate that CD is the shared wavefront, it suffices to establish that at the time the disturbance propagates from B to C or A to D, the disturbance at P arrives at L. We create a sphere with M as the center, such that CD serves as the tangent to the sphere.

From the similar triangles ACD and MCL,

`(AD)/(ML) = (AC)/(MC)`    ...(i)

Similarly, from a similar triangle ACE and MCN,

`(AE)/(MN) = (AC)/(MC)`    ...(ii)

From (i) and (ii),

`(AD)/(ML) = (AE)/(MN)`

or `(AE)/(AD) = (MN)/(ML)`

∵ AE = BC = c₁t and AD = c₂t

We can write,

`(AE)/(AD) = (c_1t)/(c_2t) = (MN)/(ML)`

or `(BC)/(AD) = (MN)/(ML) = c_1/c_2`    ...(iii)

Therefore, if AD represents the radius of the secondary wavefront for point A, then ML denotes the radius of the secondary wavefront for point M. Let i and r denote the angles of incidence and refraction, respectively.

From triangles ABC and ACD,

`(sin i)/(sin r) = (BC//AC)/(AD//AC) = (BC)/(AD) = (c_1t)/(c_2t) = c_1/c_2` = ₁n₂

₁n₂ is a constant known as the refractive index of the second medium relative to the first medium. This confirms Snell's law of refraction.