Question

Show the refraction of light wave at a plane interface using Huygens’ principle and prove Snell’s law.

Answer 1

Consider a plane wavefront incident from medium 1 (refractive index n₁, speed of light v₁) onto a plane boundary with medium 2 (refractive index n₂, speed of light v₂).

At an angle of incidence i, the incident wavefront gets closer to the boundary.

The refracted wavefront moves in medium 2 with an angle of refraction r.

Applying Huygens’ Principle:

Wavefront in Medium 1:

Consider a wavefront AC moving towards the interface.

Each point on AC acts as a secondary source, emitting wavelets.

Wavefront in Medium 2:

When point A hits the interface, it immediately starts producing secondary wavelets in the second medium.

After time t, point C reaches the interface at point F, and secondary wavelets in the second medium form the new wavefront DF.

Derivation of snell’s law:

From the geometry of the wavefront:

Distance travelled by the incident wavefront in medium 1:

AC = v₁t

Distance travelled by the refracted wavefront inmedium 2:

DF = v₂t

The incident and refracted wavefronts make angles i and r with the interface, so:

sin i = `(AC)/(AF)`

= `(v_1t)/(AF)`    ...(i)

sin r = `(AD)/(AF)`

= `(v_2t)/(AF)`    ...(ii)

On solving equations (i) and (ii):

`(sin i)/(sin r) = v_1/v_2`

Since the refractive index is defined as n = `c/v`,

Where c is the speed of light in a vacuum:

`v_1/v_2 = n_2/n_1`

Thus, we get Snell’s law:

n₁ sin i = n₂ sin r