# Use Huygens' Principle to Show the Propagation of a Plane Wavefront from a Denser Medium to a Rarer Medium. Hence Find the Ratio of the Speeds of Wavefronts in the Two Media. - Physics

Use Huygens' principle to show the propagation of a plane wavefront from a denser medium to a rarer medium. Hence find the ratio of the speeds of wavefronts in the two media.

#### Solution

Let XY be the surface separating the denser medium and the rarer medium.
Let:
v1 = Speed of light wave in the denser medium
v2 = Speed of light wave in the rarer medium
Let us consider a plane wavefront AB propagating in the direction AA'. Let this wavefront incident on the interface at an angle of incidence i with the normal to the interface.
Let τ be the time taken by the wavefront to travel the distance BC in denser medium.

BC=v1τ

To determine the shape of refracted wavefront, we will draw a sphere of radius v2τ from point A in the rarer medium. Let CD represent a tangent plane drawn from point C onto the sphere.
Now,

Here, CD would represent the refracted wavefront. Considering the triangles ABC and ADC, we get

sini=(BC)/(AC)=(v_1τ)/(AC)

 sinr=(AD)/(AC)=(v_2τ)/(AC)

sini/sinr=v_1/v-2   .....(1)
where
i = Angle of incidence
r = Angle of refraction

Since r > i (rays bend away from the normal on travelling from denser to rarer medium), the speed of light in the rarer medium (v2) will be greater than the speed of light in the denser medium (v1).
If c represents the speed of light in vacuum, then

μ1=c/v_1

 μ2=c/v_2

where
μ1 = Refractive index of denser medium ​
μ2 = Refractive index of rarer medium

Further, (1) can be written as

μ1sini=μ2sinr

This is the Snell's law of refraction.
Now, if λ1 and λ2 represent the wavelengths of light in the denser medium and rarer medium, respectively, and if the distance BC is λ1, then the distance AD will be λ2. So, we have

λ_1/λ_2=(BC)/(AD)=v_1/v_2

⇒v_1/v_2=λ_1/λ_2

Concept: Rarer and Denser Medium
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