Derive Snell’s law on the basis of Huygen’s wave theory when the light is travelling from a denser to a rarer medium.

Using Huygen’s wave theory, derive Snell’s law of refraction

#### Solution 1

Let PP′ represent the surface separating medium 1 and medium 2, as shown in the figure below.

Let v_{1} and v_{2} represent the speed of light in medium 1 and medium 2, respectively. We assume a plane wavefront AB propagating in the direction A′A incident on the interface at an angle ‘i’ as shown in the figure. Let τ be the time taken by the wavefront to travel the distance BC. Thus, BC = v_{1}t

To determine the shape of the refracted wavefront, we draw a sphere of radius v_{2}τ from the point A in the second medium (the speed of the wave in the second medium is v_{2} > v_{1}).

Let CE represent a tangent plane drawn from the point C on to the sphere. Then, AE = v_{2}τ and CE would represent the refracted wavefront. If we now consider the triangles ABC and AEC, we readily obtain

`sini=(BC)/(AC)=(v_1t)/(AC) `

`sinr=(AE)/(AC)=(v_2t)/(AC) `

where i and r are the angles of incidence and refraction, respectively. Therefore, from equations (1) and (2), we get

`sini/sinr=(v_1t)/(AC)xx(AC)/(v_2t)=v_1/v_2 `

From the above equation, if r > i (i.e. if the ray bends away from the normal), the speed of the light wave in the second medium (v_{2}) will be greater than the speed of the light wave in the first medium (v_{1}).

Now, if c represents the speed of light in vacuum, then

`n_1=c/v_1`

`n_2=c/v_2`

These are known as the refractive indices of medium 1 and medium 2, respectively. In terms of the refractive indices, equation 3 can be written as

`sini/sinr=v_1/v_2=c/n_1xxn_2/c`

∴ n_{1}sini=n_{2}sinr

This is the Snell’s law of refraction.

#### Solution 2

Laws of Refraction: Consider a plane wavefront AB incident on a surface PQ separating two

media (1) and (2). The media (1) is rarer, having refractive index n_{1}, in which the light travels with a velocity c_{1}. The medium (2) is denser, having refractive index n_{2}, in which the light travels with a velocity c_{2}.

At time t = 0, the incident wavefront AB touches the boundary separating two medium at A.The

secondary wavelets from point B advance forward with a velocity c_{1}, and after time t seconds

touches at D, thus covering a distance BD = c_{1}t. In the same time interval of t seconds, the secondary

wavelets from A, advance forward in the second an envelope is drawn to obtain a new refracted

wavefront as CD.

Consider triangle BAD and ACD,

`sin i = sin(amgleBAD) = (BD)/(AD) = (c_1t)/(AD)`

`sin r = sin(angleADC) = (AC)/(AD) = (c_2t)/(AD)`

`=> sini/sinr = (c_1t)/(c_2t) = c_1/c_2`

`=> sin i/sin r = c_1/c_2` = constant

This constant is called the refractive index of the second medium with respect to the first medium.

`c_1/c_2 = n_1/n_2`

`:. sin i/sin r =c_1/c_2 = n_2/n_1 = ""_1n_2 `

This is known as the Snell’s law.