A plane wave-front propagating in a medium of refractive index 'μ1' is incident on a plane surface making an angle of incidence ​

Question

A plane wave-front propagating in a medium of refractive index '\(\mu _1\)' is incident on a plane surface making an angle of incidence (i). It enters into a medium of refractive index \(\mu _ 2\) (\(\mu _ 2\) > \(\mu _1\)).

Use Huygen's construction of secondary wavelets to trace the retracted wave-front. Hence verify Snell's law of refraction.

OR

Using Huygen's construction, show how a plane wave is reflected from a surface. Hence verify the law of reflection.

Answer 1

Refraction of a Plane Wave: With the help of Huygens' Principle we can derive Snell’s law.

(a) Refraction from Rarer to Denser Medium : Let v₁ and v₂ represents the speed of light in medium-1 and medium-2 respectively. Consider a plane wavefront PQ propagating in the direction P′P, incident on the medium boundary at point P at an angle of incidence i. Let t be the time taken to travel from Q to B.

\(\therefore QB = v_1t\)

From the point P, draw a sphere of radius v₂t, let BR represent the forward tangent plane. It is refracted wavefront at t.

\(\therefore PR = v_2t\)

\(\text{From} \ \Delta PQB, \ \ sin i = \frac{QB}{PB} = \frac{v_1t}{PB} \Rightarrow PB = \frac{v_1t}{sini} \ \text{...(i)}\)

\(\text{Also from} \ \Delta PRB, \ sin r = \frac{PR}{PB} = \frac{v_2t}{PB} \Rightarrow PB = \frac{v_2t}{sinr} \ \text{...(ii)}\)

Equating (i) and (ii), we have

\(\frac{v_1 t}{sin i } = \frac{v_2 t}{sinr} \Rightarrow \frac{sin i }{sin r} = \frac{v_1}{v_2} \ \text{...(iii)}\)

Now, if r < i (i.e., ray bends towards the normal) sinr < sini

\(\Rightarrow \ \frac{sin i }{sin r}> 1 \Rightarrow \frac{v_1}{v_2} \Rightarrow v_1 >v_2\)

i.e., speed of light in medium 1 is greater than that in medium 2. This is exactly what we studied in Geometrical (Ray) optics.

This prediction is opposite to the prediction as per the Newton’s Corpuscular Theory.

Also, if µ1 be the refractive index of light in medium 1, then

\(\mu _1 = \frac{c}{v_1} \Rightarrow v_1 = \frac{c}{\mu_1}\)

Similarly, \(\mu _2 =\frac{c}{v_2} \ \text{and}\ v_2 = \frac{c}{\mu_2}\)

Hence, from eq. (iii), \(\frac{sin \ i}{sin \ r} = \frac{v_1}{v_2} = \frac{\mu _2}{\mu_1}\)

\(\Rightarrow \mu \ sin \ i = \mu_2 \ sin\ r\) Snell’s law of refraction

(b) Reflection of a Plane Wave by a Plane Reflecting Surface : After refraction let us now study laws of reflection from Huygens’ wave model. To prove the laws of reflection let us consider a plane wave PQ incident at an angle i on a reflecting surface AA′.

Time taken by the wave to advance to point R from point Q will be t. Hence QR = vt Now, in order to construct the reflected wavefront we draw a sphere of radius vt from point P. Let RS represent a tangent drawn from R to wavefront from P to the spherical wavefront.

Now, in order to construct the reflected wavefront we draw a sphere of radius vt from point P. Let RS represent a tangent drawn from R to wavefront from P to the spherical wavefront.