Question

State Huygens principle. Using it, draw a diagram and discuss the case of refraction of plane wave of light from a rarer medium to a denser medium at their plane interface. Hence derive Snell's law.

Hint:

Draw wavefront AB meeting the surface at C after time t, with BC = v1 t and AE = v2 t. Then sin i / sin r = v1 / v2 = n2 / n1.

Answer 1

Step 1: State Huygens principle:
Each point on a wavefront acts as a source of secondary wavelets that spread in all directions with the speed of the wave in that medium.
The new wavefront after a time \(\tau\) is the forward envelope (the common tangent) of these secondary wavelets.

Step 2: Set up the refraction:
Let XY be the plane surface between a rarer medium 1 (speed \(v_1\)) and a denser medium 2 (speed \(v_2\)). Since the second medium is denser, \(v_2 < v_1\).
A plane wavefront AB meets the surface at angle \(i\) (the angle of incidence is also the angle between AB and the surface). (Draw the surface, the incident wavefront AB, the incident rays, the refracted wavefront CE and the refracted rays.)

Step 3: Follow the wavefront for time tau:
Point A of the wavefront touches the surface first. While the point B of the wavefront travels the distance \(BC = v_1 \tau\) to reach the surface at C, the secondary wavelet from A spreads into medium 2 to a radius \(AE = v_2 \tau\).
The tangent from C to this wavelet is the refracted wavefront CE. The refracted rays are normal to CE. Because \(v_2 < v_1\), the rays bend towards the normal and the angle of refraction \(r < i\).

Step 4: Use the two right triangles:
In triangle ABC, \(\sin i = \dfrac{BC}{AC} = \dfrac{v_1 \tau}{AC}\).
In triangle AEC, \(\sin r = \dfrac{AE}{AC} = \dfrac{v_2 \tau}{AC}\).
Divide the two:
\[ \frac{\sin i}{\sin r} = \frac{v_1}{v_2} \]

Step 5: Get Snell law:
The refractive index of a medium is \(n = c/v\), so \(v_1/v_2 = n_2/n_1\).
\[ \frac{\sin i}{\sin r} = \frac{n_2}{n_1} \Rightarrow n_1 \sin i = n_2 \sin r \]
The frequency does not change, so the wavelength becomes smaller in the denser medium: \(\lambda_2 = \lambda_1 (v_2/v_1)\).

Final Answer:
A plane wave going from a rarer to a denser medium bends towards the normal. Huygens construction gives Snell law: \[ \boxed{n_1 \sin i = n_2 \sin r} \]