Huygens‘ Principle, Laws of Reflection and Refraction on the Basis of Wave Theory

Huygens‘ Principle

Huygens ’ Principle enables us to determine what its position will be at some later time. In other words, the principle gives a method to know as to how light spread out in the medium. A source of light sends out wave front is propagated forwards through a homogeneous isotropic medium, Christian Huygens made the following assumptions.

1. Each point on a wave front acts as a new source of the disturbance. The disturbances from these points are secondary wave lets. These wavelets spread out in all directions in the medium with the velocity of light.
2. The new wave front is then obtained by constructing a tangential plane to all the secondary wavelets. The new wave front is the envelope of to secondary wavelets at that instant.

Laws of Reflection on the Basis of Wave Theory

Conclusion of laws of reflection on the basis of wave theory:

1. The angle of reflection r is equal to the angle of incidence i for all wavelengths and for any pairs of materials.
   $$ r = i$$
2. The incident ray, reflected ray and normal to the reflecting surface all lie on the same plane.

Consider a plane wave front AB incident on the reflecting surface XY at an angle of incidence I as shown in the figure. The lines 1, 2 and 3 which are perpendicular to the wave font AB represent incident rays. AN is normal tot the reflecting surface. Point A of the wave front reaches the reflecting surface at times t = 0. By the time, point B, of the wave front reaches point A’ (t=t), the secondary wavelets from A spread out in the form of a sphere. There A’B’ = BA’ = ct, where c is the velocity of light. From A’ draw a tangent to the sphere. Then A’B’ represents reflected wave front. Similarly, the wavelets from C reach point D and from D reach E of the reflected wave from in time t. reflected rays must be at right angles to the wave A’B’. In the figure, reflected rays are represented by 1’ 2’ and 3’.

Draw A’N’ normal to the reflecting surface. Now, ÐB’A’A = r, is the angle of reflection. From the right angle triangles ABA’ and AB”A”, we have,

1. $$ AB’ = BA’ $$$$\angle AB’A’ = \angle ABA’ = 90^0 $$
2. $$ AA’ \: \text {is common. Therefore the two triangles are congruent.} $$

\begin{align*} \text {Hence,} \angle BAA’ = \angle B’A’A \\ \text {or,} \: i &= r \\\end{align*}

i.e the angle of incidence is equal to the angle of reflection which is the first law of reflection.

The incident wave front, AB, the reflected wave front A’B’ and the reflecting surface AA’ are all perpendicular to the plane of the paper. So the incident ray, normal to the reflecting surface and the reflected ray all lie in the same plane. This proves the second law of reflection.

Laws of Refraction on the Basis of Wave Theory

The refraction of light when light enters from a rarer medium to a denser medium. Let v and c be the velocity of the light in the denser medium and in air respectively. The laws of refraction are:

1. The ratio of the sine of angle incidence to the sine of the angle of refraction is constant for any two given media. Therefore,
   $$ \frac {\sin i}{\sin r} = \frac cv = \mu $$
   where µ is constant called the refractive index of the medium with respect to air.
2. The incident ray, the refracted rays and the normal at the point of an incident on the refracting surface lie on the same plane.

Consider a plane wave front AB incident on a refracting plane surface XY separating two different media. The lines 1, 2, and 3 which are perpendicular to the wavefront AB represent incident rays. AN is normal to the surface XY and I is the angle on incidence which is equal to the angle made by incident wavefront AB with the surface XY as shown in the figure. Let v and c be the velocity of light in the denser and rarer medium respectively, such that v<c.

According to Huygens principle, every point on the wave front AB is a source of secondary wavelets. At an instant, the wavelet to reach A’ from B, then BA’ = ct. During this time, the secondary if t is the time taken by the secondary wavelet originating at A has traveled a distance vt = AB’ in the denser medium. Similarly, the wavelet from point C of the wave front reaches D and the wavelet from D reach point E in the same time. If we draw a sphere of radius -= AB’ = vt, and draw a tangent A’B’ to this sphere, wavelets form points between A and A’ reach to the surface of spheres to which A’B’ is tangent. Then, A’B’ is the new wave front in the denser medium. The rays 1’, 2’ and 3’, normal to the wavefront AA’, are refracted rays.

Draw N’A’ normal to the refracting surface at A’. The angle, ∠AA’B’ = r is the angle of refraction.

\begin{align*} \text {In right angled triangle} \Delta \text {ABA’} \\ \sin i &= \frac {A’B}{AA’} = \frac {ct}{AA’} \dots (i) \\ \sin r &= \frac {AB’}{AA’} = \frac {vt}{AA’} \dots (ii) \\ \text {Dividing} \: (i) \: \text {by} \: (ii) \: \text {we get,} \\ \frac {\sin i}{\sin r} = \frac cv \dots {iii} \\ \text {where} \frac cv = \mu,\\ \end{align*}

is the refractive index of denser medium with respect to rarer medium.}

\begin{align*} \therefore \frac {\sin i}{\sin r} &= \mu \\ \end{align*}

The ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant for the given pair of media which proves Snell’s law.

Further, the incident ray refracted ray and the normal to the surface of separation at the point incidence all lie in the plane of the paper. This verifies the second law of refraction.

References

Manu Kumar Khatry, Manoj Kumar Thapa, Bhesha Raj Adhikari, Arjun Kumar Gautam, Parashu Ram Poudel. Principle of Physics. Kathmandu: Ayam publication PVT LTD, 2010.

S.K. Gautam, J.M. Pradhan. A text Book of Physics. Kathmandu: Surya Publication, 2003.