Question

(a) Define a wavefront. Using Huygens' principle, verify the laws of reflection at a plane surface.
(b) In a single slit diffraction experiment, the width of the slit is made double the original width. How does this affect the size and intensity of the central diffraction band ? Explain.
(c) When a tiny circular obstacle is placed in the path of light from a distant source, a bright spot is seen at the centre of the obstacle. Explain why ?

Answer 1

(a) Wavefront is defined as the continuous locus of all the particles of a medium which are vibrating in the same phase.

Verification of laws of reflection using Huygen's principle :
Let XY be a reflecting surface at which a wavefront is being incident obliquely. Let v be the speed of the wavefront and at time t = 0, the wavefront touches the surface XY at A. After time t, point B of wavefront reaches the point B' of the surface. According to Huygen's principle, each point of wavefront acts as a source of secondary waves. When the point A of wavefront strikes the reflecting surface, then due to presence of reflecting surface, it cannot advance further; but the secondary wavelet originating from point A begins to spread in all directions in the first medium with speed v. As the wavefront AB advances further, its points $A_1,A_2,A_3$... etc. strike the reflecting surface successively and send spherical secondary wavelets in the first medium.


First of all the secondary wavelet starts from point A and travels a distance AA' (= vt) in first medium in time t.
In the same time t, the point B of wavefront, after travelling a distance BB', reaches point B, (of the surface), from where the secondary wavelet now starts. Now taking A as centre we draw a spherical arc of radius AA'(= vt) and draw tangent A'B' on this arc from point B'. As the incident wavefront AB advances, secondary wavelets starting from points between A and B', one after the other, touch A'B' simultaneously. According to Huygen's principle, wavefront A'B' represents the new position of AB, i.e., A'B' is the reflected wavefront corresponding to incident wavefront AB.
Now in right-angled triangles ABB' and AA'B'
$\angle$ ABB'= $\angle$ AA'B' (both are equal to ${90}^0$)
side BB' = side AA' (both are equal to vt)
and side AB' is common
i.e., both triangles are congruent.
$\angle$ BAB'= $\angle$AB'A'
i.e, incident wavefront AB and reflected wavefront A'B' make equal angles with the reflecting surface XY.As the rays are always normal to the wavefront, therefore the incident and the reflected rays make equal angles with the normal drawn on the surface XY, i.e.,
angle of incidence i = angle of reflection r.
This is the second law of reflection.
Since AB, A'B' and XY are all in the plane of paper therefore the perpendiculars dropped on them will also be in the same plane. Therefore we conclude that the incident ray, reflected ray and the normal at the point of incidence, all lie in the same plane. This is the first law of reflection. Thus Huygen's principle explains both the laws of reflection.
(b)Width of central diffraction band = 2D. $\frac{\lambda }{a}$ where d is the width of the slit.
So, on doubling the width of the slit, the size of the central diffraction band reduces to half value. But, the light amplitude becomes double, which increases intensity four fold.
(c)If a tiny circular obstacle is kept in the path of light a bright spot is seen at the centre of the obstacle. This is because the waves which get diffracted from the edge of the circular obstacle interfere constuctively at the centre of the shadow producing a bright spot.