Question

(a) Define a wave front. Using Huygens’s principle, verify the laws of reflection in a plane surface.
(b) In a single slit diffraction experiment, the width of the slit is made twice the original width. How does this affect the size and intensity of the central diffraction bond? Explain.
(c) When a small 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

Hint: A wave front is described as the continuous locus of all the particles which are vibrating in the similar phase. The perpendicular line which is drawn at any point on the wave front represents the direction of propagation of the wave at that point.

Complete step by step answer:
(a) the wave front of a time-varying field is the locus of all points where the wave has the similar phase of the sinusoid.
Verification of laws using Huygens principle.
In the figure, AB is the wave front which is incident on a reflecting surface XY with an angle of incidence $i$ as given in figure. According to Huygens’s principle, each and every point on AB acts as a source of secondary wavelets. Firstly, wave incidents at point A and then to points C, D and E. They form a sphere with radii $A{{A}_{1}},C{{C}_{1}}$ and $D{{D}_{1}}$ as shown in figure. ${{A}_{1}}E$ shows the tangential envelope of the secondary wavelet in forward direction.

From ΔABE and ΔA$A_1$E,
$\angle ABE=\angle A{A}_{1}E=90{}^\circ$
$AE=AE$
$A{{A}_{1}}=BE$= distance travelled by wave in equal time
So, these triangles are congruent.${{A}_{1}}E$
$\begin{align}
  & \angle BAE=i \\
 & \angle BEA=r \\
\end{align}$
Therefore
$i=r$
Therefore, the angle of incidence is the same as the angle of reflection. This is the first law of reflection.
Since the incident wave front AB, normal and reflected wave front ${{A}_{1}}E$ all lies in the same plane, it proves the second law of reflection. Hence the laws of reflection is verified by Huygens’s principle.
(b) The width of the central maximum is given as
$w=\dfrac{2\lambda D}{d}$
​ Where d is slit width.
On making the width of the slit twice, the size of the diffraction band reduces to half value. If the width of the central maxima is decreased to half, the area of the central diffraction band will become quarter. And then the intensity will become four times.
(C) It is due to the wave diffracted from the edge of the circular obstacle will interfere to form constructive interference at the centre of the obstacle.

Note: A wave front is described as the continuous set of all the particles which are vibrating in the equal phase. The term wave front is generally suitable only for fields where at each point vary sinusoidal in time with a single temporal frequency. Wave fronts generally move with time.