(a) Wave Front and Huygens’s Principle in Reflection:
Wave Front:
A wave front is a surface of constant phase in a wave, where all points on the wave front oscillate in unison at any given instant. It can be visualized as an imaginary surface that connects all the points of the wave that are in phase with each other. For example:
• In light waves, the wave front could be spherical or planar, depending on the type of source (point source or plane wave).
The nature of a wave front is essential for understanding the propagation of waves and can be used to derive several important results using Huygens's principle.
Huygens's Principle:
Huygens’s principle states that every point on a wave front can be considered a source of secondary wavelets, and the new position of the wave front at any later time is the surface tangent to all these secondary wavelets. This principle is fundamental in explaining the behavior of waves, including phenomena like reflection, refraction, and diffraction.
Verification of the Laws of Reflection using Huygens’s Principle:
Let's use Huygens’s principle to verify the laws of reflection:
1. Law of Reflection: The angle of incidence θi\theta_i is equal to the angle of reflection θr\theta_r, i.e., θi=θr\theta_i = \theta_r.
2. Verification Using Huygens's Principle:
o Consider a plane wave incident on a plane surface.
o Let the incident wave front strike the reflecting surface. According to Huygens’s principle, each point on the wave front acts as a source of secondary wavelets.
o The secondary wavelets originating from points on the wave front near the surface will reflect in such a way that they form a new wave front.
o The new reflected wave front will be tangent to the secondary wavelets, and it will lie at an angle to the surface.
o By the geometry of the situation, it can be shown that the angle formed by the reflected wave front with the surface is equal to the angle formed by the incident wave front, hence verifying that θi=θr\theta_i = \theta_r.
Thus, Huygens’s principle helps explain the law of reflection by showing how the wavelets behave at the surface and how the angles of incidence and reflection are related.
(b) Effect of Doubling the Slit Width in Single Slit Diffraction:
In a single slit diffraction experiment, the pattern formed on a screen consists of bright and dark bands, with the central diffraction fringe being the brightest and widest.
1. Effect on the Size of the Central Diffraction Band:
o The angular position of the first minima in single slit diffraction is given by the formula: sinθ=λa\sin \theta = \frac{\lambda}{a} where:
λ\lambda is the wavelength of the light,
aa is the width of the slit.
o If the slit width aa is doubled, the angle θ\theta at which the first minima occurs will decrease because the sine function is inversely proportional to the slit width. This results in the central diffraction band becoming narrower.
2. Effect on the Intensity of the Central Band:
o The intensity of the central diffraction fringe is determined by the distribution of the light, with the central maximum being the brightest.
o When the slit width increases (by doubling the slit width), the intensity of the central maximum increases. This is because the amount of light passing through the slit increases, and the light is spread over a smaller angular range, leading to more concentration of light in the central maximum.
o In summary: When the slit width is doubled, the central diffraction fringe becomes narrower, and the intensity of the central fringe increases.
(c) Bright Spot at the Center of a Circular Obstacle:
When a small circular obstacle is placed in the path of light from a distant source, a bright spot is observed at the center of the obstacle. This phenomenon is explained by diffraction and interference.
1. Diffraction Around the Obstacle:
o Light waves passing around a small circular obstacle bend or diffract around the edges of the obstacle.
o The diffracted waves interfere constructively at the center of the obstacle, leading to a bright spot.
o This is a direct consequence of Huygens’s principle, where each point on the edge of the obstacle acts as a secondary source of waves. These secondary waves spread out and interfere with each other.
o The interference at the center of the obstacle is constructive, meaning the waves reinforce each other and produce a bright spot.
2. Explanation:
o The bright spot occurs at the center because the waves that diffract around the edges of the obstacle reach the center in phase with each other, reinforcing the light and creating constructive interference.
o This phenomenon is observed similarly in Young's double-slit experiment where interference patterns are formed, but here the interference occurs due to diffraction around a single obstacle.
Thus, the bright spot at the center of the obstacle is due to the constructive interference of diffracted waves.
Summary:
• (a) A wave front is a surface of constant phase, and using Huygens’s principle, we can verify the laws of reflection by considering the secondary wavelets and their behavior at the reflecting surface.
• (b) In single-slit diffraction, doubling the slit width narrows the central diffraction band and increases its intensity.
• (c) When light passes through a small circular obstacle, the diffracted waves interfere constructively at the center, creating a bright spot.
