Important Derivations: Wave Optics

Important Derivations: Wave Optics | Physics Class 12 - NEET PDF Download

(1) Using Huygens’ principle, verify the laws of reflection at a plane surface.

First the wavefront touches the reflecting surface at B and then at the successive points towards C. In accordance with Huygens’ principle, from each point on BC secondary wavelets start growing with the speed c. During the time the disturbance from A reaches the point C, the secondary wavelets from B must have spread over a hemisphere of radius DB = AC = ct, where ‘t’ is the time taken by the disturbance to travel from A to C. The tangent plane CD drawn from the point ‘C’ over this hemisphere of radius ‘ct’ will be the new reflected wavefront. Let angles of incidence and reflection be i and r respectively. In ΔABC and ΔDCB, we have

i.e., the angle of incidence is equal to the angle of reflection. This proves the first law of reflection.

(2) Use Huygens’ principle to verify the laws of refraction.

Laws of refraction: Suppose when distribution from point P on incident wave front reaches point P on the refracted wave front the disturbance from point Q reaches the point Q or the refracting surface XY. Since, A ‘Q’ P’ represents the refracted wave front the time takes by light to travel from a point on incident wave front to the corresponding point on refracted wave front would always be the same. Now, time taken by light to go from r to Q’ will be-

(Where c and v are the velocities of light in two medium)

In right angled

In right angle

Substituting eq. (ii) and (iii) in eq. (i), we get

The rays from different points or the incident wave front will take the same time to reach the corresponding points on the refracted wave front i.e., given by equation (iv) is independent of Ak. will happens so, if

However,

This is the shell’s law for refraction of light.

Introduction To Young’s Double Slits Experiment

During the year 1801, Thomas Young carried out an experiment where the wave and particle nature of light and matter were demonstrated. The schematic diagram of the experimental setup is shown below-

Figure(1): Young double slit experimental set up along with the fringe pattern.

A beam of monochromatic light is made incident on the first screen, which contains the slit S0. The emerging light then incident on the second screen which consists of two slits namely, S1, S2. These two slits serve as a source of coherent light. The emerging light waves from these slits interfere to produce an interference pattern on the screen. The interference pattern consists of consecutive bright and dark fringes. The dark fringes are the result of destructive interference and bright fringes are the result of constructive interference.

Figure(2): shows the interference pattern of two light waves to produce dark or bright fringes. Bright fringe(at P) is formed due to the overlap of two maxima or two minima. Figure 2a,2b . Dark fringe(at P) is formed due to the overlap of maxima with minima. Figure 2c.

Figure(3) Geometry of Young’s double-slit interface
• The light falls on the screen at the point P. which is at a distance y from the centre O.
• The distance between the double-slit system and the screen is L
• The two slits are separated by the distance d
• Distance travelled by the light ray from slit 1 to point P on the screen is r1
• Distance travelled by the light ray from slit 2 to point P on the screen is r2
• Thus, the light ray from slit 2 travels an extra distance of ẟ = r2-r1 than light ray from slit 1.
• This extra distance is termed as Path difference.

Refer to Figure(3) Applying laws of cosines; we can write –

Similarly,

Subtracting equation (2) from (1) we get

Let us impose the limit that the distance between the double slit system and the screen is much greater than the distance between the slits. That is L >> d. The sum of r1 and r2 can be approximated to r1 + r2 ≅2r. Thus, the path difference becomes –

In this limit, the two rays r1 and r2 are essentially treated as parallel. (See Figure(4))

Figure(4): Assuming L >> d, The path difference between two rays.
To compare the phase of two waves, the value of path difference (ẟ) plays a crucial role.
Constructive interference is seen when path difference (𝛿) is zero or integral multiple of the wavelength (λ).
(Constructive interference) 𝛿 = dSin𝜃 = mλ —-(5), m = 0, ±⁤1, ±2, ±3, ±4, ±5, ……
Where m is order number. The Zeroth order maximum (m=0)corresponds to the central bright fringe, here 𝜃=0. The first order maxima(m=±⁤1)(bright fringe) are on either side the central fringe.
Similarly, when 𝛿 is an odd integral multiple of λ/2, the resultant fringes will be 1800 out of phase, thus, forming a dark fringe.
(Destructive interference) 𝛿 = dSin𝜃 =

λ—(6), m = 0, ±⁤1, ±2, ±3, ±4, ±5, ……

Figure(5)(a) How path difference 𝛿 = λ/2 (m=0) results in destructive interference. (b) 𝛿 = λ (m=1) yields constructive inference.
Assuming the distance between the slits are much greater than the wavelength of the incident light, we get-​

Substituting it in the constructive and destructive interference condition we can get the position of bright and dark fringes, respectively. The equation is as follows-

And

The document Important Derivations: Wave Optics | Physics Class 12 - NEET is a part of the NEET Course Physics Class 12.
All you need of NEET at this link: NEET

Physics Class 12

105 videos|425 docs|114 tests

FAQs on Important Derivations: Wave Optics - Physics Class 12 - NEET

 1. What is Huygens' Principle?
Ans. Huygens' Principle is a principle in wave optics that states that every point on a wavefront acts as a source of secondary spherical wavelets, and the wavefront at any subsequent time is the envelope of these wavelets. It helps explain phenomena such as diffraction, interference, and reflection of waves.
 2. How is Huygens' Principle related to wave optics?
Ans. Huygens' Principle is a fundamental concept in wave optics. It provides a way to understand and analyze the behavior of waves in various optical phenomena, including the diffraction and interference of light. By considering every point on a wavefront as a source of secondary wavelets, Huygens' Principle helps explain the bending, spreading, and interference patterns observed in wave optics experiments.
 3. What are some derivations related to Huygens' Principle?
Ans. Some derivations related to Huygens' Principle include the derivation of the laws of reflection and refraction using the principle, the derivation of the formula for the intensity of light in Young's double slit experiment, and the derivation of the equation for the diffraction pattern produced by a single slit. These derivations use the concept of wavefronts and secondary wavelets to explain the observed phenomena.
 4. What is Young's double slit experiment?
Ans. Young's double slit experiment is a classic experiment in wave optics that demonstrates the wave nature of light. It involves shining a beam of light through two closely spaced, parallel slits, which act as sources of secondary wavelets. These wavelets interfere with each other, creating an interference pattern of alternating bright and dark fringes on a screen placed behind the slits. This experiment provides evidence for the wave nature of light and supports the wave theory of light propagation.
 5. How does Young's double slit experiment relate to wave optics?
Ans. Young's double slit experiment is a key example in wave optics as it demonstrates the phenomenon of interference. The interference pattern produced by the overlapping wavelets from the two slits provides evidence for the wave nature of light and supports the wave theory of light propagation. The experiment helps to understand the principles of constructive and destructive interference, the concept of coherence of light waves, and the wave nature of light in general.

Physics Class 12

105 videos|425 docs|114 tests

Up next

 Explore Courses for NEET exam
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
10M+ students study on EduRev
Related Searches

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

;