Wave & Optics Class 12 Physics Full Notes

                           Wave and Optics |Class 12

                             

Some Important Questions  for exam preparation:

• Define wave motion. Derive the progressive wave equations in a medium.
• Starting with the definition of stationary waves, prove that the distance between any two consecutive nodes or antinodes in a stationary wave is ƛ/2.
• Write an expression for Newton's formula for the velocity of sound in air. Explain the necessary correction made by Laplace.
• Define end correction. Show that, in an open organ pipe, both odd and even harmonics can be produced.
• Define natural frequency. Describe an experiment to determine the velocity of sound in air by the resonance air column tube method.
• Describe a sound wave as a pressure wave and deduce an expression for the pressure amplitude.
• Define beats. Obtain an expression for the beat frequency produced by the superposition of two waves of slightly different frequencies.
• What is the Doppler effect in sound? Obtain an expression for the apparent frequency of sound when both source and observer are moving towards each other.
• State Huygen's law and use it to verify Snell's law.
• Define coherent sources of light. Describe Young's double-slit experiment and show that the bright fringes and dark fringes are equally spaced.
• Define polarization of light. Show that, μ=tan𝜭p, where the symbols have their usual meanings.

Answers of Above questions.

1. Wave motion is the transfer of energy from one point to another in the form of waves. The progressive wave equation in a medium can be derived as follows:

Let's consider a disturbance traveling through a medium. The wave can be described by the displacement function y(x, t), where x is the position and t is the time. The wave is assumed to be one-dimensional, so it travels in the x direction.

The wave equation is given by:

∂²y/∂t² = (1/v²) ∂²y/∂x²

where v is the velocity of the wave in the medium.

This equation describes the motion of the wave as it propagates through the medium.

2. Stationary waves are formed by the superposition of two waves of the same frequency and amplitude traveling in opposite directions. The nodes and antinodes are the points of zero and maximum amplitude, respectively, in a stationary wave.

To prove that the distance between any two consecutive nodes or antinodes in a stationary wave is λ/2, we can consider the equation for a standing wave:

y(x, t) = 2A sin(kx) sin(ωt)

where A is the amplitude, k is the wave number, ω is the angular frequency, and λ = 2π/k is the wavelength.

The nodes occur at the points where sin(kx) = 0, which implies that kx = nπ, where n is an integer. Therefore, the distance between two consecutive nodes is λ/2.

Similarly, the antinodes occur at the points where sin(ωt) = 0, which implies that ωt = mπ, where m is an integer. Therefore, the distance between two consecutive antinodes is λ/2.

3. Newton's formula for the velocity of sound in air is given by:

v = √(γRT/M)

where γ is the ratio of specific heats, R is the gas constant, T is the temperature, and M is the molar mass of the gas.

Laplace made a correction to this formula to account for the fact that air is not a perfect gas. The corrected formula is:

v = √(γRT/M) [1 + (3/16)(P/P₀)]

where P is the pressure of the gas and P₀ is the standard atmospheric pressure.

4. End correction is the adjustment made to the effective length of an open organ pipe to account for the fact that the sound waves do not stop abruptly at the end of the pipe. In an open organ pipe, both odd and even harmonics can be produced because the open end of the pipe acts as a displacement node for odd harmonics and a displacement antinode for even harmonics.

5. Natural frequency is the frequency at which an object vibrates when it is disturbed from its equilibrium position and left to oscillate freely. An experiment to determine the velocity of sound in air by the resonance air column tube method involves filling a tube with air and adjusting the length of the tube until the standing wave is formed. The length of the tube at which the standing wave is formed is equal to λ/4, where λ is the wavelength of the sound wave.

6. A sound wave can be described as a pressure wave because it consists of compressions and rarefactions of the air molecules. The pressure amplitude of a sound wave is given by:

P = ρvωA

where ρ is the density of the medium, v is the velocity of the wave, ω is the angular frequency, and A is the amplitude of the wave.

7. Beats are the interference pattern produced when two waves of slightly different frequencies superpose. The beat frequency is given by:

fbeat

10. Define coherent sources of light. Describe Young's double-slit experiment and show that the bright fringes and dark fringes are equally spaced.

Ans. Coherent sources of light are those sources whose wavefronts maintain a constant phase difference over a period of time. In Young's double-slit experiment, a beam of coherent light is made to fall on a thin slit, which diffracts it into two coherent beams of light. These beams of light fall on a screen with two closely spaced parallel slits called double-slit. The two beams of light interfere with each other, creating a pattern of bright and dark fringes on the screen.

The spacing between the fringes can be derived as follows: Let 'd' be the distance between the two slits, 'D' be the distance between the double-slit and the screen, and 'λ' be the wavelength of the light used. When the waves from both the slits meet at a point on the screen, the phase difference between them is given by 'δ = (2π/λ) (dsinθ)', where 'θ' is the angle between the line joining the point of observation and the slits and the normal to the screen.

For constructive interference, 'δ = 2πn', where 'n' is an integer. Thus, 'dsinθ = nλ'. The distance between the bright fringes is given by 'x = Dtanθ ≈ Dsinθ ≈ nλD/d', where we have used the small angle approximation. Thus, the distance between the bright fringes is directly proportional to the wavelength and the distance between the double-slit and the screen and inversely proportional to the distance between the slits.

Since the distance between the two slits is very small, the distance between the bright and dark fringes is almost the same. This can be shown mathematically by using the above equation and substituting 'n' with 'n+1/2', which gives the distance between the dark fringes as 'x = (2n+1)λD/2d'. We can see that the distance between the bright fringes and the dark fringes is equal to 'λD/d'. Hence, the bright fringes and the dark fringes are equally spaced.

11. Define polarization of light. Show that, μ=tan𝜭p, where the symbols have their usual meanings.

Ans. Polarization of light is the phenomenon by which the oscillation of the electromagnetic wave is confined to a particular direction. A polarized light wave can be thought of as a transverse wave with the electric field vector confined to a particular plane. There are three types of polarization: linear polarization, circular polarization, and elliptical polarization.

Consider a polarized light wave with its electric field vector 'E' making an angle of 'θ' with respect to the horizontal axis. Let 'p' be the component of 'E' parallel to the horizontal axis and 's' be the component of 'E' perpendicular to the horizontal axis. The ratio of these components is called the polarization factor or the degree of polarization, given by 'μ = p/s'.

Now, consider a beam of unpolarized light incident on a polarizer with its transmission axis making an angle of 'θp' with respect to the horizontal axis. The intensity of the light transmitted through the polarizer is given by 'I = I0cos²θp'. Let 'I1' be the intensity of the polarized light transmitted through the polarizer. Then, 'I1 = I0μcos²θp'.

Since the polarization factor 'μ = p/s', we have 'p = μs'. Substituting this in the above equation, we get

 4. State the principle of superposition of waves.

Ans. According to the principle of superposition, when two or more waves are simultaneously present in a medium, the resultant displacement at any point in the medium is the algebraic sum of the individual displacements due to each wave. This means that when two or more waves pass through a region of space, the net displacement of the medium at any point and time is the sum of the individual displacements produced by each wave separately. This principle is applicable for all types of waves, including mechanical waves, electromagnetic waves, and even quantum mechanical waves.

5. Explain the term interference of waves.

Ans. Interference of waves is a phenomenon that occurs when two or more waves meet or overlap with each other at a particular point in space and time. When this happens, the individual waves combine to form a resultant wave whose amplitude and phase depend on the amplitude and phase of the individual waves. Depending on the phase difference between the waves, interference can be constructive (when the waves reinforce each other) or destructive (when the waves cancel each other). The interference of waves is an essential concept in many areas of physics, including optics, acoustics, and quantum mechanics.

6. What is diffraction of waves?

Ans. Diffraction of waves refers to the bending of waves around an obstacle or through an opening. When waves encounter an obstacle or opening whose size is comparable to the wavelength of the wave, the wavefronts bend and spread out in different directions. This is because the different parts of the wavefronts have different distances to travel to reach the other side of the obstacle or opening. Diffraction is a fundamental property of waves and is observed in many areas of physics, including optics, acoustics, and quantum mechanics. Diffraction is also one of the reasons why we can hear sounds around corners or see the edges of objects even when they are not directly in our line of sight.

7. What is polarization of light?

Ans. Polarization of light refers to the phenomenon of restricting the vibration of light waves to a particular direction. When a beam of light is polarized, the electric field vector of the light waves oscillates in a particular direction, while the magnetic field vector oscillates perpendicular to the electric field vector and in the same direction as the wave propagates. There are different ways to polarize light, including using a polarizer or passing the light through a birefringent material. Polarization is an essential concept in many areas of physics, including optics, and has many practical applications, including 3D glasses and LCD displays.

8. What is the difference between regular and irregular reflection?

Ans. Regular reflection occurs when a wave (such as light) is incident on a smooth and polished surface, and the reflected wave travels away from the surface at an angle equal to the angle of incidence. The reflected wavefronts are parallel and uniformly spaced, resulting in a clear and sharp image of the object being reflected. This type of reflection is also known as specular reflection.

Irregular reflection occurs when a wave (such as light) is incident on a rough or uneven surface, and the reflected wave travels away from the surface at various angles. The reflected wavefronts are not parallel and uniformly spaced, resulting in a distorted or blurred image of the object being reflected. This type of reflection is also known as diffuse reflection.

9. Explain the working principle of a prism.

Ans. A prism is an optical device that can refract and disperse light. The working principle of a prism is based on the fact that different colors of light have different refractive indices in a given material. When a beam of white light is incident on a prism, the different colors of light are refracted at different angles due to their different refractive indices. As a result, the beam of white light is dispersed into its constituent colors, forming a spectrum.

The shape and angle of the prism are designed to maximize the amount of dispersion of the different colors of light. The most common type of prism used for dispersing light is a triangular prism, which has two parallel triangular faces and a rectangular cross-section. The incident beam of light enters one face of the prism and exits through the other face at different angles, creating the dispersion of colors.

10. What is the difference between real and virtual images?

Ans. Real images are formed when the light rays from an object converge at a point after passing through a lens or reflecting from a mirror. Real images can be projected onto a screen and are always inverted relative to the object. Real images can also be formed by placing the screen at the focal point of a converging lens or mirror.

Virtual images are formed when the light rays from an object appear to converge at a point but do not actually converge. Virtual images cannot be projected onto a screen and are always erect relative to the object. Virtual images can be formed by using a diverging lens or mirror or by placing the object inside the focal length of a converging lens or mirror.

In summary, the main difference between real and virtual images is that real images can be projected onto a screen and are always inverted, while virtual images cannot be projected onto a screen and are always erect.

11. Define the term diffraction.

Ans. Diffraction is the bending of waves around the corners of an obstacle or through an aperture or slit. When a wave encounters an obstacle, the wavefronts are disturbed, and the wave bends around the edges of the obstacle. Diffraction is a fundamental property of waves and is a common occurrence in everyday life. The amount of diffraction depends on the size of the obstacle or aperture relative to the wavelength of the wave.

For example, when a beam of light passes through a narrow slit, it diffracts and creates a pattern of light and dark fringes on a screen. The diffraction pattern is a result of the interference of the diffracted waves with each other. Diffraction is also important in the study of sound waves, radio waves, and other types of waves.

12. What is the principle of superposition of waves?

Ans. The principle of superposition of waves states that when two or more waves are present at the same time and location, the displacement of the medium at any point is the algebraic sum of the displacements caused by each wave. In other words, the waves add together to produce a resultant wave that is the sum of the individual waves.

When two waves of the same frequency and amplitude are added together, they form a wave with a larger amplitude called constructive interference. When two waves of the same frequency and opposite amplitude are added together, they form a wave with a smaller amplitude called destructive interference.

The principle of superposition is important in the study of wave interference, standing waves, and other wave phenomena.

13. What is meant by interference of waves?

Ans. Interference of waves occurs when two or more waves are present in the same medium at the same time and location. The waves interact with each other, and the resulting wave is the sum of the individual waves.

When two waves meet, they can either reinforce each other or cancel each other out, depending on the phase relationship between the waves. If the waves are in phase, they reinforce each other and create a wave with a larger amplitude. This is called constructive interference. If the waves are out of phase, they cancel each other out and create a wave with a smaller amplitude or no wave at all. This is called destructive interference.

Interference of waves is important in the study of wave phenomena, such as standing waves, diffraction, and the behavior of light and sound waves.

By ashmin aryal