Engineering Physics Unit-IV: Interference and Diffraction

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Interference

• Coherent Sources: Two sources are said to be coherent if they emit light waves of the same frequency, nearly the same amplitude, and are always in phase with each other.
• Types of Coherence:
  • Temporal Coherence (Longitudinal Coherence): Measure of phase relation of a wave reaching a given point at two different times.
  • Spatial Coherence (Transverse Coherence): Measure of phase relation between waves reaching two different points in space at the same time.

Thin Film Interference

• Thin Film: Optical medium of thickness in the range of 0.5 μm to 10 μm.
• Interference in Thin Film:
  • Reflected Light: Interference occurs between reflected light and transmitted light.
  • Transmitted Light: Interference occurs between transmitted light and reflected light.
• Path Difference: Given by Δ = 2µt cos r - λ/2.
• Conditions for Constructive and Destructive Interference:
  • Constructive Interference: 2µt cos r = nλ (bright fringe).
  • Destructive Interference: 2µt cos r = (2n + 1) λ/2 (dark fringe).

Interference Due to Non-uniform Thin Film (Wedge Shaped Thin Film)

• Wedge Shaped Thin Film: Path difference given by Δ = 2µt cos (r + θ) - λ/2.

Newton's Ring Experiment

• Newton's Rings: Interference pattern formed by a thin film of variable thickness, such as between a plane glass plate and a convex lens.
• Experimental Arrangement: Monochromatic light is incident normally on the glass plate, and the interference pattern is viewed through a microscope.
• Theory: The air film formed is of wedge shape, and the path difference produced is Δ = 2µt cos (r + θ) - λ/2.

Diffraction

• Diffraction: Phenomenon of bending of light around the corners of an obstacle and spreading into the geometrical shadow.
• Fresnel's Diffraction: Diffraction phenomenon caused by the interference of innumerable secondary wavelets produced by the unobstructed portions of the same wave front.
• Fraunhofer's Diffraction: Diffraction phenomenon caused by the interference of light waves from different parts of the same wave front.

Fraunhofer's Diffraction at a Single Slit

• Diffraction Pattern: Consists of a central bright band with alternate dark and bright bands of decreasing intensity on both sides.
• Analysis and Explanation: The diffracted ray along the direction of the incident ray is focused at a point, and those at an angle θ are focused at a different point.
• Path Difference: Given by Δ = α = (πa sinθ) / λ.
• Intensity Distribution: Given by I = I0 (sin^2(α) / (α^2)).

Fraunhofer's Diffraction at a Double Slit

• Diffraction Pattern: Consists of a number of equally spaced interference maxima and minima.
• Theory: The resultant amplitude due to all wavelets from each slit is given by R = (A / (2i)) (e^(iα) - e^(-iα)).
• Intensity Distribution: Given by I = I0 (cos^2(φ/2)).

Plane Transmission Diffraction Grating (N-Slits Diffraction/Diffraction due to double slits)

• Definition: An arrangement consisting of a large number of close, parallel, straight, transparent, and equidistant slits, each of equal width a, with neighboring slits being separated by an opaque region of width b.
• Theory: The resultant amplitude due to all wavelets from each slit is given by R = (A / (2i)) (e^(iα) - e^(-iα)).
• Intensity Distribution: Given by I = I0 (sin^2(Nα) / (N^2 sin^2(α))).

Absent Spectra with a Diffraction Grating

• Definition: A spectrum that is not visible due to the path difference between the diffracted rays from the two extreme ends of one slit being equal to an integral multiple of λ.### Conditions for Absent Spectra

• Dividing equation (2) by equation (1) gives the condition for absent spectra in the diffraction pattern: (a+ b) /a =n/m

• When a = b (width of transparent portion equals width of opaque portion), n = 2m and 2nd, 4th, 6th, etc. orders of the spectra are absent

• When b = 2a, n = 3m and 3rd, 6th, 9th, etc. orders of the spectra are absent

Number of Orders of Spectra with a Grating

• The number of visible spectra in a grating can be calculated using the equation: (a + b) sin θ = n λ
• (a + b) is the grating element, equal to 1/N, where N is the number of lines per inch in the grating
• Maximum possible value of the angle of diffraction (θ) is 90°, so sin θ = 1
• Maximum possible order of spectra (N max) is (a+b)/λ
• If (a + b) is between λ and 2 λ, n max depends on the grating element and wavelength of light

Acoustics of Building

• Acoustics is the study of sound, its generation, transmission, and reception in the form of vibration waves in matter.

Echoes and Reverberation

• Echo: a reflection of sound that is heard distinctly after the original sound has stopped, with a time delay of at least 0.1 seconds between the original sound and its reflection.
• Reverberation: the persistence of sound in a room due to multiple reflections from walls, floor, and ceiling, even after the source of sound has stopped.

Reverberation Time

• Reverberation time is the time it takes for sound energy density to decrease to one millionth of its initial value after the source of sound has stopped.
• Sabine's Formula: t = (V / A) * (ln(10^6)), where t is the reverberation time, V is the volume of the hall, and A is the total absorption.

Requirements of an Acoustically Good Hall

• Site selection: choose a quiet location away from noisy areas.
• Volume: the hall should be large enough to allow for uniform sound distribution.
• Shape: the shape of the hall should be designed to minimize sound reflections.
• Reverberation: the reverberation time should be optimal, not too long or too short.
• Sound source: the sound source should generate sound of adequate intensity.

Ultrasonic Waves

• Ultrasonic waves: sound waves with frequencies higher than 20 kHz, beyond human hearing range.
• Applications: sonar, echo sounding, cleaning, cutting, soldering, bloodless surgery, and tumor detection.

Production of Ultrasonic Waves

• Magnetostriction method: uses a rod of ferromagnetic material and an alternating magnetic field to generate ultrasonic waves.
• Piezoelectric method: uses a crystal, such as quartz, that generates a potential difference when compressed or expanded, producing ultrasonic waves.

Fluid Dynamics

• Rate of flow: the volume of fluid that flows across a section of a pipe in unit time.
• Continuity equation: the rate of fluid entering a region is equal to the rate of fluid leaving the region.

Bernoulli's Theorem

• Statement: the sum of pressure head, velocity head, and gravitational head remains constant for a fluid in steady flow.
• Applications: atomizers, sprayers, airplane wings, velocity of efflux of liquid, venturimeter, and filter pumps.

Stokes' Law of Viscosity

• Statement: the viscous force opposing the motion of a small spherical body in a viscous fluid is proportional to the velocity of the body, the radius of the body, and the coefficient of viscosity.
• Formula: F = 6πηrv.

Viscosity and Coefficient of Viscosity

• Viscosity: the property of a fluid that opposes the relative motion of its layers.
• Coefficient of viscosity: the proportionality constant in Stokes' law of viscosity.

Poiseuille's Equation

• Derivation: assumes steady, parallel flow of a fluid through a capillary tube, with no radial flow or slip at the tube walls.
• Formula: V = (Pπr^4) / (8ηl), where V is the volume of fluid flowing per unit time, P is the pressure difference, r is the radius, η is the coefficient of viscosity, and l is the length of the tube.

Streamline Flow and Turbulent Flow

• Streamline flow: a steady, parallel flow of a fluid, with each layer moving smoothly and steadily.
• Turbulent flow: an unsteady, zigzag flow of a fluid, with many disturbances between layers.