Lecture 31: The Speed of Light, Experiments and Concepts - PDF

Lecture#31 The Reality of Light Understanding the nature of light has been a long journey, requiring the collective efforts of thousands of people over centuries. Only about 250 years ago, we began to understand its true nature. Today, we know that light is actually electromagnetic waves, produced...

Lecture#31 The Reality of Light Understanding the nature of light has been a long journey, requiring the collective efforts of thousands of people over centuries. Only about 250 years ago, we began to understand its true nature. Today, we know that light is actually electromagnetic waves, produced when a charge moves. This movement generates a current, which in turn produces these waves. The importance of light in our lives drives our desire to know everything about it. The Speed of Light One of the first questions people asked about light was about its speed. Early studies attempted to measure this, with Galileo being one of the pioneers in this field. Galileo's Experiment Galileo conducted an experiment in 1667 where he asked one of his assistants to stand on a mountain with a lantern. He stood on another mountain with his lantern. The idea was that when Galileo uncovered his lantern, the assistant would uncover his as soon as he saw the light, and they would measure the time difference. However, this method proved to be impractical due to the incredibly high speed of light. Roemer's Discovery In 1675, the Danish astronomer Ole Roemer made a significant contribution to understanding the speed of light. He observed the moons of Jupiter and noticed that the eclipses of these moons occurred at different times depending on the Earth's position relative to Jupiter as shown in fig.1. When the Earth was closer to Jupiter, the eclipses happened sooner, and when it was further, the eclipses were delayed by about 16 minutes and 6 seconds (16.6 min. late). Roemer used this time difference to estimate the speed of light (estimated c = 2.3*108 m/s), concluding it takes this amount of time for light to travel the additional distance across the Earth's orbit.  Fig. 1: Ole Roemer's observations of Jupiter's moons' eclipses in 1675 showed time variations based on Earth's position, leading to an estimate of the speed of light at 2.3 × 108 m/s. Fizeau's Experiment Overview Hippolyte Fizeau, a French scientist, conducted a pioneering experiment in 1849 to measure the speed of light. He used a rotating cogwheel apparatus as shown in the fig. 2. 1. A beam of light was directed through the gaps between the teeth of a rotating wheel. 2. The light traveled a known distance L to a distant mirror. 3. The mirror reflected the light back through the rotating wheel. 4. By adjusting the rotational speed of the wheel, Fizeau determined the speed of light with remarkable accuracy.  Fig. 2: Fizeau's 1849 experiment used a rotating cogwheel and a distant mirror to measure the speed of light by timing the light's round trip and adjusting the wheel's rotation speed. Derivation of the Formula to calculate the speed of light: The light beam travels a total distance of 2 L ( to the mirror and back ). The timet taken for the light to travel this distance is given by : 2L t= C The cogwheel has N teeth and an equal number of gaps. The angular distance between consecutive teeth ( or gaps ) is : 2  = = 2N N The wheel rotates with an angular velocity  ( radians per second ). In timet , the wheel rotates through an angle t. For the returning light beam to be observed , it must pass through the next gap. Therefore, the angle rotated by the wheel in timetshould equal to :  = t  2L Using, t = C  2L   =  C  2L  = C  2 L C=  3*108 m / s  Where, L : Distance to the mirror. c : Speed of light.  : Angular distance between the teeth of the cogwheel ( in radians ).  : Angular velocity of the rotating wheel. Foucault's Experiment In 1860, Léon Foucault improved upon Fizeau's work using a rotating mirror system. This method involved: 1. A light source directed towards an eight-sided rotating mirror as shown in the fig.3. 2. The light reflected from the mirror to a fixed mirror and back. 3. The rotation speed was adjusted so the light returned to the observer through the same path. 4. He found no signal unless mirror rotates 1/8 turn in ΔT 5. This experiment further confirmed the high but finite speed of light, c = 2.98∙108 m/s  Fig. 3: Foucault's 1860 experiment used an eight-sided rotating mirror and a fixed mirror to measure the speed of light, refining the estimate to 2.98 × 108 m/s by adjusting the rotation speed for precise light reflection. The Nature of Light and Electromagnetic Waves All theses experiments shows that light travels incredibly fast around 300,000 kilometres per second, completing multiple orbits (7-8) around the Earth in one second. This explains why we see lightning before hearing thunder since sound waves travel much slower, at 1100 meters per second. Light travels much faster than sound. For example: 1. Thunder and lightning start at the same time, but we will see the lightning first. 2. When a starting pistol is fired, we see the smoke first and then hear the bang. The Visible Spectrum Light and sound are waves, but light is part of a vast electromagnetic spectrum. The visible range for humans is from about 400 nanometers (deep blue) to 700 nanometers (deep red) as shown in fig.4. Different light sources emit varying wavelengths within this range.  Fig. 4: The visible spectrum, ranging from 400 nm (deep blue) to 700 nm (deep red), represents the portion of the electromagnetic spectrum perceivable by the human eye. White Light and Color Perception White light contains all colours in the visible spectrum. It is perceived as white when all the frequencies are approximately equal. Combining different colors can create other colors: Blue and red make turquoise. Blue and green make light blue. Red and green make yellow. Combining red, blue, and green in equal amounts produces white as shown in fig. 5.  Fig. 5: White light, containing all colors of the visible spectrum, can be split into individual colors. Combining different colors like blue and red, blue and green, and red and green can create turquoise, light blue, and yellow, respectively, while equal amounts of red, blue, and green produce white. Objects appear coloured based on the light they reflect. For example, a yellow object reflects yellow light, while a white object reflects all colours. Filters and Color Selection Colour filters can select specific colours from white light by absorbing other colours. For example, a red filter allows only red light to pass through. Light Dispersion White light is not a single colour, when white light passes through a prism, it disperses into its component colours due to different wavelengths traveling at different speeds through the medium. This results in a spectrum of colours from blue to red. Exploring Light: Reflection and Refraction Understanding the behaviour of light through simple experiments can illuminate the principles of reflection and refraction. Reflection of Light Experiment with a Laser and Mirror Using a laser, we can conduct experiments to explore how light reflects off surfaces. When a laser beam hits a mirror: 1. Part of the light passes through the mirror. 2. Another part of the light is reflected. Key Concepts of Reflection Incident Ray: The incoming light ray that strikes the mirror. Reflected Ray: The light ray that bounces off the mirror.  Angle of Incidence: The angle between the incident ray and the normal (a line perpendicular to the surface). Angle of Reflection: The angle between the reflected ray and the normal. In a smooth mirror, the angle of incidence is equal to the angle of reflection as shown in fig. 6. Fig. 6: demonstrating the law of reflection in a smooth mirror, where the angle of incidence is equal to the angle of reflection. Diffuse Reflection When light hits a rough surface, such as a coat, it scatters in many directions. Each small element of the surface reflects light according to the law of reflection, but the overall effect is diffuse reflection, where light is reflected in multiple directions as shown in fig. 7.  Fig. 7: Diffuse reflection occurs when light strikes a rough surface, causing it to scatter in various directions due to the irregularities on the surface. Refraction of Light Experiment with a Laser and Glass When a laser beam passes through a glass surface: 1. The light enters the glass and bends towards the normal. 2. As it exits the glass and re-enters the air, it bends away from the normal. Key Concepts of Refraction Refraction: The bending of light as it passes from one medium to another. Normal: A line perpendicular to the surface at the point of incidence. Angle of Incidence: The angle between the incident ray and the normal. Angle of Refraction: The angle between the refracted ray and the normal. When light travels from air to glass, it bends towards the normal as shown in fig. 8. Conversely, when it exits the glass and enters air, it bends away from the normal. The incident light and transmitted light are parallel but displaced. Fig. 8: Refraction of light at the interface between air and glass, demonstrating bending towards the normal upon entering glass and away from the normal upon exiting, with incident and transmitted light paths shown as parallel but displaced. Understanding Refractive Index and Light Behavior Definition of Refractive Index The refractive index (n) is defined as the speed of light in a vacuum (c) divided by the speed of light in a medium (v): Speed of light in vacuum Refractive index = Speed of light in material c n= v When light enters a medium, it usually slows down due to interactions with the medium's particles, causing a change in speed. Reasons for Light Slowing Down When light enters a medium, the electrons within the medium interact with the electromagnetic field of the light, causing them to vibrate and emit electromagnetic radiation. This process results in the formation of a new wave that moves forward but at a slower speed than the original wave. Refractive Index of Common Materials: Air: 1.0003 Water: 1.33 Alcohol: 1.36 Oil: 1.5 Diamond: 2.42 Light Refraction Experiment with Water 1. Shine a laser through an empty glass. 2. Gradually fill the glass with water. 3. Observe the laser beam bending towards the normal as water is added to the glass due to the higher refractive index of water (1.33). Effects of Refraction Objects submerged in water appear bent or displaced due to the change in light direction as it passes from water (higher refractive index) to air (lower refractive index). Fermat's Principle Historical Context Fermat's Principle, which is fundamental in the study of optics, states that light takes the path that requires the least time when traveling from one point to another. This principle was also mentioned by Ibn Al-Haytham (Ibne Haim) in his significant work "Kitab al-Manazir" (The Book of Optics), though the mathematics behind it was not fully developed during his time. Fermat's Principle provides an easier and more straight forward understanding of the behavior of light compared to the methods available in earlier periods. Fermat's Principle Explained Light in Free Space: In free space or a vacuum, light travels in straight lines. According to Fermat's Principle, this is because the straight-line path is the shortest and therefore the quickest route for light to travel between two points. To mathematically represent this, consider two points A and B in free space as shown in fig. 9. The time t taken for light to travel this distance L is given by: t = L/c where c is the speed of light in a vacuum. Since the distance L in a straight line is the shortest possible distance, the time t is minimized. Fig. 9: Fermat's Principle dictates that light travels in straight lines in free space or a vacuum, following the path that minimizes the time between two points A and B. A light ray traveling from one fixed point to another fixed point follows a path such that, compared with nearby paths, the time required is minimum. For a case of Reflection Consider a light wave traveling from point A to point B, reflecting off a mirror at point O. Fermat's Principle can be used to determine the path of the light.  The total distance traveled by the light is, L = a 2 + x2 + b2 + ( d − x ) 2 t = L/c Taking derivative w.r.t 'x', dt 1 dL = dx c dx where: a is the vertical distance from A to the mirror. b is the vertical distance from B to the mirror. x is the horizontal distance from A to O. d is the total horizontal distance between A and B. To minimizes the travel time, take the derivative of t with respect to x and set it to zero.  dt =0 dx 1 dL 1 d 0= = ( a 2 + x2 + b2 + ( d − x ) ) 2 c dx c dx = ( a 2 + x 2 ) ( 2 x ) + b 2 + ( d − x )  ( 2 )( d − x )( −1) 1 −1/2 1 2 1/2 2c 2c   On simplyfying, x d−x = a2 + x2 b2 + ( d − x ) 2 sin 1 = sin 1 1 = 1 This results in the equality of angles of incidence and reflection, confirming that the angle of incidence θ1 is equal to the angle of reflection θ2. Refraction and Snell's Law When light travels from one medium to another, its speed changes due to different refractive indices. This change causes the light to bend, a phenomenon known as refraction. Derivation Using Fermat's Principle Consider a light ray traveling from point A in medium 1 (with refractive index n1) to point C in medium 2 (with refractive index n2). The light refracts at the boundary between the two mediums at point O. Let: θ1 be the angle of incidence. θ2 be the angle of refraction. x be the horizontal distance from A to O. L1 be the distance from A to O. L2 be the distance from O to C. Using Fermat's Principle, the total time t taken by light to travel from A to C is the sum of the time taken in each medium: L1 L2 t= + v1 v2 where v1 and v2 are the speeds of light in mediums 1 and 2, respectively. The speeds of light in the mediums are related to the refractive indices by: c c c n = ,  v1 = ,  v2 = v n1 n2 n1 L1 + n2 L2 L t= = c c L = n1L1 + n2 L2 ( Optical path length ) L = n1 a 2 + x 2 + n2 b 2 + ( d − x ) 2 = ( a + x 2 ) ( 2 x ) + 2 b 2 + ( d − x )  ( 2 )( d − x )( −1) dt 1 dL n1 2 −1/2 n 2 1/2 0= = dx c dx 2c 2c   x d−x n1 = n2 a +x b2 + ( d − x ) 2 2 2 n1 sin 1 = n2 sin  2 (called snell's law of refraction) Total Internal Reflection When light travels from a medium with a higher refractive index to one with a lower refractive index, it can be totally internally reflected if the angle of incidence exceeds a certain critical angle. If the angle of incidence is greater than θc, total internal reflection occurs, and the light does not pass into the second medium. Critical Angle and Total Internal Reflection Consider a laser emitting light at different angles. When the angle of incidence reaches the critical angle (θc), the light travels along the surface of the medium. If the angle of incidence exceeds θc, total internal reflection occurs, and the light remains within the medium as shown in fig. 10.  One can calculate the critical angle by following: n1 sin c = n2 sin 900 n2 c = sin −1 n1 Fig. 10: Total internal reflection occurs when light within a medium strikes the boundary at an angle exceeding the critical angle θc, causing it to reflect back into the medium rather than refracting out. Importance of Total Internal Reflection Critical Angle: The angle beyond which light is totally internally reflected within the medium. Applications: o Optical fibers for communication. o Diamond brilliance due to a low critical angle and high refractive index. Optical Fibers: Optical fibers use total internal reflection to transmit light signals over long distances with minimal loss. An optical fiber consists of a core with a high refractive index surrounded by cladding with a lower refractive index. Light entering the fiber is reflected internally, ensuring it travels through the core without escaping. For effective transmission, the optical fiber must be very thin, and the refractive index must be controlled to ensure total internal reflection. Optical Fiber Communication and the Nature of Light Structure of Optical Fiber: Core: Thin glass thread, thinner than a human hair, with a high refractive index as shown in fig. 11. Cladding: Surrounds the core and has a lower refractive index to ensure total internal reflection. Protective Covering: Encases the cladding for protection. Fig. 11: Representation of an optical fiber, comprising a core with a high refractive index, surrounded by cladding with a lower refractive index, and a protective covering, essential for efficient light transmission through total internal reflection. Optical Fiber in Communication 1. Historical Context: o 100 years ago, cables could only support 50-60 telephone calls simultaneously. o Modern optical fibers can handle millions of calls due to total internal reflection. 2. Mechanism: o Light travels through the core, reflecting internally due to the difference in refractive indices. o This prevents signal loss, even if the fiber is bent or under slight pressure. 3. Advantages:  o High capacity for data transmission. o Minimal signal loss. The Nature of Light: Particle or Wave? Historical Debate Newton's Particle Theory: Light is made of particles. Wave Theory: Light behaves as waves. Modern Understanding Light exhibits both particle and wave properties, known as wave-particle duality. A wave is a moving pattern that is revealed by its interaction with particles Concept of Wavefronts 1. Wavefronts: o Circular Wavefront: Originates from a point source, spreads out in circles. o Plane Wavefront: Forms when circular wavefronts travel far and flatten. 2. Interaction with Obstacles: o When waves hit an obstacle, they can change direction and create new wavefronts. o This results in regions of light and dark, demonstrating wave behavior. Huygens' Principle Basic Concept: o When light encounters an obstacle, each point on the wavefront acts as a source of new wavelets. o These wavelets spread out and form new wavefronts as shown in fig. 12. Implications: o Light can bend around corners. o This bending and spreading of light waves are due to their wave nature. Fig. 12: Huygens' Principle illustrates how each point on a wavefront acts as a source of secondary wavelets, which propagate and form new wavefronts, allowing light to bend around corners and demonstrating its wave nature. Light as Electromagnetic Radiation Generation of Light: o Light is generated by accelerating charges. o Examples: Heated filament in a light bulb, or electrons moving in atoms. o Power radiated by a single charge is: 2 e2 a 2 P= 3 c3  o Electrons in atoms emit radiation when they accelerate. o Classical physics could not explain why electrons don't lose energy and spiral into the nucleus. Quantum Mechanics: o Introduced to explain atomic behavior and energy quantization. Electromagnetic Waves: o Composed of electric and magnetic fields oscillating perpendicular to each other. o Energy carried by electromagnetic waves depends on the amplitude of these fields. o Electromagnetic waves transport linear momentum as well as energy o For complete absorption of energy U, U p= c o For complete reflection of energy U, 2U p= c o Radiation pressures can be determined experimentally Emax Bmax E2 B2 = max2 = max 2 o c o c 2 o Colors and Temperature Color of Light: o The color of light from an object indicates its temperature. o Hotter objects emit light of shorter wavelengths (e.g., blue stars). o Cooler objects emit light of longer wavelengths (e.g., our sun appears yellow). Radiation Pressure Concept: o The electric field of an electromagnetic wave that strikes a surface act on an electron, giving it a velocity v as shown in fig. 13. The magnetic field then exerts a  force on the moving charge in the direction of propagation of the incident light. That is the origin of the “light pressure”. o Light exerts pressure on objects when it is absorbed or reflected. o This radiation pressure is typically very small and not noticeable in everyday life. Fig. 13: Radiation pressure arises from the momentum carried by electromagnetic waves. When light is absorbed or reflected, it exerts a small but measurable pressure on surfaces, stemming from the interaction of the electric and magnetic fields with the surface of electrons. Solar Radiation Pressure: o Within the sun, radiation pressure counteracts gravitational pull, maintaining equilibrium. Radiant Flux: Total amount of energy passing through a surface or region of space per unit time o Typically denoted by  (Phi) o Also called power o Measured in watts (W) or joules/second (J/s) ▪ These are metric quantities (rather than e.g., calories/second) Typically used for a light’s total output.