2024 Physics End-Term Study Material PDF

Simple Harmonic Motion: A Periodic motion possessing an equilibrium position, such that the Restoring force is directly proportional to the displacement from the mean position which also happens to be the equilibrium position. De-Broglie Hypothesis: To any body or material particle having momentum...

Simple Harmonic Motion: A Periodic motion possessing an equilibrium position, such that the Restoring force is directly proportional to the displacement from the mean position which also happens to be the equilibrium position. De-Broglie Hypothesis: To any body or material particle having momentum ‘p’, a fictitious wave (matter-wave) of wavelength ‘λ’ is intrinsically associated according to the following relation; ℎ λ= 𝑝 Gauss’s law for Magneto-statics: 𝐵 ∙ 𝑑𝑎 = 0 The LHS accounts for the total magnetic-flux through an arbitrary closed surface ‘A’, the RHS implies that the total magnetic-flux through an arbitrary closed surface ‘A’ will always be zero. Effectively, this means that the magnetic fields do not have any divergence whatsoever. This rules out the possibility of existence of any independent magnetic charges/poles. Transducers and Sensors: A transducer is a device that converts a signal in one form of energy to a signal in another. A sensor is a transducer that receives and responds to a signal or stimulus from a physical system. Pressure sensor is a type of smart-sensor that are used to monitor the pressure of gases or fluids in a pipeline. A sudden drop in pressure might indicate a leak causing the smart-sensor to take a pre-defined action of closing valves and issue warning. Energy-bands in a Solid: Valence-band and Conduction-band. For insulators 𝐸𝑔 ≫ 𝑘𝑇 Derivation of Acceptance angle for step-index Optical Fiber. Refer to Class-Notes. De-Broglie wavelength of an electron that has been accelerated from rest through a potential difference of 650V. From Work-Energy theorem, 𝑚𝑣 2 𝑝2 𝑉𝑞 = = 2 2𝑚  𝑝2 650 V 1.6 × 10−19 C = 2(9.11 × 10−31 kg) 𝑝= 2 9.11 × 10−31 kg 650 V 1.6 × 10−19 C ℎ λ= 𝑝 ℎ λ= 2 9.11 × 10−31 kg 650 V 1.6 × 10−19 C 6.63 × 10−34 J s λ= 2 9.11 × 10−31 kg 650 V 1.6 × 10−19 C ∴ λ = 4.81 × 10−11m From De-Broglie relation, ℎ λ= 𝑚𝑣 It is evident that ‘λ’ is inversely proportional to ‘mv’ For macroscopic object ‘m’ is on the order of kg, making its ‘λ’ quite small. Consider, least eligible macroscopic particle of 𝑚 = 0.001 kg and 𝑣 = 0.1 m/s Its 6.63 × 10−34 J s λ= (0.001 kg)(0.1 ms −1 ) λ = 6.63 × 10−30 m This wavelength is even smaller than the width of a proton (1.70 × 10−15 m) ! Therefore, on macroscopic scales (cm to m) the λ of matter-wave of an object is almost non-existent compared to its dimensions. Hence, particle nature overwhelms wave nature on macroscopic scales. Integral Form of Maxwell-Heaviside Equations Gauss’s law for Electro-statics: 𝑄𝑒𝑛𝑐 𝐸 ∙ 𝑑𝑎 = 𝜀0 This law establishes a relation between electric charges and electric fields. It defines what positive, negative, neutral charge is based on Electric-flux through ‘A’. The LHS accounts for the total Electric-flux through an arbitrary closed surface ‘A’, the RHS implies that the nature of net charge enclosed within ‘A’ depends on the total Electric-flux through an arbitrary closed surface ‘A’. If the total flux is positive then, ‘A’ contains source of Electric field/net positive charge. If the total flux is negative then, ‘A’ contains sink of Electric field/net negative charge. If the total flux is zero then, ‘A’ contains no net charge it is neutral. Gauss’s law for Magneto-statics: 𝐵 ∙ 𝑑𝑎 = 0 The LHS accounts for the total magnetic-flux through an arbitrary closed surface ‘A’, the RHS implies that the total magnetic-flux through an arbitrary closed surface ‘A’ will always be zero. Effectively, this means that the magnetic fields do not have any divergence whatsoever. This rules out the possibility of existence of any independent magnetic charges/poles. Maxwell-Faraday’s Law: 𝜕 𝐸 ∙ 𝑑𝑙 = − 𝐵 ∙ 𝑑𝑎 𝜕𝑡 This law establishes the relation between Electric and Magnetic fields; and describes how changing Magnetic flux through arbitrary surface ‘A’ bounded by contour ‘L’ results in circulation of Electric-field through contour ‘L’. Stronger Magnetic-flux results in greater circulation of Electric field. Greater rate of change of Magnetic-flux results in greater circulation of Electric field. The minus sign is the embodiment of Lenz’s law. Ampere-Maxwell law: 𝜕 𝐵 ∙ 𝑑𝑙 = 𝜇0 𝐼𝑒𝑛𝑐 + 𝜇0 𝜀0 𝐸 ∙ 𝑑𝑎 𝜕𝑡 This law also establishes the relation between Magnetic and Electric fields; and describes how changing Electric flux through arbitrary surface ‘A’ bounded by contour ‘L’ results in circulation of Magnetic-field through contour ‘L’. It also indicates that circulation of Magnetic-field through contour ‘L’ can also result from flow of electric-charges (current) through surface ‘A’ bounded by contour ‘L’ Rate of change of E-flux, strength of E-field, and amount of 𝐼𝑒𝑛𝑐 are all in proportion to the strength of circulation of B-field.  Conditions for a Physically valid wave-functions. a) The wave-function 𝜳(𝒙, 𝒕) must be continuous. b) All its partial derivatives must also be continuous. c) The wave function Ψ 𝑥, 𝑡 must be quadratically integrable. This means that the below integral must exist. Ψ ∗ Ψ 𝑑τ d) The above integral must be single-valued. e) The wave-function 𝜳(𝒙, 𝒕) must be normalised. +∞ Ψ𝑖 ∗ Ψ𝑖 𝑑τ = 1 −∞ f) The wave-function 𝜳(𝒙, 𝒕) must be orthogonal. +∞ Ψ𝑖 ∗ Ψ𝑗 𝑑τ = 0 −∞ g) The wave-function 𝜳(𝒙, 𝒕) must be finite everywhere. h) The wave function 𝜳(𝒙, 𝒕) must satisfy the boundary conditions of the quantum mechanical system it represents. (g) Heisenberg’s Uncertainty Principle: ℎ σ𝑥 σ𝑝 ≥ 4𝜋 It signifies that there is a limit to the precision with which certain pairs of physical properties, such as position 𝑥 and momentum 𝑝, can be simultaneously known. The more accurately one property is measured, the less accurately the other property can be known. Energies of SHM: 1 2 𝐸𝑘 = 𝑘𝐴 𝑐𝑜𝑠 2 (𝜔𝑡 + 𝜑) 2 1 𝑈 = 2 𝑘𝐴2 𝑠𝑖𝑛2 (𝜔𝑡 + 𝜑) 𝐸𝑇 = 𝐸𝑘 + 𝑈 1 1 = 𝑘𝐴2 𝑐𝑜𝑠 2 𝜔𝑡 + 𝜑 + 𝑘𝐴2 𝑠𝑖𝑛2 𝜔𝑡 + 𝜑 2 2 1 1 = 𝑘𝐴2 𝑐𝑜𝑠 2 𝜔𝑡 + 𝜑 + 𝑠𝑖𝑛2 𝜔𝑡 + 𝜑 = 𝑘𝐴2 (1) 2 2  1 2 ∴ 𝐸𝑇 = 𝑘𝐴 2 1 𝑑(𝐸𝑇 ) 𝑑(2 𝑘𝐴2 ) = =0 𝑑𝑡 𝑑𝑡 𝑑(𝐸𝑇 ) ∵ =0 𝑑𝑡 Total Energy 𝐸𝑇 does not vary with time 𝑡. Doping of Semi-Conductors: Doping is the intentional introduction of impurities (trivalent or pentavalent) into an intrinsic semiconductor for the purpose of modulating its electrical, optical and structural properties. Two types of extrinsic semiconductor are, p-type (trivalent doping) and n-type (pentavalent doping). Optical Fibres rely on AIR Total Internal Reflection (TIR): TIR is the phenomenon in which waves arriving at the interface from one medium to another are not refracted into the external medium, but completely reflected back into the internal medium. It occurs when WATER the external medium has a lower refractive index than the internal medium, and the waves are incident at a sufficiently oblique angle (Critical Angle) on the interface. A Diode is a p–n junction connected to two electrical terminals. TDSE: TISE: ħ2 𝑑2 𝜑 𝑥 − + 𝑉 𝑥 φ 𝑥 = 𝐸φ(𝑥) 2𝑚 𝑑𝑥 2 Probability of observing a particle between 𝒙𝟏 and 𝒙𝟐 𝒙𝟐 𝒙𝟐 𝑷 𝒙𝟏 < 𝑿 < 𝒙𝟐 = 𝜳∗ 𝒙 𝜳 𝒙 𝒅𝒙 = 𝜳(𝒙) 𝟐𝒅𝒙 𝒙𝟏 𝒙𝟏 Normalisation Condition: +∞ +∞ 𝜳∗ 𝒙 𝜳 𝒙 𝒅𝒙 = 𝜳(𝒙) 𝟐𝒅𝒙 = 𝟏 −∞ −∞ Plane Progressive Wave: Wavelength (λ) : The amount of distance per cycle or The minimum distance between particles which are vibrating in phase with each other. SI Unit is metres (m) Amplitude : The maximum displacement of a particle in the wave from its rest/equilibrium position. For EMWs, it is the maximum disturbance in the Electric or Magnetic Field strength. Frequency (ν) : The number of oscillations (cycles) per unit time. 1 𝜔 ν= = 𝑇 2𝜋 SI unit, 𝑠 −1 or Hz Time-Period (T) : The amount of time per cycle and has units of time, seconds (s) in SI. 1 2𝜋 𝑇= = ν 𝜔 Path Difference (d) : Constructive Interference 𝑑 = 𝑛λ 1 Destructive Interference 𝑑 = (𝑛 + 2)λ Where, 𝑛 = 0, 1, 2, 3, … Theoretical speed of EMWs and the EM-Spectrum: Radio Waves Ultraviolet Rays Used in radio and television UV lamps are used to kill germs in water communication systems. purifiers. Cellular phones use radio waves to Used in anti-counterfeiting designs. transmit voice communication in the (Currency notes/Passport) ultrahigh frequency (UHF) band. X-Rays Microwaves X-Ray imaging to detect bone fractures. Used in radar systems for aircraft X-ray crystallography to determine navigation. lattice arrangement of atoms. Used in Microwave oven to heat food. Gamma Rays Infrared Rays Used in Radiotherapy to kill cancerous Used in TV remote controls. cells. Used in Night-vision cameras for Sterilisation of medical equipment. surveillance/Thermal imaging. Visible Rays Visible light emitted or reflected from objects around us provides us information about the world. For illumination during night/darkness. Historical Key-points on the development of wave-particle duality of Radiation/Light: Wave Nature of Radiation(Light) Particle Nature of Radiation(Light) Huygens’s Principle Newton’s Corpuscular Theory Young’s double slit experiment Planck’s Law for Black-Body Radiation Arago’s Spot Photo-Electric Effect Foucault’s experimental measurement of Compton Effect speed of Light Maxwell’s theoretical derivation of speed of EMWs Theoretical Speed ≈ Experimental Speed Hertz experimentally proving existence of EMWs Energy Band Diagram (Insulator, Semiconductor, Conductor): a) The highest occupied energy band is only partially filled at 0 K. Sodium is an example of this kind. They are good conductors of electricity because as an electric field is applied, the electrons in the partially filled band can receive energy from the field and drift accordingly. b) The highest occupied energy level is completely filled at 0 K and the next higher level is completely empty when the atoms are well-separated. But as the atoms come closer and these levels split into bands, the bands overlap with each other. There are empty states at energies close to the occupied states and hence these solids are also good conductors. c) The highest occupied energy band is completely filled and the next higher band, which is empty, is well above it. The band gap between these two bands is large. The electrons do not have empty states at an energy slightly above or below their existing energies. If an electric field is applied by connecting the two ends of such a solid to a battery, the electrons will refuse to receive energy from the field. This is because they do not find an empty state at a slightly higher energy. Diamond is an example of this kind. d) The highest occupied band is completely filled at 0 K but the next higher band, which is empty, is only slightly above the filled band. An example is silicon. At temperatures well above 0 K, thermal collisions may push some of the electrons from the highest occupied band to the next empty band. These few electrons, have a large number of empty states just above their existing energy and hence can participate in electric conduction. The total number of electrons that can receive energy from the electric field is small, the conductivity is quite small as compared to common conductors. Longitudinal and Transverse Waves: A transverse wave is one in which the vibrations of the particles or disturbances in the wave are at right angles to the direction in which the energy of the wave is travelling. A longitudinal wave is one in which the direction of the vibrations of the particles or disturbances in the wave is along or parallel to the direction in which the energy of the wave is travelling. Interference by Amplitude Splitting: The amplitude of the incident beam is divided into two or more parts either by partial reflection or refraction. Thus we have coherent beams produced by division of amplitude. These beams travel different paths and are finally brought together to produce interference. Characteristics of a Good Sensor: it is sensitive to the measured property it is insensitive to any other property likely to be encountered in its application, and it does not influence the measured property. Characteristics of a Semi-Conductor: Conductivity increases as the temperature is increased. Intermediate electrical conductivity, conductivity greater than Insulators but less than Conductors. Given, Ψ 𝑥 = 𝐴𝑒 −𝛼 𝑥 Ψ∗ 𝑥 = 𝐴𝑒 −𝛼 𝑥 2 Ψ(𝑥) = Ψ∗ 𝑥 Ψ 𝑥 = 𝐴2 𝑒 −2𝛼 𝑥 Normalisation Condition, +∞ +∞ 2 2 Ψ(𝑥) 𝑑𝑥 = 𝐴 𝑒 −2𝛼 𝑥 𝑑𝑥 −∞ −∞ −∞ < 𝑥 < 0, 𝑥 = −𝑥 𝑎𝑛𝑑 0 ≤ 𝑥 < +∞, 𝑥 = 𝑥 0 +∞ = 𝐴2 𝑒 2𝛼𝑥 𝑑𝑥 + 𝑒 −2𝛼𝑥 𝑑𝑥 −∞ 0 Do Integral yourself ! 1 = 𝐴2 𝛼 By Normalisation Condition, 2 1 𝐴 =1 𝛼 ⇒𝐴= 𝛼 Normalised Wave-function is, Ψ 𝑥 = 𝛼𝑒 −𝛼|𝑥| 𝑃(0 < 𝑥 < 1/𝛼) 1 1 𝛼 𝛼 |Ψ(𝑥)|2 𝑑𝑥 = 𝛼 𝑒 −2𝛼|𝑥| 𝑑𝑥 0 0 1 0 ≤ 𝑥 < 𝛼, 𝑥 = 𝑥 1 𝛼 =𝛼 𝑒 −2𝛼𝑥 𝑑𝑥 0 Do Integral yourself ! 1 𝑃 0