Tiêu đề 20. Electromagnetic Waves and Matter Waves.pdf ✅

Electromagnetic Waves and Matter Waves Electromagnetic Waves 449 20 and Matter Waves QUICK LOOK Electromagnetic waves are waves that can travel through a vacuum, like in space. This is possible because they're not vibrations in an actual material; they're fluctuations in electric and magnetic fields. Examples of electromagnetic waves include radio waves, microwaves, infrared, visible light, ultraviolet, x-rays, and gamma rays. Radio waves have the lowest energy and frequency, and the longest wavelength. Gamma rays have the highest energy and frequency, and the shortest wavelength. Figure 20.1: The Electromagnetic Spectrum In electromagnetic wave, electric and magnetic field vector vibrate sinusoidally perpendicular to the direction of propagation of wave and also they are mutually perpendicular i.e., vectors E H k , , are mutually perpendicular. Electromagnetic waves are. The energy of electromagnetic waves is carried by electric and magnetic fields and transverse in nature. In free space electromagnetic wave travels with speed of light. Maxwell’s Equation (Integration Form) 0 q E ds ε ⋅ = ∫ (Gauss’s law in electrostatics) E ds ⋅ = 0 ∫ (Gauss’s law of magnetism) E ds t ∂φ ⋅ = ∂ ∫ (Maxwell’s law of electromagnetic induction) 0 0 E d B ds I dt φ μ ε   ⋅ = +     ∫ (Modified Ampere’s law) Maxwell’s Equations (Differential Form) I s f φ dS Q= Qf = total free charge inside surface S Div D = ρ f ρ f = charge density other than that of polarization div 0 E ρ ε = (in polarization – free space) II 0 φS B dS = Div B = 0 III L S d E dr B ds dt φ = − ∫ curl B E t ∂ = ∂ IV L S d H dr I D ds dt φ = + ∫ D = electric displacement = electric flux density curl D H j t ∂ = + ∂ curl 0 0 E B j t μ ε μ ∂ = + ∂ (when P = M = 0) Law of Charge Conservation 0 S S d j dS D ds dt φ φ + = div j 0 t ∂ρ + = ∂ Electromagnetic Waves Properties Frequency of oscillation in LC circuit 1 2 f π LC = Speed of electromagnetic wave in free space (vacuum) 0 0 1 c μ ε = Relation between 0 ε and μ0 2 0 0 0 ε μ = 1/ c Speed of electromagnetic wave in isotropic medium 0 0 1 1 r r c v με μ ε μ ε n = = Refractive index r r n = μ ε Yellow-green Infrared (IR) Violet Blue Green Yellow Orange Red Long wavelength Short wavelength Wavelength in nm 103 1 104 105 106 102 10 300 400 500 600 700 800 X rays Visible light Ultraviolet (UV)
450 Quick Revision NCERT-PHYSICS Pointing vectors (power flux density): Pointing vector represents power flow per unit area along the direction of wave propagation. S E H = × 2 0 0 = × c E B ε Wave impedance 0 0 r r E Z H μ μ μ ε ε ε = = = ⋅ For free space wave impedance, 376.6 r r Z μ ε = = Ω Relation between E and H of and B in free space r r E H μ ε = and E Bc = Energy density in an electric and magnetic field respectively when r ε and μr are constant 2 0 1 1 . 2 2 w D E E e r = = ε ε 2 0 1 1 1 2 2 m r w H B B μ μ = ⋅ = Energy density: In free space 2 2 0 0 1 1 2 2 B u E ε μ = + ⋅ Electric energy density 2 0 1 2 e u E = ε Magnetic energy density 2 0 2 m B u μ = Covering range of antenna of height h is d hR = (2 ) R = radius of earth In particular, total electrostatic energy of a charge distribution 1 1 2 2 W D E dv f Vdv = ⋅ = ρ ∫ ∫ v = volume Effective penetration depth (skin effect) 0 1/ r δ πμ μ σ = f σ = Electric conductivity Vector field (at point P) from a point charge Q1 with velocity 0 v 1 0 2 0 1 1 0 1 1 grad grad 4 Q v A E V πε r r t c     ∂ = − − = − −     ∂     0 0 0 1 curl r B E A c r = × = The Photoelectric Effect: The Einstein proposed quantum theory of light and was awarded Nobel Prize for it. Photoelectric effect was discovered by Hallwach in 1888, but explained by Einstein. The photoelectric current is directly proportional to intensity, but is independent of frequency of incident light. The maximum kinetic energy (or cut off voltage) increases with increases of frequency of incident light, but is independent of intensity. Einstein’s photoelectric equation was verified by Millikan. Photoelectric cell is device in which light energy is converted into electric energy. Einstein photoelectric equation gives maximum kinetic energy of photoelectron and is given by E hv k, max 0 = −φ φ0 = work function ( ) 5 p 7 / 2 Z hv σ ≥ σ p = cross section of photoelectric effect Work function 0 0 0 hv hv λ Φ = = Where 0 v is threshold frequency and λ0 is threshold wavelength. Kinetic energy of photoelectron varies from 0 to Ek 2 0 max 1 2 E eV mv k = = (V0 = cut off potential) For photoelectric effect to take place 0 0 hv ≥ Φ or 0 v v ≥ Energy and Momentum E h hv = = ω 0 v p n c = ħ n = refractive index p k = ħ 0 2 | | n k c π ω λ = = k = wave vector Photon: A photon is the quantum of electromagnetic radiation. The term quantum is the smallest elemental unit of a quantity, or the smallest discrete amount of something. Photons travel through empty space at a speed of approximately 186,282miles (299,792 kilometers) per second. This is true no matter what Electron Light Metal Figure: 20.2
Electromagnetic Waves and Matter Waves 451 the electromagnetic wavelength. The energy contained in a single photon does not depend on the intensity of the radiation. Energy of a photon hc ε hv λ = = Momentum of a photon hv h p c λ = = Rest mass of a photon = 0 Kinetic mass of a photon 2 hv h c cλ = = Number of photons in given energy energy energy of one photon E E N hc λ ε = = Compton Effect: In photoelectric effect the electron is bond, while in Compton effect the electron is free. Compton shift is independent of wavelength of incident light. Figure: 20.3 Compton shift in wavelength, ( ) 0 1 cos h m c ∆ = − λ φ Wavelength of scattered photon, ( ) 0 ' 1 cos h m c λ λ λ φ = + ∆ + − Kinetic energy of recoil electron, ' k hc hc E λ λ = = Direction of recoil electron, sin tan ' cos λ φ θ λ λ φ = − Compton wavelength of electron 0 0.024Å h m c = = Maximum Compton shift, max 0 2 ( ) 0.048Å h m c ∆ = = λ ( ) ' ' 1 1 cos E E hv α θ = = + − ; 2 0 0 E m c α = 2 2 2 2 0 0 0 0 0 ' E E E c p m c m c k c = − = + − ( )2 2 2 cos 1 1 tan 1 θ α φ = − + + 'sin 'sin cos cos 'cos p p Pc p p θ θ φ θ θ = = − c Z E σ ∝ σ c = cross-section of Compton effect De Broglie’s Relation: The wave nature of material particles was first proposed by de-Broglie. All material particles, charged or uncharged show wave nature. de-Broglie wavelength associated with a moving particle ( ) 2 k h h h p mv m E λ = = = Where n = masses of particle, v = velocity of particle, Ek = kinetic energy For a charged particle accelerated through a potential difference V ( ) 2 h mqV λ = where q = charge on particle. For electron accelerated through of p.d. of V volts ( ) 2 h mqV λ = meter 150 Å V       For thermal particle at absolute temperature T ( ) 1 2mkT λ = Where k = Boltzmann constant. For electron revolving in Bohr’s circular orbit of radius r s r n = = 2π λ Momentum h p k λ = = ħ , k = wave vector Group velocity and phase velocity , g d dE v v dk dp ω = = = p E v k p ω = = Dispersion law and de Broglie wavelength of a relativistic particle Figure: 20.4 Wavelength radius a λ θ φ Scattered electron Scattered x-ray photon X-ray photon electron
452 Quick Revision NCERT-PHYSICS 2 2 2 2 E p c E = +0 0 , 2 E m c 0 0 0 = ( ) 0 1/ 2 2 0 2 k k hc E E E λ = + Dispersion law and de Broglie wavelength of a non- relativistic particle 2 0 2 p E E m = + ( )1/ 2 2 k h mE λ = The first experimental evidence for wave nature of matter was given by Davisson and Germer. Figure: 20.5 Note A material particle is not equivalent to a single wave but it is equivalent to a wave packet (or group of waves). The velocity of group of waves is called group velocity = particle velocity. Wave-particle Duality: Light Does light consist of particles or waves? When one focuses upon the different types of phenomena observed with light, a strong case can be built for a wave picture: Table 20.1 Phenomena can be Explained by Waves or Particles Nature Phenomenon Can be explained in terms of waves. Can be explained in terms of particles. Reflection Yes Yes Refraction Yes Yes Interference Yes No Diffraction Yes No Polarization Yes No Photoelectric effect No Yes Compton effect No Yes Heisenberg’s Uncertainty Principle: The uncertainty principle holds for all systems (microscopic or macroscopic) and is independent of experimental technique. Impossible to know exactly: where something is how fast it is going Figure: 20.6 1 2 ∆ ∆ ≥ x Px ħ A more rigorous treatment gives       ≥∆∆ 4π h or 2 .. ħ px . If ∆x = 0 then ∆p = ∞ if ∆ = p 0 then ∆ = ∞ x i.e., if we are able to measure the exact position of the particle (say an electron) then the uncertainty in the measurement of the linear momentum of the particle is infinite. Similarly, if we are able to measure the exact linear momentum of the particle i.e., ∆ = p 0 , then we can-not measure the exact position of the particle at that time. Figure: 20.7 An electron cannot be observed without changing it's momentum; Uncertainty principle successfully explains Non-existence of electrons in the nucleus Finite size of spectral lines. Note: For Energy and time: 1 2 ∆ ∆ ≥ E t ħ ∆t is the time required to observe an energy to within the uncertainty ∆E. Incident photon Viewer Reflected photon Original momentum of electron Final momentum of electron ∆p ∆x 2 ∆ ∆ ≥ p x ħ Electron gun +54 V 50° Nickel crystal Electron scattering peak at 50° Theory 1.67 h mv λ = = Å for 54 V Experiment Path length difference d Å sin 2.15sin50 1.65 θ λ = ° = = for constructive interference Accelerating electrode hot filament to release electrons Not bad for a three year old idea! Nickel lattice Spacing d Å =2.15 d θ 1942 De Broglie’s hypothesis 1927 Davisson Germer experiment 1929 Nobel Prize for De Brodlie