Chapter 19: Dawn of Modern Physics

• Classical Physics vs. Modern Physics

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  Classical Physics	Modern Physics
  Based on classical/Newtonian mechanics.	Based on quantum mechanics.
  Deals with macroscopic level, ordinary everyday life.	Deals with microscopic level, and macro levels as special cases.
  Span includes: Newton's laws of motion, Newton's law of gravitation, Four laws of thermodynamics, Gas laws, Maxwell's equations of E.M. waves.	Span includes: Michelson's and Morley's findings, Schrodinger wave equations, De-Broglie hypothesis, Bohr's ideas, Heisenberg's uncertainty principle, Planck's theory, Special theory of relativity, General theory of relativity.

• Ideas of Modern Physics

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  • Light is an electromagnetic wave.
  • Light consists of photons, each having energy $E=hf$.
  • Speed of light is a universal constant in vacuum, independent of motion of source and observers.
  • Absorption and emission of electromagnetic waves take place in packets of energy called photons.
  • True laws of physics retain their mathematical form in all inertial frames of reference.
  • All motions and rest are relative; nothing like absolute rest or motion.
  • Wave-particle duality.
  • Reaction takes some time to replay, after action.
  • Matter waves are associated with moving particles.
  • Einstein's Mass-Energy relation: $E=mc^2$.
  • Pair production: Photon splits into electron-positron pair.
  • Positron and electron annihilate to photons.
  • Position, momentum, energy, and time cannot be measured accurately at the same time (Heisenberg's Uncertainty Principle).

• Relative Motion & Frame of Reference

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  • Concept of direction is purely relative.
  • Experiments give same results in same frame of reference; different results in different frames.
  • Frame of Reference: Coordinate system (x, y, z axes) from where observation is made.
  • Inertial Frame: At rest or moving with uniform velocity. Law of inertia is valid. Earth can be considered an inertial frame for motion on Earth.
  • Non-inertial Frame: Moving with accelerated motion.

• Theory of Relativity (Special Theory)

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  • Proposed by Einstein in 1905.
  • Deals with inertial frames of reference.
  • Important Results:
    • Relativistic Mass: $m=\frac{m_o}{\sqrt{1-\frac{v^2}{c^2}}}$. ($m_o$: rest mass, $v$: relativistic speed, $c$: speed of light).
    • Lorentz Contraction (Length Contraction): $l=l_o\sqrt{1-\frac{v^2}{c^2}}$. ($l_o$: proper length).
    • Time Dilation: $t=\frac{t_o}{\sqrt{1-\frac{v^2}{c^2}}}$. ($t_o$: proper time).
    • Mass-Energy Equivalence: $E=mc^2$.
    • No body moves with velocity greater than velocity of light.
    • At ordinary speeds ($v \approx 0$), $m \approx m_o$, $l \approx l_o$, $t \approx t_o$. This is why relativistic effects are not observed in daily life.
    • If $v = 0.999c$, then $m = 22.4 m_o$.
    • If $v \ge 2.6 \times 10^8~m/s$, then $l = l_o/2$ and $m = 2 m_o$.
  • NAVSTAR Navigation System uses results of special theory of relativity for accurate location and speed measurements.

• Black Body Radiation

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  • Black body: Ideal radiator and ideal absorber. Emissive/absorptive power is 1. Very bad reflector.
  • Emits electromagnetic waves.
  • Intensity Distribution (Lummer and Pringsheim): Energy is not uniformly distributed across wavelengths at a given temperature.
  • Wien's Displacement Law: $\lambda_{max} T = \text{constant} = 2.9 \times 10^{-3}~m K$. So, $T \propto \frac{1}{\lambda}$. (Holds for shorter wavelengths).
  • Stefan-Boltzmann Law: Total energy (area under curve) $E = \sigma T^4$. ($\sigma = 5.67 \times 10^{-8}~W m^{-2} K^{-4}$ is Stefan's constant).
  • Shape of black body radiation curve is independent of material.
  • Classical theories (Wein's, Rayleigh-Jeans) failed to explain the full spectrum. Rayleigh-Jeans holds for longer wavelengths.
  • Planck's Quantum Theory: Absorption and emission of radiation takes place in packets of energy called quanta ($E=hf$). Explains black body radiation at all wavelengths. Max Planck received Nobel Prize in 1918.

• The Photons

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  • Einstein: Light energy consists of packets of electromagnetic energy (photons).
  • Planck: Emission and absorption by atoms is in form of indivisible packets (quanta).
  • Light has discontinuous (granular) nature.
  • Beam of light with wavelength $\lambda$ consists of stream of photons traveling at speed $c$ and carrying energy $hf$.
  • Energy: $E = hf = \frac{hc}{\lambda}$.
  • Momentum: $p = mc = \frac{hf}{c} = \frac{h}{\lambda}$.
  • Emission or absorption of energy applies to any oscillating system.

• Interaction of Electromagnetic Radiation with Matter

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  • Three types depending on photon energy:

  • Energy Range	Type of Interaction
    $hf < 0.1~MeV$	Photoelectric effect
    $0.1~MeV < hf < 1~MeV$	Compton effect
    $hf > 1~MeV$	Pair production

  • Photoelectric Effect & Photon Theory of Light

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    • Emission of electrons from a metal surface when light is incident on it.
    • Supports quantum nature of light. Emitted electrons are photoelectrons.
    • Based on energy conservation. Explained by Einstein (Nobel Prize 1921).
    • Photoelectric current ($i$) is proportional to intensity of light ($I$). ($i \propto I$).
    • Saturation current: Maximum photoelectric current at fixed positive potential.
    • Stopping potential ($-V_o$): Negative potential at which photoelectric current becomes zero.
    • Stopping potential is proportional to maximum kinetic energy ($K.E_{max}$) of photoelectrons.
    • Stopping potential depends on frequency of incident light, NOT on intensity.
    • Einstein's Photoelectric Equation: $K.E_{max} = hf - \phi_o = hf - hf_o = h(f-f_o)$.
    • Also, $eV_o = K.E_{max} = \frac{1}{2}mv_{max}^2$.
    • Laws of Photoelectric Effect:
      1. Rate of emission $\propto$ intensity of incident light.
      2. Existence of threshold frequency ($f_o$): Minimum frequency below which no electrons are ejected, regardless of intensity.
      3. Work function ($\phi_o$): Minimum energy to eject an electron. $\phi_o = hf_o = \frac{hc}{\lambda_o}$. Depends on nature of metal and impurities. Unit: eV, Joule, erg.
      4. Maximum K.E. of photoelectrons $\propto$ frequency of incident light, independent of intensity.
      5. Instantaneous process: Time interval between light incidence and electron emission is negligible ($<10^{-8}s$).
    • Failure of Classical Theory: Could not explain independence of K.E. from intensity, existence of threshold frequency, and instantaneous emission.

    Phenomenon	Explained by Waves	Explained by Particles
    Reflection	✓	✓
    Refraction	✓	✓
    Interference	✓
    Diffraction	✓
    Polarization	✓
    Photoelectric effect		✓

  • Photocell

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    • Based on photoelectric effect.
    • Consists of cathode (photo-emissive surface) and anode.
    • Sodium/Potassium for visible light, Cesium coated oxidized silver for infrared, Selenium/Silicon for UV/X-rays/gamma-rays.
    • Applications: Security systems, Counting systems, Automatic doors/street lighting, Exposure meters, Sound tracks of movies, Solar panels.

  • Compton Effect

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    • When a photon hits an electron, it scatters with a frequency less than incident photon (wavelength increases).
    • Usually X-ray photons used ($E \ge 17.5~KeV$).
    • Compton shift ($\Delta\lambda$): Increase in wavelength.
    • $\Delta\lambda = \lambda_f - \lambda_i = \frac{h}{m_o c}(1-\cos\theta)$.
    • Compton wavelength: $\frac{h}{m_o c} = 2.43 \times 10^{-12}~m$.
    • $\Delta\lambda = 0$ when $\theta = 0^\circ$.
    • $\Delta\lambda = \frac{h}{m_o c}$ when $\theta = 90^\circ$.
    • $\Delta\lambda = \frac{2h}{m_o c}$ when $\theta = 180^\circ$.
    • Strong evidence for particle (photon) nature of E.M. waves.

  • Pair Production

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    • Decomposition of a photon into an electron-positron pair.
    • Requires photon energy $hf > 1.02~MeV$.
    • Energy equation: $hf = 2m_o c^2 + K.E_{e^-} + K.E_{e^+}$.
    • Rest mass energy of electron/positron = $m_o c^2 = 0.51~MeV$.
    • Cannot take place in vacuum; usually occurs near a heavy nucleus (for momentum conservation).
    • Materialization of energy.

  • Annihilation of Matter

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    • Reverse process of pair production. Conversion of mass into energy.
    • Electron and positron annihilate to produce two photons.
    • Two photons move in opposite directions to conserve momentum.
    • Each photon has energy $0.51~MeV$ (equivalent to rest mass energy of electron).

  • Antimatter

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    • P.A.M. Dirac theoretically predicted in 1928.
    • Anderson discovered positron in 1932.
    • Every antiparticle has same mass, same spin, but opposite magnetic moment and charge to its respective particle.

    Particle	Antiparticle
    Electron	Positron
    Proton	Antiproton
    Neutron	Antineutron
    Neutrino	Antineutrino

• De-Broglie's Hypothesis (Wave-Particle Duality)

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  • All moving particles behave as waves (matter waves or particle waves).
  • Wavelength associated with moving particles: $\lambda = \frac{h}{mv} = \frac{h}{p}$.
  • $\lambda \propto \frac{1}{m}$, $\lambda \propto \frac{1}{v}$.
  • For large mass and ordinary speed objects, wavelength is too small for wave effects (interference, diffraction) to be observable.
  • Prince Louis Victor de Broglie received 1929 Nobel Prize.

  • Davisson and Germer Experiment

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    • Experimental confirmation of de-Broglie's hypothesis.
    • Showed electrons are diffracted from metal crystals like X-rays.
    • Wavelength associated with moving electrons: $\lambda = \frac{h}{\sqrt{2mVe}}$. ($V$: accelerating potential).
    • For $V=54~V$, $\lambda = 1.66 \times 10^{-10}~m$.
    • This matches Bragg's equation for 1st order diffraction from nickel crystal ($d=0.91 \times 10^{-10}~m$), which gives $\lambda = 1.65 \times 10^{-10}~m$.
    • Davisson and Thomson shared Nobel Prize in 1937.
    • Electron microscope is a practical application of wave-particle duality.

• Heisenberg's Uncertainty Principle

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  • Certain pairs of quantities cannot be measured with perfect accuracy at the same time.
  • Conjugate quantities: Linear momentum and position ($\Delta p \cdot \Delta x \approx h$). Energy and time ($\Delta E \cdot \Delta t \approx h$).
  • To increase accuracy in measuring position, use short wavelength.
  • To increase accuracy in measuring momentum, use long wavelength.

• Electromagnetic Spectrum

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  S.#	Name	Wavelength Range (metre)	Frequency Range (hertz)	How Produced
  1.	Gamma rays	$6 \times 10^{-13} - 1 \times 10^{-10}$	$5 \times 10^{20} - 3 \times 10^{19}$	Nuclei of atoms
  2.	X-rays	$1 \times 10^{-10} - 3 \times 10^{-8}$	$3 \times 10^{19} - 1 \times 10^{16}$	Bombardment of high Z target by electrons
  3.	Ultraviolet rays	$3 \times 10^{-8} - 4 \times 10^{-7}$	$1 \times 10^{16} - 8 \times 10^{14}$	Excitation of atoms and vacuum spark
  4.	Visible light	$4 \times 10^{-7} - 8 \times 10^{-7}$	$8 \times 10^{14} - 4 \times 10^{14}$	Excitation of atoms, spark and arc flame
  5.	Infrared rays	$8 \times 10^{-7} - 3 \times 10^{-5}$	$4 \times 10^{14} - 3 \times 10^{13}$	Excitation of atoms and molecules
  6.	Heat radiation	$1 \times 10^{-5} - 1 \times 10^{-1}$	$3 \times 10^{13} - 3 \times 10^{9}$	Heating
  7.	Microwaves	$1 \times 10^{-3} - 3 \times 10^{-1}$	$3 \times 10^{11} - 1 \times 10^{9}$	Oscillating currents in special vacuum tubes
  8.	Ultra high radio frequencies	$1 \times 10^{-1} - 1$	$3 \times 10^{9} - 3 \times 10^{8}$	Oscillating circuits
  9.	Very high radio frequencies	$1 - 10$	$3 \times 10^{8} - 3 \times 10^{7}$	Oscillating circuits
  10.	Radio frequencies	$10 - 10^4$	$3 \times 10^{7} - 3 \times 10^{4}$	Oscillating circuits
  11.	Power frequencies	$5 \times 10^6 - 6 \times 10^4$	$60 - 50$	Weak radiations from A.C. circuits

  • Human body emits E.M. waves in infrared region at only 310K.

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