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Calculate α ν and j ν from the Einstein coefficients

Jan. 31, 2011 Einstein Coefficients Scattering E&M Review: units Coulomb Force Poynting vector Maxwell’s Equations Plane Waves Polarization. Calculate α ν and j ν from the Einstein coefficients. (1). Consider emission: the emitted energy is. where.

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Calculate α ν and j ν from the Einstein coefficients

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  1. Jan. 31, 2011Einstein CoefficientsScatteringE&M Review: unitsCoulomb ForcePoynting vectorMaxwell’s EquationsPlane WavesPolarization

  2. Calculate αν and jν from the Einstein coefficients (1) Consider emission: the emitted energy is where

  3. Each emission event produces energy hν0 spread over 4π steradians so (2) From (1) and (2): Emission coefficient

  4. Absorption coefficient: Total energy absorbed in In volume dV is or Let Recall: so

  5. Stimulated Emission Repeat as above for absorption, but change sign, and level 1 for level 2 So the “total absorption coefficient” or the “absorption coefficient corrected for simulated emission” is

  6. Equation of Radiative Transfer The Source Function

  7. Recall the Einstein relations,

  8. In Thermodynamic Equilibrium:

  9. And The SOURCE FUNCTION is the PLANCK FUNCTION in thermodynamic equilibrium

  10. LASERS and MASERS When the populations are inverted Since increases along ray, exponentially HUGE amplifications

  11. Scattering termin equation of radiativetransferRybicki & Lightman, Section 1.7 Consider the contribution to the emission coefficient from scattered photons • Assume: • Isotropy: scattered radiation is emitted equally in all angles •  jνis independent of direction • 2. Coherent (elastic) scattering: photons don’t change energy • ν(scattered) = ν(incident) • 3. Define scattering coefficient:

  12.  Scattering source function An integro-differential equation: Hard to solve. You need to know Iν to derive Jν to get dIν/ds

  13. Review of E&M Rybicki & Lightman, Chapter 2

  14. Qualitative Picture:The Laws of Electromagnetism • Electric charges act as sources for generating electric fields. In turn, electric fields exert forces that accelerate electric charges • Moving electric charges constitute electric currents. Electric currents act as sources for generating magnetic fields. In turn, magnetic fields exert forces that deflect moving electric charges. • Time-varying electric fields can induce magnetic fields; similarly time-varying magnetic fields can induce electric fields. Light consists of time-varying electric and magnetic fields that propagate as a wave with a constant speed in a vacuum. • Light interacts with matter by accelerating charged particles. In turn, accelerated charged particles, whatever the cause of the acceleration, emit electro-magnetic radiation After Shu

  15. Lorentz Force A particle of charge q at position With velocity Experiences a FORCE = electric field at the location of the charge = magnetic field at the location of the charge

  16. Law #3: Time varying E  B Time varying B  E

  17. Lorenz Force

  18. More generally, let current density charge density Force per Unit volume

  19. Review Vector Arithmetic

  20. Cross product Is a vector

  21. Direction of cross product: Use RIGHT HAND RULE

  22. NOTE:

  23. Gradient of scalar field T is a vector with components

  24. scalar

  25. A vector with components

  26. THEOREM:

  27. THEOREM

  28. Laplacian Operator: T is a scalar field Can also operate on a vector, Resulting in a vector:

  29. UNITS • R&L use Gaussian Units convenient for treating radiation • Engineers (and the physics GRE) use MKSA (coulombs, volts, amperes,etc) • Mixed CGS electrostatic quantities: esu electromagnetic quantities: emu

  30. Units in E&M We are used to units for e.g. mass, length, time which are basic: i.e. they are based on the standard Kg in Paris, etc. In E&M, charge can be defined in different ways, based on different experiments ELECTROSTATIC: ESU Define charge by Coulomb’s Law: Then the electric field is defined by

  31. So the units of charge in ESU can be written in terms of M, L, T: [eESU]  M1/2 L-3/2 T-1 And the electric field has units of [E]  M1/2 L-3/2 T-1 The charge of the electron is 4.803x10-10 ESU

  32. In the ELECTROMAGNETIC SYSTEM (or EMU) charge is defined in terms of the force between two current carrying wires: Two wires of 1 cm length, each carrying 1 EMU of current exert a force of 1 DYNE when separated by 1 cm. Currents produce magnetic field B:

  33. Units of JEMU (current density): Since [jEMU] = M1/2 L1/2 T -1 current [JEMU] = [jEMU] L-2 = M1/2 L-3/2 T-1 So [B]  M1/2 L-1/2 T-1 Recall [E]  M1/2 L-1/2 T-1 So E and B have the same units

  34. EMU vs. ESU Current density = charge volume density * velocity So the units of CHARGE in EMU are: [eEMU] = M1/2 L1/2 Since M1/2 L-3/2 T-1 = [eEMU]/L3 * L/T Thus, Experimentally,

  35. MAXWELL’S EQUATIONSWave Equations

  36. Maxwell’s Equations Let Charge density Current density

  37. Maxwell’s Equations Gauss’ Law No magnetic monopoles Faraday’s Law

  38. We will be mostly concerned with Maxwell’s equations In a vacuum, i.e.

  39. Dielectric Media: E-field aligns polar molecules, Or polarizes and aligns symmetric molecules

  40. Diamagnetic: μ < 1 alignment weak, opposed to external field so B decreases Paramagnetic μ > 1 alignment weak, in direction of field Ferromagnetic μ >> 1 alignment strong, in direction of external field

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