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 Consider emission: the emitted energy is where (1)

Each emission event produces energy hν 0 spread over 4π steradians so (2) From (1) and (2): Emission coefficient

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

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

Equation of Radiative Transfer The Source Function

Recall the Einstein relations,

In Thermodynamic Equilibrium:

And The SOURCE FUNCTION is the PLANCK FUNCTION in thermodynamic equilibrium

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

Scattering term in equation of radiative transfer Rybicki & Lightman, Section 1.7 Consider the contribution to the emission coefficient from scattered photons Assume: 1.Isotropy: scattered radiation is emitted equally in all angles  j ν is independent of direction 2. Coherent (elastic) scattering: photons don’t change energy ν(scattered) = ν(incident) 3. Define scattering coefficient:

 Scattering source function An integro-differential equation: Hard to solve. You need to know I ν to derive J ν to get dI ν /ds

Review of E&M Rybicki & Lightman, Chapter 2

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

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

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

Lorenz Force

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

Review Vector Arithmetic

Cross product Is a vector

Direction of cross product: Use RIGHT HAND RULE

NOTE:

Gradient of scalar field T is a vector with components

scalar

A vector with components

THEOREM:

THEOREM

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

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

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

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

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:

Units of J EMU (current density) : Since [j EMU ] = M 1/2 L 1/2 T -1 current [J EMU ] = [j EMU ] L -2 = M 1/2 L -3/2 T -1 So [B]  M 1/2 L -1/2 T -1 Recall [E]  M 1/2 L -1/2 T -1 So E and B have the same units

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

MAXWELL’S EQUATIONS Wave Equations

Maxwell’s Equations Let Charge density Current density

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

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

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

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