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Calculate α ν and j ν from the Einstein coefficientsPowerPoint Presentation

Calculate α ν and j ν from the Einstein coefficients

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### MAXWELL’S EQUATIONS L, T:Wave Equations

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

Calculate αν and jν from the Einstein coefficients

(1)

Consider emission: the emitted energy is

where

Each emission event produces energy hν0

spread over 4π steradians

so

(2)

From (1) and (2):

Emission coefficient

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

When

the populations are inverted

Since

increases along ray, exponentially

HUGE amplifications

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:

Scattering source function

An integro-differential equation: Hard to solve.

You need to know Iν to derive Jν to get dIν/ds

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

Time varying B E

Force

Is a vector

is a vector with components

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

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:

[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

In the L, T: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 L, T:EMU (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

EMU vs. ESU L, T:

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,

We will be mostly concerned with Maxwell’s equations L, T:

In a vacuum, i.e.

Dielectric Media L, T:: E-field aligns polar molecules,

Or polarizes and aligns symmetric molecules

Diamagnetic: L, T: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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