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Calculate α ν and j ν from the Einstein coefficients

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

(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

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

the populations are inverted

Since

increases along ray, exponentially

HUGE amplifications

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

Review of E&M

Rybicki & Lightman, Chapter 2

- 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

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

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

Operator:

T is a scalar field

Can also operate on a vector,

Resulting in a vector:

- 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:

[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 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 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

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,

MAXWELL’S EQUATIONSWave Equations

Maxwell’s Equations

Let

Charge density

Current density

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