Remove["Global`*"]
{grad,div,curl}={"∇", "∇•","∇×"};
{tdE,tdB}={"((∂(Overscript[E, ->]))/(∂t))","((∂(Overscript[B, ->]))/(∂t))"};
{vecE,vecB,vecJ,vecF,vecv}={"(Overscript[E, ->])","(Overscript[B, ->])","(Overscript[J, ->])","(Overscript[F, ->])","(Overscript[v, ->])"};
{vecx,vecxt}={"(Overscript[x, ->])","((Overscript[x, ->]),t)"};
M1=StringJoin[div,vecE,"=","4π ρ"];
M2=StringJoin[curl,vecE,"= –\!\(\*FractionBox[\(\(1\)\(\\ \)\), \(c\)]\)",tdB];
M3=StringJoin[div,vecB,"=","0"];
M4=StringJoin[curl,vecB,"=","((4  π)/c)", vecJ ," + ",tdE];
TF=TraditionalForm;

SetDirectory["/home/stump/rs842"];
import1=Import["chap4.png","PNG"];

import1

Lecture 1 - Review of Microscopic Electrodynamics

Units

In PHY 842 we will use Gaussian units.  The base units are cm, g, s.

The unit of electric charge is esu (electrostatic unit).

1 esu = 1 statcoulomb = 1 franklin
1 (esu^2) = 1 dyne (cm^2)
1 esu = 1 (g^(1/2)) (cm^(3/2)) (s^(–1/2)) .
The proton charge is e = 4.8 × (10^(–10)) esu.

Other electric and magnetic units:
[E] = esu/(cm^2) = statvolt/cm
[B] = esu/(cm^2) = gauss

See the Appendix in Jackson to convert between SI units and gaussian units.

Microscopic Electrodynamics

Microscopic electrodynamics means the theory of charges, currents, and fields in “empty space”.
Of course the space is not empty, because of the charges, currents and fields.
However, there are no macroscopic materials present.

Field equations

Grid[{{Style["micro electrodynamics [gaussian units]",24],SpanFromLeft},{M1,M2},{M4,M3}},Frame->All]//TF

Vacuum permeability and permittivity

In SI units we need two parameters, (Subscript[μ, 0]) and (Subscript[ϵ, 0]). These parameters don't exist (or, we could say they are equal to 1) in Gaussian units.

The only parameter in the microscopic field equations is c = 2.998 × (10^10) cm/s = the speed of light in vacuum.

The continuity equation for charge

StringJoin[div,vecJ," ="," – ((∂ρ)/(∂t))"]//TF

This equation says that charge is locally conserved.

◼ The forces on a test charge q

StringJoin[vecF," = ","q(",vecE,"+",{"((Overscript[v, ->])/c)"},"×" ,vecB, ") ;" ]//TF

i.e.,

StringJoin[vecF,vecxt," = ","q [ ",vecE,vecxt," + (1/c) ","(Overscript[v, ->])(t)"," × " ,vecB,vecxt, " ]" ]//TF

These forces acting on a test charge q may be taken to define the fields.

Energy and Momentum

You already know these quantities from PHY 841.

The field energy in a volume Ω is

U = StringJoin["U(t) = "," \!\(TraditionalForm\`\*SubscriptBox[\(∫\), \(\(\\ \\ \\ \)\(Ω\)\)]\) u" ,vecxt ,"  (d^3)x" ]//TF

StringJoin["u" ,vecxt ," = (1/(8  π)) (E^2)",vecxt, "  +  ","(1/(8  π)) (B^2)",vecxt]//TF

The work per unit time per unit volume is

(Overscript[J, ->])( (Overscript[x, ->]),t) •  (Overscript[E, ->])( (Overscript[x, ->]),t) ;
this is the work done on the charge by the fields.

The energy flux is

(Overscript[S, ->]) = (Overscript[E, ->]) × (Overscript[B, ->]) ;

this is the Poynting vector  ( units: [S]  = erg/(cm^2)/s )

Poynting’s theorem, for microscopic electrodynamics,

eq

Note:
Poynting' s theorem is the continuity equation for energy.
It says that energy is locally conserved.

THREE SPECIAL CASES

1. Electrostatics
2. Magnetostatics
3. E.M. Waves in vacuum

ELECTROSTATICS

⟹  a static system with

StringJoin["ρ(",vecx, ")  and  ",vecE ,"(",vecx,")"]//TF
StringJoin[vecJ," =0   and   " ,vecB, " =0"]//TF

∴ The field equations are

StringJoin[div,vecE," = ","4π ρ","   and   ",curl,vecE, " =0 "]//TF

Or, we might write

StringJoin[ vecE ,"= –" ,grad,"Φ" ,"   and    ", "–(∇^2)Φ = 4π ρ"]//TF

An important issue will be, what are the boundary conditions?

MAGNETOSTATICS

⟹  a static system with

StringJoin["ρ = 0  and  ",vecE ," = 0"]//TF
StringJoin[vecJ,"(",vecx,")" ,"  and  " ,vecB, "(",vecx,")"]//TF
StringJoin[div,vecJ," = 0"]//TF

∴ the field equations are

StringJoin[curl,vecB,"  = ","((4  π)/c) ",vecJ,"  and  ",div,vecB, " = 0 "]//TF

Or, we might write

StringJoin[ vecB ," = ",curl,"(Overscript[A, ->])" ,"   and  ", div, "(Overscript[A, ->])"," = 0"]//TF

i.e., using the Coulomb gauge.

ELECTROMAGNETIC WAVES

For free waves in vacuum, there are no charges or currents.

(Charge or current somewhere else must have created the waves in the first place; that is called "radiation". But waves propagate through vacuum, conserving energy and momentum, and no charges are necessary.)

The solutions of Maxwell’s equations are

StringJoin[vecE,vecxt," = ","(Subscript[(Overscript[E, ->]), 0]) cos((Overscript[k, ->])•(Overscript[x, ->])–ωt + φ)"]//TF
StringJoin[vecB,vecxt," = ","(Subscript[(Overscript[B, ->]), 0]) cos((Overscript[k, ->])•(Overscript[x, ->])–ωt + φ)"]//TF

with ω = ck   and    (Overscript[k, ->]) × (Subscript[(Overscript[E, ->]), 0]) = k (Overscript[(Subscript[B, 0]), ->]) .