TF=TraditionalForm;

FUNDAMENTAL FORMULAS OF PHYSICS
(out of print)

Chapter 13 : ELECTROMAGNETIC THEORY

Nathaniel H. Frank and William Tobocman

Table of Contents
1. Definitions and Fundamental Laws
2. Electrostatics
3. Magneto statics
4. Electric Circuits
5. Electromagnetic Radiatiopn
Tables

/1/ Definitions and Fundamental Laws of Macroscopic Electrodynamics

/1.1/ Primary definitions

text

eq1 = Grid[{
{"SI", "Gaussian"},
{"d (Overscript[F, ->])/dq = (Overscript[E, ->]) + (Overscript[v, ->]) × (Overscript[B, ->])","d (Overscript[F, ->])/dq = (Overscript[E, ->]) + (Overscript[v, ->])/c × (Overscript[B, ->])"}},
ItemSize->{15,2},Spacings->{0,0.5},Frame->All]//TF

text

eq2=Grid[{{"SI","Gaussian"},
{" (Overscript[E, ->]) = (Overscript[D, ->])/ϵ = (Overscript[D, ->])/ (κ ϵ0)"," (Overscript[E, ->]) = (Overscript[D, ->])/ϵ = (Overscript[D, ->])/κ"}},
ItemSize->{15,2},Spacings->{0,0.5},Frame->All]//TF

text

eq3=Grid[{{"SI","Gaussian"},
{" B = μH = κμH "," B = μH = κ H "}},
ItemSize->{15,2},Spacings->{0,0.5},Frame->All]//TF

text

eq4=Grid[{{"vacuum","SI","Gaussian"},
{"permittivity","(Subscript[ϵ, 0])=8.854((▫10)^(-19)) F/m","(Subscript[ϵ, 0])=1"},
{"permeability","(Subscript[μ, 0])=4((π▫10)^(-7)) H/m","(Subscript[μ, 0])=1"}},
ItemSize->{{9,12,6},2},Spacings->{0,0.5},Frame->All]//TF

text

eq57=Grid[{{"","SI","Gaussian"},
{"polarization","P=D-(Subscript[ϵ, 0])E","P=1/4π ( D=ϵ0 E}"},
{"magnetization","M=B/(Subscript[μ, 0])-H","P=1/4π ((B/μ0-H)}"},
{"magn susceptibility","χm = M/H","χm=M/H"},
{"elec susceptibility","χe=P/(Subscript[ϵ, 0])E","χe=P/ϵ0 E"}},
ItemSize->{{12,8,12},2},Spacings->{0,0.5},Frame->All]//TF

/1.2/ Conductors

text

eq121=" (Overscript[J, ->]) = σ (Overscript[E, ->])"//TF
eq122=" R = (L/(σA)) = ((Lρ)/A)"//TF

Clearly it is impossible to maintain a static potential difference betaeen twon points of a conductor without flow of charge ...
... The tangential component of E must vanish at the surface of a conduction for static situations.

/1.3/ Ferromagnetic materials

text

/1.4/ Fundamental laws

Ele ctrromagnetic theory can be beased on the definitions above and the following fundamental laws

Grid[{
{"fundamental laws","SI","Gaussian"},
{"conservation of charge","divJ=-∂ρ/∂t","divJ=-∂ρ/∂t"},
{"field equations"," ",SpanFromLeft},
{"Faraday's law","curl (Overscript[E, ->]) = - ∂ (Overscript[B, ->])/∂t","curl (Overscript[B, ->]) = - (1/c) ∂ (Overscript[E, ->])/∂t"},
{"Ampere's law","curl (Overscript[H, ->]) = (Overscript[J, ->]) + ∂(Overscript[D, ->])/∂t","curl (Overscript[H, ->]) = ((4  π)/c) (Overscript[J, ->]) + (1/c) ∂(Overscript[D, ->])/∂t"},
{"Gauss's law","div D = ρ","div D = 4π ρ"},
{"Gauss's law","div B = 0","div B = 0"}},
ItemSize->{{12,12,15},2},Spacings->{0,0.5},Frame->All]//TF

The 4 field equations are called Maxwell’s equations.

/1.5/ Boundary conditions

Applications of Maxwells' equations to infinitesimal contours and volumes at the boundary between two regions in space yield the following boundary conditiopns for electromagnetic fields

eq151=Grid[{{"SI","Gaussian"},
{"(OverHat[n])•(B2-B1) = 0","(OverHat[n])•(B2-B1) = 0"},
{"(OverHat[n])×(H2-H1) = K","(OverHat[n])×(H2-H1) = 4π/c K"},
{"(OverHat[n])×(E2-E1) = 0","(OverHat[n])•(E2-E1) = 0"},
{"(OverHat[n])▪(D2-D1) = σ","(OverHat[n])▪(D2-D1) = 4π σ"}},
ItemSize->{{12,12,15},2},Spacings->{0,0.5},Frame->All]//TF

where (OverHat[n]) is a unit vector normal to the boundary and pointing from region 1 into region 2. Also,  σ is surface density of free charge;  and (Overscript[K, ->]) is surface current density of free current.

/1.6/ Vector and scalar potentials

We always have curl (Overscript[B, ->]) = 0. Therefore we can write (Overscript[B, ->]) = curl (Overscript[A, ->]). The function (Overscript[A, ->])((Overscript[x, ->]),t) is called the vector potential.

Then from Faraday' s law, we can write E = – grad Φ – ∂A/∂t (SI units) or
 E = – grad Φ  – (1/c) ∂A/∂t (Gaussian units). The function Φ(x,t) is called the scalar potential.

The functions A and Φ are not unique, because of gauge invariance. Therefore we may impose additional conditions on A and/or Φ; this process is called “choosing a gauge”.

The equations B = curl A and E = -grad Φ  - ∂A/∂t  guaranteethat the homogeneous  Maxwell equations are obeyed. Now substituting these equatiopn into the inhomogeneous Maxwell equations (without specifying a gauge choice) we obtain general equations for Φ and Α.

eq153=Grid[{{"SI",
"(∇^2)Φ = ϵμ (∂^2)(∂^2) - ∂∂(divA+ϵμ∂∂)-ρ/ϵ"},
{"Gaussian",
"(∇^2)Φ = ϵμ/(c^2) (∂^2)(∂^2) - 1/c ∂∂(divA+ϵμ/c ∂∂)-4πρ/ϵ"}},
ItemSize->{{8,30},2},Spacings->{0,0.5},Frame->All]//TF

eq154=Grid[{{"SI",
"(∇^2)A = ϵμ \!(∂^2)(∂^2) + ∇(divA+ϵμ∂∂)-μJ"},
{"Gaussian",
"(∇^2)A = ϵμ/(c^2) (∂^2)(∂^2) + ∇(divA+ϵμ/c ∂∂)-4πμ/c J"}},
ItemSize->{{8,30},2},Spacings->{0,0.5},Frame->All]//TF

1.7/ Bound charge

text

eq171=Grid[{
{"SI","Gaussian"},
{"div (Overscript[E, ->]) =(1/(Subscript[ϵ, 0])) ( ρ - div (Overscript[P, ->]))","div E = ((4  π)/(Subscript[ϵ, 0])) ( ρ - div (Overscript[P, ->]))"}},
ItemSize->{15,2},Spacings->{0,0.5},Frame->All]//TF

This result has led to the following nomenclature

StringJoin["     ","– div P = bound charge density"]//TF
StringJoin["     "," (OverHat[n]) ▫ (Overscript[P, ->]) = bound surface charge density"]//TF

where (OverHat[n]) is hte outward normal to the saurface.

1.8/ Amperian currents

text

eq181=Grid[
{{"SI","Gaussian"},
{"curl (Overscript[B, ->]) = (Subscript[μ, 0]) ( (Overscript[J, ->]) + curl (Overscript[M, ->]))","curl (Overscript[B, ->]) = 4π (Subscript[μ, 0]) ( (Overscript[J, ->]) + curl (Overscript[M, ->]))"}},
ItemSize->{{15,18},2},Spacings->{0,0.5},Frame->All]//TF

when ∂(Overscript[D, ->])/∂t = 0. This result provides justification fo the folloing nomenclature

StringJoin["     "," curl (Overscript[M, ->]) = amperian current density"]//TF
StringJoin["     "," (Overscript[M, ->]) × (OverHat[n]) = amplerian surface current density"]//TF

where (OverHat[n]) is the outward normal to the surface.

/2/ Electrostatics