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 Maxwell's equations
§ 26. Maxwell's Equations
With the method developed in the derivation of the equations of the atomic orbitals allows for the derivation of Maxwell's equations and Maxwell's structure of light. Maxwell's electric curl equation is derived using Faraday's wire loop induction effect represented with the magnetic flux (fig 24),
emf =  ʃʃ (dB/dt)· dA...........................................93
A second wire loop emf equation is used that represents the internal electric field E that forms the wire loop emf,
emf = ʃ E · dl................................................ .......94
Equating equations 89 and 90,
ʃ E · dl =  ʃʃ (dB/dt)· dA.......................................95
Using Stokes' theorem (Hecht, p. 649),
ʃ E · dl =  ʃʃ (∇ x E)· dA......................................96
Equating equations 91 and 92,
 ʃʃ(dB/dt)· dA = ʃʃ (∇ x E)· dA.............................97
Maxwell electric curl equation is derived using equation 93,
∇ x E =  dB/dt................................................ ...98
Faraday's induction effect depicts an internal electric current that only forms within the conduction wire represented in equation 90 yet Maxwell's electric curl equation (equ 94) is used to represent an electric field of an electromagnetic light wave that exists in the space outside the conduction wire. Faraday's induction effect is not luminous yet Maxwell's equations are used to represent the structure of light, and, the magnetic flux of Faraday's induction effect is pointing in the direction of the propagating magnetic field which represents a longitudinal magnetic wave yet Maxwell's electric curl equation is used to derive equations that depict electromagnetic transverse waves.
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Maxwell's magnetic curl equation is derived using Ampere's law (Hecht, p. 42),
ʃ B · dl = ui................................................ ..........99
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Maxwell electric current (dE/dt), that forms in the space between a varying capacitor (fig 25), is added to Ampere's law,
ʃ B · dl = ʃʃ (J + ε dE/dt) · dA ..............................100
Using Stokes' theorem, on the left side of equation 96 forms (Hecht, p. 649),
ʃ B · dl = ʃʃ (∇ x B) · dA........................................101
Equating equations 96 and 97 then using J = 0,
ʃʃ (ε dE/dt)· dA = ʃʃ (∇ x B) · dA............................102
Maxwell's magnetic curl equation is derived using equation 98,
∇ x B = 1/c (dE/dt)..............................................1 03
Hecht's electric current (dE/dt) forms in the open space between the plates of a varying capacitor which conflicts with Faraday's induction effect that electric current only forms within the current wire. In Maxwell's derivation of Maxwell's equations, Maxwell only uses Faraday's induction effect to derive Maxwell's equations (Maxwell, Part III) yet Hecht's derivation of the magnetic curl equation is using a varying capacitor. Hecht is using Stokes' theorem to derive equations 92 and 97 that depict the equating of a line integral with a surface integral which is physically and mathematically invalid.
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§ 27. Maxwell's Structure of Light
The electromagnetic transverse wave equations of light are derived using Maxwell's equations,
∇ x E =  dB/dt........................∇ x B = 1/c (dE/dt).....................................104a,b
Maxwell's curl equations (equ 100a,b) are expanded to form,
dEz/dy  dEy/dz =  dBx/dt................................................ ...........................105
dEx/dz  dEz/dx =  dBy/dt................................................ ..........................106
dEy/dx  dEx/dy =  dBz/dt................................................ ...........................107
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dBz/dy  dBy/dz = 1/c (dEx/dt)............................................... .....................108
dBx/dz  dBz/dx = 1/c (dEy/dt)............................................... ....................109
dBy/dx  dBx/dy = 1/c (dEz/dt)............................................... ...... ..............110
The zdirection electric transverse wave equations is derived using equations 101 and 105 by eliminating dEy/dz and dBz/dx to form (Jenkins, p. 410),
dEy/dz = 1/c (dBx/dt)..............................dBx/dz = 1/c (dEy/dt)...................111a,b
Differentiating equation 107a, with the respect to d/dz, and equation 107b with respect to d/dt produces (Condon, p, 1108),
d2Ey/d2z = 1/c (d2Bx/dtdz)......................d2Bx/dtdz = 1/c (d2Ey/d2t)...........112a,b
Equating equations 108a,b,
d2Ey/d2z = 1/c2 (d2Ey/d2t).............................................. ............................113
Differentiating equation 107a, with the respect to d/dt, and equation 107b with respect to d/dz produces ,
d2Ey/dtdz = 1/c (d2Bx/d2t)......................d2Bx/d2z = 1/c (d2Ey/dtdz)...........114a,b
Equating equations 110a,b forms,
d2Bx/d2z = 1/c2 (d2Bx/d2t).............................................. ............................115
Equations 109 and 111 are used to derive the z direction electromagnetic transverse wave equations of light (fig 17),
Ey = Eo cos(kz  wt) ĵ .................................................. ............................116
Bx = Bo cos(kz wt) î .................................................. ..............................117
To test the derivation, the zdirectional electric and magnetic transverse wave equations of light (equ 112 & 113) are used in equation 107a,
d/dz[Eo cos(kz  wt)] ĵ =  (1/c) d/dt[Bo cos(kz  wt)] î.........................118
Equation 114 forms,
Eo ĵ = Bo î .................................................. ..........................................119
Equation 107a that is used to derive the electromagnetic transverse wave equations of light produces a unite vector catastrophe since equation 115 depicts the equating of the î and ĵ unit vectors which is produced since Maxwell's equations represents electromagnetic longitudinal waves.
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Part B
In an alternative gradient identity method, the electromagnetic transverse wave equations of light are derived using Maxwell's equations,
∇ x E =  dB/dt...........................∇ x B = 1/c (dE/dt)....................120a,b
A curl operator is applied to Maxwell's electric curl equation (equ 116a) to form,
∇ x (∇ x E) =  d/dt (∇ x B)................................................ ..........121
Using equation 116b, in equation 117, then rearranging forms,
∇ x (∇ x E) =  1/c (d2E/d2t).............................................. ............122
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A second equation is derived using the gradient identity (Klein, p. 523),
∇ x (∇ x E) = E(∇ · E)  ∇2 E................................................. .......123
and ∇ · E = 0 to form,
∇ x (∇ x E) = ∇2 E................................................. .........................124
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Equating equations 118 and 120 forms (Hobson, p. 23),
d2E/d2t  c2 ∇2E = 0................................................. .........................125
A similar equation is derived using the 116b,
d2B/d2t  c2 ∇2B = 0................................................. .........................126
The electromagnetic wave equations of light (fig 17) are derived using equations 121 and 122,
E = Eo ei(kr  wt) .................................................. ..................................127
B = Bo ei(kr  wt) .................................................. ..................................128
The gradients ∇2 E and ∇2 B of equations 121 and 122 denote a volume that depicts electromagnetic longitudinal waves.
