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Old Apr 23rd 2018, 03:14 PM   #1
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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.



.................................................. .................................................. .................................................. .................................................. .................................................. ..............




Maxwell's magnetic curl equation is derived using Ampere's law (Hecht, p. 42),





ʃ B dl = ui................................................ ..........99

..



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



.................................................. .........




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 z-direction 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, 1-108),





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) .................................................. ..............................117



To test the derivation, the z-directional 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)] .........................118





Equation 114 forms,





Eo ĵ = Bo .................................................. ..........................................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 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



.................................................. .................................................. ................................





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



.................................................. .................................................. ................................





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.
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Old Apr 23rd 2018, 03:54 PM   #2
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You seem to like to write a lot. But can you at least write an abstract of sorts to tell us what your point is? All I see here is a commentary on the Maxwell equations. Is there a question here? If not, what is the point you are trying to make?

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Old Apr 23rd 2018, 04:17 PM   #3
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Is Maxwell's equations that are used to represent the structure of light in a cosmic sense of the word for real or not for real or maybe totally rad or just tantie which is it?
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Old Apr 23rd 2018, 04:31 PM   #4
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Originally Posted by lovebunny View Post
Is Maxwell's equations that are used to represent the structure of light in a cosmic sense of the word for real or not for real or maybe totally rad or just tantie which is it?
Remember that it is not the depiction of the photon from a Classical sense that matters, but from the Quantum. The properties are the same but the idea behind where is comes from and why comes from Quantum.

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Old Apr 28th 2018, 04:59 PM   #5
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You justification of my theory is surely awesome and I hope to here more from you. Thank you for your encouraging replies. Also, Maxwell's equations are based on Faraday's induction effect that depicts an expanding electromagnetic field that conflicts with a light particle that has a discrete structure. If we were to look at it a macroscopic sense an expanding field negates all particle structure.
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Old Apr 28th 2018, 09:02 PM   #6
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Originally Posted by lovebunny View Post
You justification of my theory is surely awesome and I hope to here more from you. Thank you for your encouraging replies. Also, Maxwell's equations are based on Faraday's induction effect that depicts an expanding electromagnetic field that conflicts with a light particle that has a discrete structure. If we were to look at it a macroscopic sense an expanding field negates all particle structure.
You keep mentioning how wonderful it is that I agree with you, but I don't. You need to read the contents of my posts, not just what you want to understand.

And a photon does not have any structure that we know of, discrete or not. It's a neutral spin 1 boson that is the quanta of the EM field. It also plays a role in electro-weak theory at energies above the critical energy. It also must play a role in quantum gravity at some level (since it carries energy) but I don't know if anyone has been able to work that part of the theory out yet.

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Old Apr 29th 2018, 06:16 AM   #7
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Originally Posted by topsquark View Post
Remember that it is not the depiction of the photon from a Classical sense that matters, but from the Quantum. The properties are the same but the idea behind where is comes from and why comes from Quantum.

-Dan
Dan M'man. I don't envy these tasks you've taken upon yourself.
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Old May 1st 2018, 12:12 PM   #8
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How do you know for certain that an electron has a spin since QM is based on the gauge that is massless. How can a massless structure represent an electron that has a mass?
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Old May 1st 2018, 08:17 PM   #9
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Originally Posted by lovebunny View Post
How do you know for certain that an electron has a spin since QM is based on the gauge that is massless. How can a massless structure represent an electron that has a mass?
Clearly you don't understand what a gauge is... There is no such thing as a gauge that would render all particles in a theory massless.

And I think I already gave you this link for spin but here it is again. This is for the Stern-Gerlach experiment.

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Old May 6th 2018, 10:58 AM   #10
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When you are represent an entity with a gauge (Maxwell's equations) that entity becomes massless. Also, in Stern-Gerlach experiment electron are being destroyed which violates energy conservation.
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