. . 1.
Maxwell's
equations in classic field theory
. 1.a) Maxwell's equations (in rest - no movement of objects) .
Equivalent notation for operations in vector analysis, i.e. rot H = curl H = ∇ x H etc. In the above shown 4 equations is Nr. 1. Ampere's law (extended by Maxwell with displacement current), Nr. 2. Faraday's law, Nr. 3: electric Gauss' law, Nr. 4: magnetic Gauss' law (source definition). . Entities in these 4 Maxwell's equations: electric flux density D (-> Maxwell's "displacement") in [As / m²], electric field strength E [V / m], magnetic flux density B [Vs / m²], magnetic field strength H [A / m], electric current density J [A / m²], electric current I [A], electric volume charge density ρ [As / m³], electric flux ψel [As] = electric charge Q [As], magnetic flux ψmag[Vs] , displacement current density ∂ D / ∂ t [A / m²] and the equivalent in magnetics ∂ B / ∂ t [V / m²], differential length element dl [m], differential volume element dv [m³], NOTE: differential area element is ds [m²], ( with s = "surface" and in our Maxwell case not the normally used A = "area", because the same character A - derived from B = curl A - is the important magnetic vector potential A [Vs/m] ) . The constitutive relations between the classical field terms D, E, B, H and J, ( including both polarisations P & BP and external current sources Je ) are defined by : .
P [As
/ m²] is electric polarisation,
Bp = µ0 Mp
[Vs/m²] is magnetic Polarisation,
Ma is the magnetization of iron caused by an external magnetic field, without permanent magnets Bp = µ0 Mp = 0, without iron Ma = 0, too. Note: Ma is sometimes written as M and Bp is equivalent to Jp . . 
1.b) Maxwell's equations (extended for moved bodies) .
( * ) 1. EXAMPLE for evaluation with magnetic flux density Terms : Derivation of Faraday-Lorentz' Law using equation ( * ) : Assuming special conditions/restrictions (-> in literature often not mentioned) i.e. incompressible materials -> div v = 0, space independent constant movements -> (B grad) v = 0 and in magnetic fields directly from magnetic Gauss' law always -> div B = 0 the remaining term on the right side in equation ( * ) yields -> rot ( B x v ) = - rot ( v x B ) = - curl ( v x B ). Inserting this result in Faraday's Law eq. (2a) or (2*) we can simply derive the extended 2. Maxwell's equation for moved bodies: .
using equation ( ** ) with same condition mentioned above you get eq. (2b) with ∇ x A = Nabla x A = curl A = B The first term on the right side of this equation (2b) was proved by Faraday, the second one by Lorentz. NOTE: using this vector analytical formulation you get the Lorentz-Term E = v x B automatically ! The famous Lorentz law is therefore a (very important) vector IDENTITY ... but not really a separate physical LAW. . ( * ) 2. EXAMPLE for evaluation with electric flux density Terms : Derivation of Ampere-Maxwell's Law using equation ( * ) : Assuming special conditions/restrictions (-> in literature often not mentioned) i.e. incompressible materials -> div v = 0, space independent constant movements -> (D grad) v = 0 and in electric fields directly from electric Gauss' law -> div D = ρ the remaining term on the right side in equation ( * ) yields for the cross product -> rot ( D x v ) = - rot ( v x D ) = - curl ( v x D ) and in opposite to Faraday-Lorentz' Law in this Ampere-Maxwell's Law an additional term v ∙ div D = v ∙ ρ Inserting these results in Ampere-Maxwell's Law eq. (1a) or (1*) we can derive the extended 1. Maxwell's equation for moved bodies or particles: .
The first term on the right side of this equation (1b) was proved by Ampere, the third term by Rowland, the second term by Hertz (suggested and introduced by Maxwell), the fourth term by Roentgen. NOTE: using this vector analytical formulation you get the "dualism" of Lorentz-Term H = - v x D automatically ! The Rowland and Roentgen terms are therefore (important) vector IDENTITIES ... but not really separate physical LAWS. . ( * ) 3. EXAMPLE: Proof of extended 1. + 2. Maxwell's equations using famous HELMHOLTZ' formula Helmholtz derived following formula in his curl laws for any arbitrary vector flux X in physics (i.e. hydrodynamics) through a moved ( v ) and simultaneously deformable area element - as a subset of ( * ) - : .
Inserting this Helmholtz' formula ( *** ) in the Maxwell equations (1a) or (1*) and (2a) or (2*) - prerequisiting both the same above mentioned conditions/restrictions and X = B alternatively X = D - we immediately get the extended Maxwell's equations (1b) and (2b) in the 1. and 2. example ! . NOTE: using ( *** ) the extended Maxwell's equations are derivable without any knowledge in vector analysis. The Helmholtz' formula is ingenious and the basis for Lorentz and Minkowski, too. Helmholtz derived his formula visualizing - like a "mnemonics artist" - moved and deformable geometric elements. But nevertheless Helmholtz' d X / dt neglects the LAST term (X ∇) v inside (v ∇) X , refer to ( * ) and ( ** ) ! . .
. 2. Maxwell's equations considering quantum field theory 2.a) Considering relativistic quantum mechanics PROCA has developed following extended maxwell's equations in quantum field theory -> the so-called proca's equations .
The difference between maxwell's equations in classic field theory and quantum field theory is shown in red boxes. The additional "red box" terms consist of magnetic vectorpotential A, electric scalar potential PHI, material properties in vacuum both permeability mue & permittivity eps. The magnetic flux density B is equal to rot A = curl A . The special term Term k² = Kapa² = (m0 c / S)² is famous in quantum mechanics, because Kapa is the Compton frequency devided by velocity c of light ... or Einstein's energy in view of quantum mechanics. The mass in rest (no relativistic movements) is m0, the universal Planck's constant in quantum mechanics is S. Note: These PROCA equations (8) can be also derived from unified equation Re + i Im = 0 in section 4. 2.b) The well known relativistic Schrödinger equation, the so-called Klein-Gordon equation is:  2.c) Extending this homogeneous Klein-Gordon wave equation ( f = 0 ) , applying a probability function Y ("PSI") we get the non homogeneous Proca wave equations ( f not equal 0 ) .
. Introducing magnetic Vectorpotential A and scalar Potential PHI we can derive following wave equations : ..
Using B = curl A and the constitutive relations these equations (8a), (8b) or (8c) can be derived directly from equations (8) ... or can be also evaluated from unified equation Re + i Im = 0 in section 4 .
. 3. "Right Hand Rules as Memo Maps for all central derivations from Maxwell's equations" Central derivations from Maxwell's equations with respect to all important phenomena inside electrodynamics are developed and visualized as a new Mind Map with 10 memorable Memo Maps based on variations of famous Maxwell's "Right Hand Rule". . Starting from differential equations we can formulate all central equations governing electrodynamics and interdisciplinary physics. These Memo Maps are valuable mnemonics for necessary derivations. Using these 10 special pictures it's easy to bear all derivations in mind. . 
. Variations of Maxwell's Hand as a MIND MAP with MEMO MAPS (C) 2005 by Wolfram Stanek .
.
Additionally to Maxwell's equations (in rest and with moving bodies) also Lorentz-Einstein's relativistic energy relations, Klein-Gordon's equation -> (relativistic) Schrödinger's equation, Proca's extended Maxwell's equations in quantum mechanics, Bohm-Aharanov effects and Newton's impuls mechanics (using mass m - "function") are integrated in this one equation " Re + i Im = 0 " : . 
. SOME INTERPRETATIONS: . 1. Real part Re of this equation shows the DUALITY of classic waves (left side) and quantum particles (right side). 2. Using the light field gauge term 5) = 0 (PHIs jS = c ∙ A) and applying quantum flux (hbar / charge q) to the imaginary part Im the re-formulated term 6) yields the total electric field strength E' = - ∂ A / ∂ t - grad js (9a) with all influences (i.e. arbitrary translation, rotation, distortion movements) in classic electrodynamics: .
By this simple re-formulation of the imaginary part Im we've automatically changed the term 7) so that we also see the DUALITY of classic electric field strength E and quantum flux based alternative formulation in quantum electrodynamics. The 2. Maxwell equation = Faraday-Lorentz' law you get from (9) applying the operator "curl". A good excercise for you: Where are the other Maxwell equations "hidden" in Re + i Im = 0 ? . 3. Furthermore we have 6 possible gauges for electrodynamic field defined by div A = ... (eq. Ib) as combination i.e. the simplest gauges are Coulomb's gauge div A = constant and Lorentz' gauge div A = -[d(PHI)/dt] / c^2 The important 7th gauge (in literature often not mentioned) is the light field gauge term 5) = 0. This light field gauge is a basis for classic electrodynamics. But a further conclusion is, that PHOTONS (and GRAVITONS) must have a rest mass m0 = 0, propagating with the velocity of light in vacuum. . 4. From real part < Re > we see a surprising (but measured) phenomenon, that electrodynamics can influence the mass of rigid bodies! The speed of rigid bodies v is always less then speed of light c. From eq. (1a) we can conclude that the gravitational masses of atoms in a material can be changed, especially reduced or nullified or even inverted under special electrodynamic field conditions ! 5. Using no classic field terms, i.e. E, H, D, B or mixed terms like in Proca's equations the equation Re + i Im = 0 in quantum electrodynamics shows a general result: magnetic vector potential A and scalar potential PHIs are superior, classic field terms i.e. E, H, D, B are secondary because those are simply to derive from primarily A and PHIs equations. . 6. Term 6) = 0 in imaginary part < Im > in above shown equation Re + i Im = 0 implies not only the Lorentz' gauge but also - choosing light field gauge term 5) = 0 - the central basis for classical interdisciplinary physics. . Some examples for interdisciplinary evaluations of imaginary part < Im > . a) Directly from eq.(Ib) yields the special Lorentz gauge: div A = - { d (PHIs) / dt/ } / c^2 ___(11a)___ or from eq.(Ib) unified equations i.e. using naturics UNIT checks: div J = - d (RHO) / dt _______(11b)___ or directly from (1b) with J = v RHO (with electric charge density RHO r in electrodynamics) .
It's clear, that you can get eq. (11b) directly from Ampère-Maxwell's law applying operator "div": div curl H = 0 = div J + div [ d D /dt ] with electric Gauss' law div D = RHO: Because of J = (current) FLOW and RHO = (electrical charge) DENSITY = charge q / m^3 (analogous quick derivation as for the basic equation in mechanics in the following lines) eq. (11a) or (11b) can be written as an universal law: . div (FLOW DENSITY) = - d (specific DENSITY) / dt (11)_ _ => CONTINUITY law . FLOW DENSITY is based on electrodynamics, thermodynamics or hydrodynamics etc., too ... Specific DENSITY means the equivalent medium (charge, mass etc). Eq. (11) can be re-formulated i.e. with respect to mechanics .
where RHOm = Mass Density in mechanics as formal equivalence with charge density in electrodynamics. A separate quick development of new formulas can be related to naturics based UNIT checks, i.e. directly from eq. (Ib) for electrodynamics leading to eq. (11c) governing mechanics: RHO. A [As/m^3 Vs/m = Ws/m^3 s/m = Nm/m^3 s/m = kg/m^3 m/s] = RHOm. v __(11d) RHO. PHIs / c^2 [As/m^3 V s^2/m^2 = Nm/m^3 s^2/m^2 = kg m / s^2 m /m^3 s^2/m^2 = kg / m^3] = RHOm_(11e)_ Eq. (11d) and (11e) in eq. (1b) yields eq. (11c) . . b) additionally applying light field gauge: d (q ∙ A) / dt = - grad (q ∙ PHIs)___(12a) because of q ∙ A = Impuls and q ∙ PHIs = (potential) energy eq. (12a) can be written as an universal law: . d (IMPULS) / dt = - grad (ENERGY) ___(12)__ => FORCE law__ . eq. (12) or directly (11c) can be re-formulated i.e. with respect to mechanics: d (m ∙ v ) / dt = m ∙ dv / dt + v ∙ dm / dt = - grad ( Wpot )___(12b)__ where m = Mass, v = speed of body, Wpot = m ∙ g ∙ h (i.e. and/or other force-"sources") . NOTE: Regarding (11b), (11c) and (12) we can formally switch between Newton's and Maxwell's relations using Hamiltonian vector gradient formulation eq. (II), (III) shown above in unified equation Re + i Im = 0 ... always thinking in analogies. HINTS: Start from grad (A ∙ v) = (v Nabla) A + (A Nabla) v + v x curlA + A x curlv _____(12c) = ( ** ) or simplify with A = v as needed i.e. for NAVIER-STOKES equations in hydrodynamics you directly get : .
Using (12c), (12d) we can compare important features of Maxwell equations and Newton's law, too. ... it's a good excercise for you to test your capabilities in handling vector analytic operations! . NOTE: 1. Though the physicist Heinrich Hertz thought that Maxwell's equations are not derivable from Newton's equations, you can prove it ... at least in a formal analogy ... useful for multiphysics applications in engineering. 2. But never forget thinking in analogies: In real mechanics nothing is identical with electric charge in electrodynamics. . c) Integrating eq. (12) yields the well known universal energy law in general form: . W total = W kinetic + W potential = constant___(13)___ => ENERGY law . NOTE: Kinetic energy derived from relativistic Energy with Taylor approximation: W (kinetic) = W (total) - W (restmass) = m ∙ c^2 - m0 ∙ c^2 = 0.5 ∙ m0 ∙ v^2 + ... tiny terms (x) (x) can be neglected in non-relativistic applications . Further details: discussion about maxwell's equations combined with quantum mechanics
. These topics are based on lectures at other universities in
EU and Asia, i.e. Swiss German University (Java) in 2002 + 2003,
.
|
|||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
