Background of coloured characters in Maxwell's equations: 
.

Coulour
MEMO
&
Dash (')
entities

- 1. Green characters are maGnetic field entities, Red characters are electRic field entities
- 2. rot H ≠ rot H  &  rot E ≠ rot E etc, H & E are transformation equations in the left side of Maxwell's equations
      H in rot H is the dashed field entity H' , E in rot E is the dashed field entity E' etc (attention: "rot" = "curl")
→ It is a fundamental difference between view points: ... moving with the object/field or only observing this movement
→ Transformation equations in general formulation are:   E = E' = E + v x B + ... (*) ,   H = H' = H - v x D + ... (*)  etc
→ Transformation equations in basic formulation: E = E' = E + v x B  &  H = H' = H - v x D  (= Lorentz' Transformation)
→ The additional field entities i.e. v x B & v x D caused by moved objects → ( ' ) are only 1 of 4 possible terms (*), (**)
→ Dashed entities → ( ' ) are related to the moved system (i.e. space ship, rotor of a motor etc), without ( ' ) to the system "in rest"
→ Using only basic Lorentz-Transformation : E' = E + v x B  &  H' = H - v x D -> transformed current is then J = J' = J - v r
→ As color information is lost in black-white-copies, it's better using dashed entities ( ' ) in transformation equations (1*) - (4*)

THE NECESSITY TO APPLY TRANSFORMATION EQUATIONS ON MAXWELL'S EQUATIONS CAN BE EASILY SEEN FROM
LORENTZ' EXTENSION OF FARADAY'S LAW CONSIDERING MOVED OBJECTS WITH A LINEAR, UNIFORM RELATIVE VELOCITY v:

Based on Faraday's Law = 2. Maxwell's equation: curl E = - ∂ B / ∂ t => Lorentz' (v x B)-extension yields: curl E' = - d B / dt = - ∂ B / ∂ t + curl (v x B)
It's definitely clear that E' is NOT equal to E ... IF the linear, uniform relative velocity v of the moved object is NOT equal to 0 ! In this case
the extremely simple Lorentz' transformation is => E' = E + v x B and the additional electric Lorentz' field, caused by velocity v is => E_V = v x B

Generally we shall state the relativity postulate as follows (i.e. ref. to STRATTON, JACKSON, EINSTEIN etc): When properly formulated,
the laws of physics are invariant to a transformation from one reference system to another moving with a linear, uniform relative velocity v:

Notation (A) of extended Maxwell's equations, using dashed characters H' etc

1. Ampere-Maxwell's Law

(1*)

2. Faraday-Lorentz's Law

(2*)

3. electric Gauss' Law      

(3*)

4. magnetic Gauss' Law   

(4*)

.

Simple examples for derivations and interpretations with respect to extended Maxwell's equations:

============================================================

Notation (B) of extended Maxwell's equations including material properties

extended Maxwell's equations (1*) - (4*),
If we are replacing the constitutive equations (5) - (7) by the entities D, B and J
and the fields in the left sides of eq. (1a) - (4a) by the "hidden" Lorentz' transformation equations

As color information is lost in black-white-copies, it's better using dashed entities (1*)-(4*)

1. extended maxwell's equation (fields)   
Ampere-Maxwell's Law

(1a)

2. extended maxwell's equation (fields)   
Faraday-Lorentz'-Law

(2a)

3. extended maxwell's equation (sources)
electric Gauss' Law

(3a)

4. extended maxwell's equation (sources)
magnetic Gauss' Law

(4a)

NOTE:
Symmetric structure
of Maxwell Equations

This shown re-formulation leads to a symmetric structure
of Maxwell's equations (1a), (2a), (3a) and (4a).
Similar to Dirac's suggestion constructing a set
of symmetric Maxwell equations we automatically get
both the electric charge density (r - ∇∙P)
and the magnetic charge density (-∇∙M)

( * ) 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 ( ** ) !

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