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Maxwell's Equations: basic formulation
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Maxwell's equations in rest - no moving objects

DIFFERENTIAL form of Maxwell's equations (1) - (4) -> NO moved objects:
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NOTE: in English publications the operator "rot" is equivalent to "curl"
		x
James C. MAXWELL 1831 - 1879 (father of famous equations in "quaternion" formulation)	NOTE: We can quickly derive and memorise the first Maxwell's equations only using logic + unit checks ( )**	(*) Oliver HEAVISIDE 1850 - 1925 (streamlined re-design of Maxwell's equations in "vector algebra" form)

INTEGRAL form of Maxwell's equations (1) - (4) -> NO moved objects:

Eq. (1) and (2) are FIELD equations__
Eq. (3) and (4) are SOURCE equations

... with the scalar entities
electric and magnetic flux,
all electric currents and charges

Note: Using mathematical laws of Stokes and Gauss we can derive
Maxwell's equations in integral form directly from the differential Maxwell's equations.

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).
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Entities in these 4 Maxwell's equations in differential and integral formulation:
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The combination of all forms of electric currents I [A]
(including permanent magnets, convection currents etc)
and Maxwell's time changing electric flux ψ_el [As]
represents a huge variety of different possible excitations producing magnetic fields
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The further entities are: 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 volume charge density ρ [As / m³],
electric flux ψ_el [As] = electric charge Q [As], magnetic flux ψ_mag[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 & B_P and external current sources J_e ) are defined by :

D = ε E + P

(5)

B = μ H + B_p

(6)

J = γ E +J_e

(7)

P [As / m²] is electric polarisation, B_p= µ₀ M_p[Vs/m²] is magnetic Polarisation,
where M_p is the magnetization of the used permanent magnets,
J_e[A / m²] are external current sources
Material properties are permittivity ε [As / Vm], permeability μ [Vs / Am] and electrical conductivity γ [A / Vm].

Optional Formulation of eq. (6):

B = μ H + B_P= µ₀ H +µ₀ M_a + µ₀ M_P= µ₀ (H +M_a) + µ₀ M_P

(6a)

M_a is the magnetization of iron caused by an external magnetic field,
without permanent magnets B_p= µ₀ M_p = 0, without iron M_a = 0, too.
Note: M_a is sometimes written as M and B_pis equivalent to J_e

( ** ) NOTE: We can easily derive the Maxwell's equations (1) + (2) + (3) only using logic and unit checks ...but a "little" problem for eq. (4):

(1a) What will produce a curl of a magnetic field H? -> unit of curl is 1/m & of H is A/m -> yields unit A/m² = current density J -> curl H = J
(1b) Which physical entity can also produce a magnetic field H? -> unit of J is A/m² = 1/s As/m² = ∂ / ∂t (Displacement D)    -> curl H = J + ∂D / ∂t
1. Maxwell equation = Eq. (1a) is Ampere's Law, in Eq.(1b) the additional term "displacement current" by J. Maxwell -> Ampere-Maxwell's Law
(2) What will produce a curl of an electric field E? -> unit of E is V/m -> yields unit V/m² = 1/s Vs/m² -> with actio = reactio      -> curl E = - ∂B / ∂t
A curl E can be produced by other physical phenomena, too: i.e. curl (v x B) -> unit-check 1/m m/s Vs/m² = V/m² -> o.k. -> ref. website with v > 0.
2. Maxwell equation = Equation (2) is Faraday's Law, if extended with (v x B) influence -> Faraday-Lorentz' Law
(3) What will produce a div of a displacement D? -> unit of div is 1/m & of D is As/m² -> yields As/m³ = electric charge/m³        -> div D = ρ
3. Maxwell equation = Equation (3) is the electric Gauss' Law
(4a) What will produce a divergence div of a magnetic field B or H? -> Gauss and until now no one has measured a div of B -> div B = 0
A similar unit check would provide a "magnetic charge density" in Vs/m³ but it was never proved ... a not finished discussion in our world.
4. Maxwell equation = Equation (4) is the magnetic Gauss' Law
(4b) But considering permanent magnets with magnetization M_p and constitutive relation (6a) without iron we get a div of H    -> div H = - div M_p

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