We assume both are correct.
div D = ρ applies to emw propagation in infinite boundless space.
And Div D = ρ + α dV/dt applies to emw scattering under Diriclet boundary conditions on the voltage V in bounded space.
But the question arises, can it be proven theoretically or experimentally?
Belirtilen koşulun deneysel ve teorik olarak gerçekleşebilmesi için sınır şartlarının elektromanyetik dalga koşulları varlığının gerçek ve herhangi bir zamana, mekana, varlığa ihtiyacı vardır.
As a general rule, any physical equation must contain time.
Classical time-independent PPDEs and LPDEs where the electromagnetic energy transfer is assumed to be instantaneous are incomplete.
They only take place because the speed of transfer of their em energy is close to that of light.
Moreover, it is quite surprising that the solution of time-dependent PPDEs and LPDEs is more accessible than that of time-independent solutions.
In a related question/answer, it was assumed that:
A time-dependent Poisson PDE expressed by:
dU/dt)partial=D Nabla^2 U+S . . .(1)
And also assumed
Laplace time dependent PDE as:
dU/dt)partial=D Nabla^2 U . . . (2)
(In normal conventions.)
Subject to Dirichlet boundary conditions or any other suitable BC can replace classical PDE.
Note that Eq 1 or Eq 2 implies that:
Div D = ρ + α. dV/dt . . . (3)
It is clear that equation (3) applies to emw scattering under Dirichlet boundary conditions on the voltage V in bounded space.
As for the proof of , Equation 3 can be proven theoretically via an appropriate statistical technique and experimentally via thermal cooling curves or via sound reverberation curves in audio rooms.
The above statement is valid because of the analogy between PPDE and LPDE with Diriclet boundary conditions and heat diffusion equation and sound diffusion equation in audio rooms with the same Diriclet boundary conditions.
In general spacetime formulation we have dF=0 and dH=J; in the usual Heaviside notation this second equation has its "charge" part written
div.D = rho
However, we might (also) choose to split D into a vacuum part (essentially epsilon_0 E) and a macroscopic-material part P, and/or we might divide the charge rho into free and bound parts, or some other scheme. That is, there are a variety of ways you can rewrite div.D=rho, and your second equation is one of them.
The main issue will be the limitations on the validity of the model generating the (alpha dV/dt) term, and whether or not they affect the particular situation you consider.
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