On the Electric Displacement Vector D=E+4*pi*P

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I have seen a couple of posts on this website of physics students who were puzzled by what the electric displacement vector D in electromagnetic theory represents. This quantity also had me rather puzzled when I was a student and I think I can provide some insight into its significance. The equations div E=4*pi*(rho total)=4*pi*(rho free +rho p) and div P=- rho p are both on rather solid footing. A little algebra gives div (E+4*pi*P)=4*pi*(rho free). Thereby if we define D=E+4*pi*P, then div D=4*pi*(rho free). There is however no measurement device of any kind that can distinguish between the electric field created by free electrical charges from the electric field created by polarization charges. Thereby, after much careful thought, I have come to the conclusion that the electric displacement vector "D" is a very useful mathematical construction, but it is not a physically measurable quantity.

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DrDu

The point is that you can define P without distinguishing between free and polarisation charges. The definition of P and M is not unique. In solid state physics (and specially in the optical region) one uses a convention where M=0. Then P is defined as $P=\int_{-\infty}^t j(t')dt'$, where j(t) is the current density due to all charges in of the material, whether bound or free. Hence $\nabla\cdot D=4 \pi \rho_\mathrm{ext}$, where $\rho_\mathrm{ext}$ is the density of external charges e.g. charges on a condenser into which you brought your material.

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Thank you DrDu. Your response helped to clarify the matter. A related item that I put a lot of effort into is the "pole method of magnetostatics" and after some detailed calculations, I was able to show/prove that the pole description along with B=H+4*pi*M actually originates from the magnetic surface currents and Biot-Savart. I did a write-up of my calculations and one conclusion that came out of the calculations is that the H from the poles in the material is simply a subtractive correction term to B of the 4*pi*M for non-infinite cylinder geometries. The 4*pi*M actually comes from surface currents (which are ignored in the pole formalism, and the 4*pi*M in the pole formalism is introduced as a local contribution of M to B. The calculations of the pole method for the magnetic field B are in complete agreement with those computed from surface currents, but can often be misinterpreted. The equation B=H+4*pi*M is first proved in the absence of currents in conductors. The H (and B) from the currents in conductors is included as an add-on to both sides of the equation. Perhaps you might find my paper of interest. It has been difficult to find physics people with any complete expertise in the magnetostatics subject. https://www.overleaf.com/read/kdhnbkpypxfk I welcome your feedback.

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The point is that you can define P without distinguishing between free and polarisation charges. The definition of P and M is not unique. In solid state physics (and specially in the optical region) one uses a convention where M=0. Then P is defined as $P=\int_{-\infty}^t j(t')dt'$, where j(t) is the current density due to all charges in of the material, whether bound or free. Hence $\nabla\cdot D=4 \pi \rho_\mathrm{ext}$, where $\rho_\mathrm{ext}$ is the density of external charges e.g. charges on a condenser into which you brought your material.
DrDu Thank you. Your response helped clarify the matter. Please see my response above.

DrDu

My understanding is the following: Polarisation and magnetisation are a consequence of the conservation of the charges inside the medium, $\nabla\cdot j=\partial \rho/\partial t$. This can be written as a 4-dimensional divergence of the charge density current vector. Now by some general theorems from calculus, a vector whose divergence vanishes can always be written as the "differential" of another quantity. In the language of differential forms dX=0 can be fulfilled with X=dY as ddY=0 holds for all differential forms. Y is not unique, as X remains unchanged if we add dZ to Y.

In our case this other quantity is the tensor made up of polarisation and magnetisation. The splitting into magnetisation and polarisation is not unique, as a current j can be represented as rot M or $\partial P/\partial t$. However, in magnetostatics, j is constant, and if one wants P and M to be constant in time, too, then $\partial P/\partial t$ vanishes and j can only depend on rot M. This seems to be what you are looking for: the magnetisation is determined by the currents inside the material.
If fields vary rapidly in time, the splitting into magnetisation and polarisation becomes arbitrary and one usually chooses M=0, i.e. j=$\partial P/\partial t$.

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My understanding is the following: Polarisation and magnetisation are a consequence of the conservation of the charges inside the medium, $\nabla\cdot j=\partial \rho/\partial t$. This can be written as a 4-dimensional divergence of the charge density current vector. Now by some general theorems from calculus, a vector whose divergence vanishes can always be written as the "differential" of another quantity. In the language of differential forms dX=0 can be fulfilled with X=dY as ddY=0 holds for all differential forms. Y is not unique, as X remains unchanged if we add dZ to Y.

In our case this other quantity is the tensor made up of polarisation and magnetisation. The splitting into magnetisation and polarisation is not unique, as a current j can be represented as rot M or $\partial P/\partial t$. However, in magnetostatics, j is constant, and if one wants P and M to be constant in time, too, then $\partial P/\partial t$ vanishes and j can only depend on rot M. This seems to be what you are looking for: the magnetisation is determined by the currents inside the material.
If fields vary rapidly in time, the splitting into magnetisation and polarisation becomes arbitrary and one usually chooses M=0, i.e. j=$\partial P/\partial t$.
div J="-" d (rho)/dt (with a minus sign). The magnetostatics was actually my major focus, but the pole method of magnetostatics is mathematically analogous to the pole method of electrostatics, and a better understanding of what the electric displacement vector "D" represents was one of the results of studying these methods in detail.

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