The introduction of electric polarization and magnetization—the density of electric and magnetic dipole moments respectively—into Maxwell's equations requires establishing their respective relation to polarization charges and magnetization currents. Using a method introduced by Feynman in his famous lectures on physics and considering statistically distributed dipoles on the microscopic scale, the desired relations can be established in a manner that may be more intuitive to undergraduate students.

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J. D.
Jackson
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Wiley
,
New York
,
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D. T.
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Dynamic electromagnetic model for material media
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M. N. O.
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Oxford, New York
,
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),
Section 5.
5
.
5.
The question remains how the requirement of uniformity is to be tested for a particularly chosen volume ΔV. One method would be to consider the polarization for a neighboring volume element of the same size, which should yield virtually the same value for the polarization.
6.

See Ref. 1, Section 4.3.

7.
E. M.
Purcell
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Mcgraw-Hill
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8.
R. P.
Feynman
,
M.
Sands
, and
R. B.
Leighton
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Pearson-Addison Wesley
,
San Francisco, Boston, New York
,
2006
), Vol.
2
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10.
We note that Hofmann has pointed this problem out and analyzed it rigorously in
H.
Hofmann
,
Das elektromagnetische Feld
, 3rd ed. (
Springer
,
Vienna
,
1999
), Section 1.2.
11.
We note that Feynman considers the movements of charges across the surface as an external field applied, which is equivalent to considering already present dipoles being cut by a virtual surface.
12.
Note that in previous considerations, it is also possible to avoid the application of the divergence theorem if the surface integral is taken over an infinitesimal cube with volume dV. Expanding P in a Taylor series and ignoring terms of 2nd order and higher (which is permissible due to the infinitesimal character of the volume), the total charge in the cube ρPdVcan be readily obtained as PdV.
13.

See Ref. 7, Chapter 11.

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