Can someone help me in understanding where I am wrong when thinking about the derivation of the Lorentz field in a dielectric. I give the derivation in italics (although the familiar reader should not need to read it) and after that I present the paradox.
The basic idea is to consider a spherical zone containing the dipole under study, immersed in the dielectric.
The sphere is small in comparison with the dimension of the condenser, but large compared with the molecular dimensions.
We treat the properties of the sphere at the microscopic level as containing many molecules, but the material outside of the sphere is considered a continuum.
The field acting at the center of the sphere where the dipole is placed arises from the field due to
(1) the charges on the condenser plates
(2) the polarization charges on the spherical surface, and
(3) the molecular dipoles in the spherical region.
The field due to the polarization charges on the spherical surface, ## E_{sp} ##, can be calculated by considering an element of the spherical surface defined by the angles ## \theta ## and ## \theta + d \theta ##.
The area of this elementary surface is: ## 2 \pi r^2 \sin \theta d \theta ##.
The density of charge on this element is given by ## P \cos \theta ##, and the angles between this polarization and the elementary surface is ## \theta ##. Integrating over all values of angle formed by the direction of the field with the normal vector to spherical surface at each point and dividing by the surface of the sphere we obtain
[tex]E_{sp}=\frac{1}{r^2}\int_0^{\pi} 2 \pi r^2 P \sin \theta \cos^2 \theta d \theta = \frac{4 \pi P}{3}[/tex]
Now suppose the dielectric is a sphere of radius R and that the smaller sphere of radius r is in the middle of this bigger sphere. Since the Surface charges are reversed compared to the smaller sphere one can then evaluate the field in a similar way from the bigger sphere as
[tex]E_{SP}=-\frac{1}{R^2}\int_0^{\pi} 2 \pi R^2 P \sin \theta \cos^2 \theta d \theta = -\frac{4 \pi P}{3}[/tex]
The field acting at the centre of the sphere then is
[tex]E_{tot}=E_{sp}+E_{SP}+E_{external}=\frac{4 \pi P}{3}-\frac{4 \pi P}{3}+E_{external}=E_{external}[/tex]
This is, by the way also, in line with that the displacement field is constant everywhere but that there is no polarization inside the region inside the smaller sphere so that the field there should be the external field.
Now did Lorentz totally miss this?
(The Picture did not appear good against a White background.)
The basic idea is to consider a spherical zone containing the dipole under study, immersed in the dielectric.
The sphere is small in comparison with the dimension of the condenser, but large compared with the molecular dimensions.
We treat the properties of the sphere at the microscopic level as containing many molecules, but the material outside of the sphere is considered a continuum.
The field acting at the center of the sphere where the dipole is placed arises from the field due to
(1) the charges on the condenser plates
(2) the polarization charges on the spherical surface, and
(3) the molecular dipoles in the spherical region.
The field due to the polarization charges on the spherical surface, ## E_{sp} ##, can be calculated by considering an element of the spherical surface defined by the angles ## \theta ## and ## \theta + d \theta ##.
The area of this elementary surface is: ## 2 \pi r^2 \sin \theta d \theta ##.
The density of charge on this element is given by ## P \cos \theta ##, and the angles between this polarization and the elementary surface is ## \theta ##. Integrating over all values of angle formed by the direction of the field with the normal vector to spherical surface at each point and dividing by the surface of the sphere we obtain
[tex]E_{sp}=\frac{1}{r^2}\int_0^{\pi} 2 \pi r^2 P \sin \theta \cos^2 \theta d \theta = \frac{4 \pi P}{3}[/tex]
Now suppose the dielectric is a sphere of radius R and that the smaller sphere of radius r is in the middle of this bigger sphere. Since the Surface charges are reversed compared to the smaller sphere one can then evaluate the field in a similar way from the bigger sphere as
[tex]E_{SP}=-\frac{1}{R^2}\int_0^{\pi} 2 \pi R^2 P \sin \theta \cos^2 \theta d \theta = -\frac{4 \pi P}{3}[/tex]
The field acting at the centre of the sphere then is
[tex]E_{tot}=E_{sp}+E_{SP}+E_{external}=\frac{4 \pi P}{3}-\frac{4 \pi P}{3}+E_{external}=E_{external}[/tex]
This is, by the way also, in line with that the displacement field is constant everywhere but that there is no polarization inside the region inside the smaller sphere so that the field there should be the external field.
Now did Lorentz totally miss this?
(The Picture did not appear good against a White background.)