Paradox in evaluating the Lorentz field in a dielectric

[Order]

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

$$E_{sp}=\frac{1}{r^2}\int_0^{\pi} 2 \pi r^2 P \sin \theta \cos^2 \theta d \theta = \frac{4 \pi P}{3}$$

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
$$E_{SP}=-\frac{1}{R^2}\int_0^{\pi} 2 \pi R^2 P \sin \theta \cos^2 \theta d \theta = -\frac{4 \pi P}{3}$$
The field acting at the centre of the sphere then is
$$E_{tot}=E_{sp}+E_{SP}+E_{external}=\frac{4 \pi P}{3}-\frac{4 \pi P}{3}+E_{external}=E_{external}$$
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.)

 

[Meir Achuz]

The molecules inside do not have surface charge producing an E field.
The field due to the inside molecules vanishes for a regular distribution.

 

[Order]

The molecules inside do not have surface charge producing an E field.
The field due to the inside molecules vanishes for a regular distribution.

Vielen dank for giving my problem attention Meir Achurz. I thought it would engage more phycisists when I am saying something is wrong. I Think it should be easy to correct a misstake. But I don't know if you have really understood my problem. (If you cannot imagine Surface charges inside the dielectric then how do you appreciate the local field in a non-polar ditto?)

What is the problem then? The problem is to understand the subtleties in working out the Lorentz field and eventually the Clausius-Mossotti formula. Let me clarify.

Suppose it is a correct assumptions to imagine a spherical cavity in the dielectric. (Obviously Lorentz was kinda right.) And let me also assume that the macroscopic Surface charges on the Surface of the dielectric itself does not contribute to the local field. (This is obviously so, although I can't understand it, and this is what my question is all about: how can you ignore this?!) Then the total local field E_i will be
$$E_i = E_{sp} + E_{0}$$ where E₀ is the original field applied to the dielectric. Now using the definition of polarization (in Gaussian units I Think)
$$P=\frac{1}{4 \pi}(\varepsilon - 1) E_{0}$$
and using the expression for E_sp given in my original post this leads to the Lorentz field$$E_i = E_0 + \frac{\varepsilon - 1}{3}E_0=\frac{\varepsilon + 2}{3}E_0$$
And with a Little more work you can derive the Claussius-Mossotti formula. This is obviously correct, so in what way am I wrong?