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Suppose a small solid sphere made of an isotropic dielectric is introduced in the field of some given system of remote charges. Assuming that the polarization in the dielectric does not affect the distribution of the remote charges, would the total dipole moment induced on the sphere depend also upon its volume, or it would only depend upon the Electric field strength (due to the remote charges) at the point where the sphere is introduced?

$\boldsymbol{D} = \boldsymbol{E} + 4\pi \boldsymbol{P}$ and $\boldsymbol{D}=\epsilon \boldsymbol{E}$ would together imply that $$\boldsymbol{P}= \frac{\epsilon -1}{4\pi}\boldsymbol{E}.$$Now it would follow that the total dipole moment should depend upon volume, as $\boldsymbol{P}$ is the dipole moment per unit volume. However, a certain problems-book (Sidney B Cahn, Guide to Physics problems) states that the total dipole moment would depend only on the field strength (Problem No 3.2). I would like to know how is that conclusion possible?

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  • $\begingroup$ @vnd Do you mean problem 3.2 in this edition? Otherwise, can you point to a specific instance of the volume-independence claim, or reproduce it here in its entirety? $\endgroup$ – Emilio Pisanty Nov 18 '15 at 21:31
  • $\begingroup$ You should also cite the source of the problem you quote. $\endgroup$ – Emilio Pisanty Nov 18 '15 at 21:35
  • $\begingroup$ @Emilio Yes, the problem 3.2 in the link you posted is the one I was referring to. In the solution to that problem (Page 201), the authors state, "The charged sphere will polarize the neutral one, which acquires a dipole moment proportional to the electric field created by the charged sphere". According to the argument I posted in my question, the dipole moment acquired should also depend upon the volume of the sphere, because of which the answer would change from what is given in the book. I wanted to know how is my argument wrong? $\endgroup$ – vnd Nov 18 '15 at 22:30
  • $\begingroup$ Don't discount the possibility that Cahn's claim is incorrect. Let me think about it and I'll try to write something up if I have time. $\endgroup$ – Emilio Pisanty Nov 19 '15 at 0:01
  • $\begingroup$ Going by the well-known result that the dipole moment induced in a dielectric ellipsoid placed in a uniform electric field is indeed dependent upon the volume of the ellipsoid (Eq:8.10, Landau and Lifshitz, Electrodynamics of Continuous Media, 2nd Ed), it seems that Cahn's claim would be incorrect. Additionally, it is doubtful that the identical sizes of the two spheres would have a role here as they are far separated. $\endgroup$ – vnd Nov 19 '15 at 0:27

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