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In Irodov's Basic Laws of Electromagnetism, D is derived as D=ε°E+P. Later substituting the polarization vector for an isotropic dielectric as P=Χε°E, the result reached is D=ε°εD where ε is dielectric constant of the dielectric.

However later in an example he says that the second result is not valid (in that particular example). enter image description here See Example 3. Does the invalidity of the second result mean the sphere is not made of a isotropic material? If yes, how do we infer that? Ifnot,then what is the reason behind the invalidity?

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He is probably thinking of ferroelectric body, which manifests electric polarization even without external field. Such material media manifest non-linear relation between electric field and polarization, and hysteresis. So there is no general useful relation between $\mathbf D$ and $\mathbf E$. It is like with ferromagnets - there, there is no general useful relation between $\mathbf B$ and $\mathbf H$.

Ferroelectrics are usually crystalline media, so they are indeed not isotropic. Whether this is necessary for ferroelectricity, I am not sure. Certainly when the body is polarized, it is no longer isotropic.

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  • $\begingroup$ I now understand that the polarization inside has absolutely no relation with the external E field but depends only on how the permanent charge inside is stored. Just wanted to clarify, is there any reason why the E field and D field are shown to coincide (although they are opposite)? $\endgroup$ Aug 7, 2018 at 19:30
  • $\begingroup$ Yes, the E field is generated by the sphere, so in the simplest case where the spherehas uniform P, the charge distribution on sphere is such that the resulting E field is uniform inside and in the opposite direction as the P field(this component is present even in ordinary dielectric sphere, where it is called depolarizing field). The uniformity and antiparallel direction to P is a special property of uniformly polarized sphere. By definition of D, it is parallel or antiparallel to E. $\endgroup$ Aug 7, 2018 at 20:01
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The second result is invalidated because for an electret the polarization is no longer directly dependent on an external electric field. An electret's polarization is "frozen" in when an electret is formed. To remove the apparent confusion the result is invalid for this particular example because it doesn't apply here. As for your first question on how to infer whether the material is isotropic or not would be by observing the nature of the susceptibility term. If that reduces to a scalar then it is isotropic. If not and it is a tensor with varying components then it is anisotropic. I hope this answers your question.

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