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It is well known that complex polarizability of uniform dielectric sphere with radius $r$ and complex permittivity $\hat\epsilon_{in}(\omega)$ placed in a medium with complex permittivity $\hat\epsilon_{out}(\omega)$ under homogeneous electric field with circular frequency $\omega$ is defined by (in the CGS system of units):

$$\hat\alpha(\omega)=r^3 {\hat\epsilon_{in}(\omega)-\hat\epsilon_{out}(\omega)\over \hat\epsilon_{in}(\omega)+2\hat\epsilon_{out}(\omega)}$$

This relation is derived for the static case in many textbooks on electrostatics (see here related demonstration). What about the dynamic case?

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What do you mean by "static case" ? The quantities dependent on $\omega$ describe response of the medium to oscillating harmonic field. –  Ján Lalinský Jan 14 at 13:57
    
@Ján The static case is the case of static field, in this case the equation of course should be written without $\omega$. This case is considered in many textbooks and places, for example here. –  Alexey Popkov Jan 14 at 14:40
    
Mie scattering deals with high-frequency fields in the vicinity of dielectric sphere: en.wikipedia.org/wiki/Mie_scattering (just in case you did not came across it yet). –  Ján Lalinský Jan 14 at 17:20
    
@Ján I consider the case when no scattering occurs. –  Alexey Popkov Jan 14 at 18:10

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