Relation between Polarization and electric field for instantaneous response

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SUMMARY

The discussion focuses on the relationship between polarization (P) and electric field (E) in materials that respond instantaneously to external fields, expressed as P(r,t) = χE(t). It establishes that susceptibility (χ) can be time-independent, allowing for the derivation of frequency domain equations through Fourier or Laplace transforms. The Kramers-Kronig relation is highlighted as a fundamental principle, emphasizing the causal nature of susceptibility. Historical context is provided by referencing Noll's general restrictions on susceptibility from the 1960s and relevant literature, including the Encyclopedia of Physics.

PREREQUISITES
  • Understanding of polarization and electric fields in materials
  • Familiarity with susceptibility and its implications in electromagnetism
  • Knowledge of Fourier and Laplace transforms
  • Awareness of the Kramers-Kronig relations in physics
NEXT STEPS
  • Study the Kramers-Kronig relations in detail
  • Learn about the implications of Noll's restrictions on susceptibility
  • Explore the Fourier transform techniques in the context of electromagnetism
  • Review the Encyclopedia of Physics, Vol III, focusing on non-linear field theories
USEFUL FOR

Physicists, electrical engineers, and materials scientists interested in the instantaneous response of materials to electric fields and the mathematical foundations of polarization and susceptibility.

Algarion
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Dear all

In case of a material that instantaneous responds to an external applied field is

P(r,t)=χE(t)

Is the suseptibility then time independent? And if so, how it is possible to derive from the above equation an equation in the frequency domain?

Yours
 
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The most simple relation (the Kramers-Kronig relation) is found by only demanding the susceptibility be causal- P(t) can only depend on E(t'), where t' < t (extend the results to a lightcone for spatially-varying E).

The general restrictions on the susceptibility (or any constitutive relation) were laid down by Noll in the '60s, and are fairly general. AFAIK, there is no requirement that the susceptibility *may not* be time-dependent. This is covered in several books, including the Encyclopedia of Physics (Vol III, Non-linear field theories).

For the frequency domain, the usual procedure is to Fourier transform (or Laplace transform) the equation.
 

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