If a soft material has a dielectric response characterized by a single relaxation time t, then the polarization obeys the equation:=- [P(t) - XoE (t)], where xs is the static electric susceptibility. (a) Under an applied sinusoidal field given, in complex notation, by E(t) = E。e-iw verify by direct substitution that P = XoEoe-i satisfies the equation P(t) = - [P(t) − x¸€oE (t)]. Show that the complex dt T hence calculate the real and imaginary parts of the complex susceptibility obeys the Debye law x = = Xs 1-iw 2 susceptibility, and sketch what they look like. = (c) Apply this model to the case of water at room temperature, for which Xs 80 and t 9.4 ps. Compute the frequency in Hz at which the imaginary part of the susceptibility is maximum. What is the real part of the susceptibility of water for quasistatic fields (low frequency)? And for high frequency? When is water more polarizable by the electric field, and why? Is water a Newtonian liquid at low frequency? And at high frequency? At what time scales is viscoelasticity observed?
If a soft material has a dielectric response characterized by a single relaxation time t, then the polarization obeys the equation:=- [P(t) - XoE (t)], where xs is the static electric susceptibility. (a) Under an applied sinusoidal field given, in complex notation, by E(t) = E。e-iw verify by direct substitution that P = XoEoe-i satisfies the equation P(t) = - [P(t) − x¸€oE (t)]. Show that the complex dt T hence calculate the real and imaginary parts of the complex susceptibility obeys the Debye law x = = Xs 1-iw 2 susceptibility, and sketch what they look like. = (c) Apply this model to the case of water at room temperature, for which Xs 80 and t 9.4 ps. Compute the frequency in Hz at which the imaginary part of the susceptibility is maximum. What is the real part of the susceptibility of water for quasistatic fields (low frequency)? And for high frequency? When is water more polarizable by the electric field, and why? Is water a Newtonian liquid at low frequency? And at high frequency? At what time scales is viscoelasticity observed?
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Your Question:
![If a soft material has a dielectric response characterized by a single relaxation time t, then the polarization
obeys the equation:=- [P(t) - XoE (t)], where xs is the static electric susceptibility.
(a) Under an applied sinusoidal field given, in complex notation, by E(t) = E。e-iw verify by direct
substitution that P = XoEoe-i satisfies the equation P(t) = - [P(t) − x¸€oE (t)]. Show that the complex
dt
T
hence calculate the real and imaginary parts of the complex
susceptibility obeys the Debye law x =
=
Xs
1-iw
2
susceptibility, and sketch what they look like.
=
(c) Apply this model to the case of water at room temperature, for which Xs 80 and t 9.4 ps. Compute the
frequency in Hz at which the imaginary part of the susceptibility is maximum. What is the real part of the
susceptibility of water for quasistatic fields (low frequency)? And for high frequency? When is water more
polarizable by the electric field, and why? Is water a Newtonian liquid at low frequency? And at high frequency?
At what time scales is viscoelasticity observed?](https://content.bartleby.com/qna-images/question/529d3be1-7d0c-4d3a-9611-4db985a703ef/a39f358a-d709-47ce-a3b5-e4dade2bcd60/h3vizwv_processed.jpeg)
Transcribed Image Text:If a soft material has a dielectric response characterized by a single relaxation time t, then the polarization
obeys the equation:=- [P(t) - XoE (t)], where xs is the static electric susceptibility.
(a) Under an applied sinusoidal field given, in complex notation, by E(t) = E。e-iw verify by direct
substitution that P = XoEoe-i satisfies the equation P(t) = - [P(t) − x¸€oE (t)]. Show that the complex
dt
T
hence calculate the real and imaginary parts of the complex
susceptibility obeys the Debye law x =
=
Xs
1-iw
2
susceptibility, and sketch what they look like.
=
(c) Apply this model to the case of water at room temperature, for which Xs 80 and t 9.4 ps. Compute the
frequency in Hz at which the imaginary part of the susceptibility is maximum. What is the real part of the
susceptibility of water for quasistatic fields (low frequency)? And for high frequency? When is water more
polarizable by the electric field, and why? Is water a Newtonian liquid at low frequency? And at high frequency?
At what time scales is viscoelasticity observed?
