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(Pr. 4) If a soft material has a dielectric response characterized by a single relaxation time t, then the polarization - [P(t)- XoE (t)], where X is the static electric susceptibility. obeys the equation: dP dt T (a) Under an applied sinusoidal field given, in complex notation, by E(t) = Ege-iw verify by direct substitution that P = XEE e-i satisfies the equation dP(t) = − ² [P(t) = x₂ €₁E (t)]. Show that the complex - dt hence calculate the real and imaginary parts of the complex Xs 1-iw susceptibility obeys the Debye law x = 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?

(Pr. 4) If a soft material has a dielectric response characterized by a single relaxation time t, then the polarization - [P(t)- XoE (t)], where X is the static electric susceptibility. obeys the equation: dP dt T (a) Under an applied sinusoidal field given, in complex notation, by E(t) = Ege-iw verify by direct substitution that P = XEE e-i satisfies the equation dP(t) = − ² [P(t) = x₂ €₁E (t)]. Show that the complex - dt hence calculate the real and imaginary parts of the complex Xs 1-iw susceptibility obeys the Debye law x = 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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(Pr. 4) If a soft material has a dielectric response characterized by a single relaxation time t, then the polarization - [P(t)- XoE (t)], where X is the static electric susceptibility. obeys the equation: dP dt T (a) Under an applied sinusoidal field given, in complex notation, by E(t) = Ege-iw verify by direct substitution that P = XEE e-i satisfies the equation dP(t) = − ² [P(t) = x₂ €₁E (t)]. Show that the complex - dt hence calculate the real and imaginary parts of the complex Xs 1-iw susceptibility obeys the Debye law x = 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?
Transcribed Image Text:(Pr. 4) If a soft material has a dielectric response characterized by a single relaxation time t, then the polarization - [P(t)- XoE (t)], where X is the static electric susceptibility. obeys the equation: dP dt T (a) Under an applied sinusoidal field given, in complex notation, by E(t) = Ege-iw verify by direct substitution that P = XEE e-i satisfies the equation dP(t) = − ² [P(t) = x₂ €₁E (t)]. Show that the complex - dt hence calculate the real and imaginary parts of the complex Xs 1-iw susceptibility obeys the Debye law x = 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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