Question

(Pr. 4) If a soft material has a dielectric response characterized by a single relaxation time $\tau$, then the polar obeys the equation: $\frac{dP}{dt} = -\frac{1}{\tau}[P(t) - \chi_s \epsilon_0 E(t)]$, where $\chi_s$ is the static electric susceptibility. (a) Under an applied sinusoidal field given, in complex notation, by $E(t) = E_0 e^{-i\omega t}$, verify by substitution that $P = \tilde{\chi} \epsilon_0 E_0 e^{-i\omega t}$ satisfies the equation $\frac{dP(t)}{dt} = -\frac{1}{\tau}[P(t) - \chi_s \epsilon_0 E(t)]$. Show that the c susceptibility $\tilde{\chi}$ obeys the Debye law $\tilde{\chi} = \frac{\chi_s}{1 - i\omega \tau}$, hence calculate the real and imaginary parts of the c susceptibility, and sketch what they look like. (c) Apply this model to the case of water at room temperature, for which $\chi_s \approx 80$ and $\tau = 9.4$ ps. Comp frequency in Hz at which the imaginary part of the susceptibility is maximum. What is the real par susceptibility of water for quasistatic fields (low frequency)? And for high frequency? When is wate polarizable by the electric field, and why? Is water a Newtonian liquid at low frequency? And at high freq

          (Pr. 4) If a soft material has a dielectric response characterized by a single relaxation time $\tau$, then the polar
obeys the equation: $\frac{dP}{dt} = -\frac{1}{\tau}[P(t) - \chi_s \epsilon_0 E(t)]$, where $\chi_s$ is the static electric susceptibility.
(a) Under an applied sinusoidal field given, in complex notation, by $E(t) = E_0 e^{-i\omega t}$, verify by
substitution that $P = \tilde{\chi} \epsilon_0 E_0 e^{-i\omega t}$ satisfies the equation $\frac{dP(t)}{dt} = -\frac{1}{\tau}[P(t) - \chi_s \epsilon_0 E(t)]$. Show that the c
susceptibility $\tilde{\chi}$ obeys the Debye law $\tilde{\chi} = \frac{\chi_s}{1 - i\omega \tau}$, hence calculate the real and imaginary parts of the c
susceptibility, and sketch what they look like.
(c) Apply this model to the case of water at room temperature, for which $\chi_s \approx 80$ and $\tau = 9.4$ ps. Comp
frequency in Hz at which the imaginary part of the susceptibility is maximum. What is the real par
susceptibility of water for quasistatic fields (low frequency)? And for high frequency? When is wate
polarizable by the electric field, and why? Is water a Newtonian liquid at low frequency? And at high freq

(Pr. 4) If a soft material has a dielectric response characterized by a single relaxation time τ, then the polar
obeys the equation: (dP)/(dt) = -(1)/(τ)[P(t) - ϵ0 E(t)], where is the static electric susceptibility.
(a) Under an applied sinusoidal field given, in complex notation, by E(t) = E0 e^-iω t, verify by
substitution that P = χ̃ϵ0 E0 e^-iω t satisfies the equation (dP(t))/(dt) = -(1)/(τ)[P(t) - ϵ0 E(t)]. Show that the c
susceptibility χ̃ obeys the Debye law χ̃ = ()/(1 - iωτ), hence calculate the real and imaginary parts of the c
susceptibility, and sketch what they look like.
(c) Apply this model to the case of water at room temperature, for which ≈ 80 and τ = 9.4 ps. Comp
frequency in Hz at which the imaginary part of the susceptibility is maximum. What is the real par
susceptibility of water for quasistatic fields (low frequency)? And for high frequency? When is wate
polarizable by the electric field, and why? Is water a Newtonian liquid at low frequency? And at high freq

Added by Brenda A.

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