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luckreez
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Homework Statement
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Let us assume that neutral atoms or molecules can be modeled as harmonic oscillators in some cases. Then, the equation of the displacement between nucleus and electron cloud can be written as
$$\mu\left(\frac{d^x}{dt^2}+\gamma\frac{dx}{dt}+\omega_0^2x\right)=qE.$$
where ##x## is the displacement in the direction of the external electric field ##E##, ##\mu## is the effective mass, ##\gamma## is the damping constant, ##\omega_0## is the oscillation frequency of the harmonic potential, and ##q## is the effective charge of the neutral atom or molecule. Assuming that E is a single frequency electric field with angular frequency ##ω##, answer the following questions:
1. Obtain the amplitude of the stationary solution for the equation of the displacement.
2. Assuming that the material contains such atoms or molecules with density ##n_0## , obtain the current density.
3. Obtain the dielectric constant ##\epsilon(\omega)##, which will be in complex form.
4. Obtain the real and imaginary parts of the dielectric constant. Assume that ##\gamma << \omega## and calculate approximate solutions.
5. Introducing the following constants, ##\epsilon_{st} = \epsilon(\omega=0)## and ##\epsilon_\infty=\epsilon(\omega=\infty)##, draw the curves for the real and imaginary parts of the dielectric constant as functions of the frequency ##\omega##.
Homework Equations
1. Tricks of finding solution to 2nd order ODE
2. Definition of current density?
3. 4. 5. Totally no idea.
The Attempt at a Solution
1. So I assume that stationary solution is the steady-state solution, which represent the particular solution of the differential equations. As the driving function is constant (not a function of the displacement) then the I choose the particular solution to be constant,
$$ x_P=\frac{qE}{\omega_0^2\mu}.$$
2. 3. 4. 5.
So what I do not understand this question is how to relate the damped oscillation model of nucleus and electron clouds to the electromagnetic variables in matter, like current density, or the dielectric constant.
I thought I could relate the current density with the driving force through drift velocity in this formula,
$$ j=nqv$$
but I cannot figure it out.
I am sorry if it seems I still did not do much, but I have spent a whole day trying to read, start from the definition of current density, complex dielectric constant in internet or EM books, but still not get the slightest idea on how to solve it.
I would be very grateful for any hints given.
Thank you in advance.
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