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    ---
    title: Problem Set 9
    ---
    
    ## (9.6)
    
    {:.question}
    Solve the periodically forced Lorentz model for the dielectric constant as a function of frequency,
    and plot the real and imaginary parts.
    
    
    The periodically forced Lorentz model is
    
    $$
    m \left( \ddot{x}(t) + \gamma \dot{x}(t) + \omega_0^2 x(t) \right) = -e E(t)
    $$
    
    It models the motion of a particle of mass $$m$$ and charge $$-e$$ subjected to a time-varying
    electric field $$E(t)$$. Assuming a bulk material composed of such particles, we can use this model
    to find a relation between the dielectric constant and frequency of incoming radiation.
    
    To start, the [polarization density](https://en.wikipedia.org/wiki/Polarization_density) can be
    expressed in terms of the number of particles per unit volume, their charge, and their displacement:
    
    $$
    P = -N e x
    $$
    
    But it can also be expressed using the electric field and dielectric constant:
    
    $$
    P = \epsilon_0 E(\epsilon_r - 1)
    $$
    
    Thus the dielectric constant for this material is
    
    $$
    \epsilon_r = \frac{-N e x}{\epsilon_0 E} + 1
    $$
    
    So now let's solve the model. Let's assume a simple sinusoidal solution.
    
    $$
    \begin{align*}
    x(t) &= A e^{i \omega t} \\
    \dot{x}(t) &= i \omega A e^{i \omega t} \\
    \ddot{x}(t) &= - \omega^2 A e^{i \omega t}
    \end{align*}
    $$
    
    Then the Lorentz model reduces to
    
    $$
    m A e^{i \omega t} \left( - \omega^2  + i \omega \gamma + \omega_0^2 \right) = -e E(t)
    $$
    
    or
    
    $$
    \frac{x(t)}{E(t)} = \frac{-e}{m \left( \omega_0^2 - \omega^2 + i \omega \gamma \right)}
    $$
    
    So this solution is valid for a sinusoidally varying electric field.
    
    Finally we just plug this in to find
    
    $$
    \epsilon_r = \frac{N e^2}{\epsilon_0 m \left( \omega_0^2 - \omega^2 + i \omega \gamma \right)} + 1
    $$
    
    
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    ## (12.1)
    
    ### (a)
    
    {:.question}
    How many watts of power are contained in the light from a 1000 lumen video projector?
    
    ### (b)
    
    {:.question}
    What spatial resolution is needed for the printing of a page in a book to match the eye’s limit?
    
    
    ## (12.2)
    
    ### (a)
    
    {:.question}
    What is the peak wavelength for black-body radiation from a person? From the cosmic background
    radiation at 2.74 K?
    
    ### (b)
    
    {:.question}
    Approximately how hot is a material if it is “red-hot”?
    
    ### (c)
    
    {:.question}
    Estimate the total power thermally radiated by a person.
    
    
    ## (12.3)
    
    ### (a)
    
    {:.question}
    Find a thickness and an orientation for a birefringent material that rotates a linearly polarized
    wave by $$90^\circ$$. What is that thickness for calcite with visible light ($$\lambda \approx 600
    \si{nm}$$)?
    
    ### (b)
    
    {:.question}
    Find a thickness and an orientation that converts linearly polarized light to circularly polarized
    light, and evaluate the thickness for calcite.
    
    ### (c)
    
    {:.question}
    Consider two linear polarizers oriented along the same direction, and a birefringent material
    placed between them. What is the transmitted intensity as a function of the orientation of the
    birefringent material relative to the axis of the polarizers?