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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.
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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?