Problem Set 9

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(9.6)

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 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

$m A e^{i \omega t} \left( - \omega^2 + i \omega \gamma + \omega_0^2 \right) = -e E(t)$

$\frac{x(t)}{E(t)} = \frac{-e}{m \left( \omega_0^2 - \omega^2 + i \omega \gamma \right)}$

So this solution is valid for a sinusoidally varying electric field.

Finally we just plug this in to find

$\epsilon_r = \frac{N e^2}{\epsilon_0 m \left( \omega_0^2 - \omega^2 + i \omega \gamma \right)} + 1$

(12.1)

(a)

How many watts of power are contained in the light from a 1000 lumen video projector?

(b)

What spatial resolution is needed for the printing of a page in a book to match the eye’s limit?

(12.2)

(a)

What is the peak wavelength for black-body radiation from a person? From the cosmic background radiation at 2.74 K?

(b)

Approximately how hot is a material if it is “red-hot”?

(c)

Estimate the total power thermally radiated by a person.

(12.3)

(a)

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)

Find a thickness and an orientation that converts linearly polarized light to circularly polarized light, and evaluate the thickness for calcite.

(c)

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?