Homework 5

4/29/02

Physics 110B

Griffiths 9.22, 9.34 but to simplify the algebra, take $n_1 = n_3 = 1$.

1. Consider a parallel plate capacitor with a distance between the plates of $L$. The capacitor is filled with a dielectric material that has a dielectric constant as a function of angular frequency:

$$\epsilon ( \omega ) ~=~ \epsilon_o + {K \over { 1 - i \omega } } .$$

A voltage difference $V(t)$ is applied to the plates as a function of time.

(a) If $V(t) = Re ( e^{- i \omega t } )$, calculate the polarization ${\bf P}(t)$.

(b) If a voltage pulse $V(t) = \delta (t)$ is applied, calculate the polarization.

In this problem, ignore the magnetic field (i.e. set ${\bf B} = 0$). Hint: revisit 114B Homework 9 problem problem 3.

2. Consider a plasma with a dispersion relation

$$k^2 ~=~ {1 \over {c^2}} ( \omega^2 - {\omega_p}^2)$$

Calculate the time averaged Poynting vector of a linear plane wave for $\omega ~ < ~ \omega_p$. Assume $\omega_p$ is given. Take $\mu ~=~ \mu_o$.

3. A FM (frequency modulated) wave travels in a medium with dispersion so that $\omega (k)$ is not a linear function. To be concrete, take the initial signal to have the form

$$f(t) ~=~ A \cos((\omega_0 + \Delta\omega \cos(\Omega t))t).$$

The interpretation of this expression is that the wave has a carrier frequency of $\omega_0$ which varies over a range of frequencies of $\Delta\omega$ which is assumed to be very small. The rate at which frequency is modulated, $\Omega$ is also very small. Estimate the order of magnitude of the distance that this signal propagates beyond which the degradation becomes significant. Assume $\Omega << \Delta\omega << \omega_0$.

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