THE UNIVERSITY OF BRITISH COLUMBIA

*Physics 401 *
Assignment #
**7:**

** WAVES IN MEDIA **

Wed. 22 Feb. 2006 - finish by Wed. 1 Mar.

Wed. 22 Feb. 2006 - finish by Wed. 1 Mar.

- (p. 395, Problem
**9.18**) -**Practical Questions:**^{1}- Suppose you embedded some free charge in a piece of glass. About how long would it take for the charge to flow to the surface?
- Silver is an excellent conductor, but it's expensive.
Suppose you were designing a microwave experiment to operate
at a frequency of 10
^{10}Hz. How thick would you make the silver coatings? - Find the wavelength and propagation speed in copper for radio waves at 1 MHz. Compare the corresponding values in air (or vacuum).

- (p. 396, Problem
**9.19**) -**Skin Depth:**- Show that the skin depth in a poor conductor
(
) is
(independent of frequency).
Find the skin depth (in meters) for (pure) water.
^{2} - Show that the skin depth in a good conductor
(
) is
(where is the wavelength
*in the conductor*. Find the skin depth (in nanometers) for a typical metal [ (m)^{-1}] in the visible range ( s^{-1}), assuming and . Why are metals opaque? - Show that in a good conductor the magnetic field lags the electric field by 45, and find the ratio of their amplitudes. For a numerical example, use the "typical metal" in the previous question.

- Show that the skin depth in a poor conductor
(
) is
(independent of frequency).
Find the skin depth (in meters) for (pure) water.
- (p. 398, Problem
**9.21**) -**Silver Mirror:**Calculate the reflection coefficient for light at an air-to-silver interface [ (m)^{-1}], at optical frequencies ( s^{-1}).

- (p. 413, Problem
**9.37**) -**TIR:***n*_{1}>*n*_{2}) the propagation vector bends*away*from the normal (see Figure). In particular, if the light is incident at the**critical angle**, then , and the transmitted ray just grazes the surface. If*exceeds*, there is no refracted ray at all, only a reflected one. This is the phenomenon of**total internal reflection**,^{3}on which light pipes and fiber optics are based. But the*fields*are not zero in medium 2; what we get is a so-called**evanescent wave**, which is rapidly attenuated and transports no energy into medium 2.^{4}

A quick way to construct the evanescent wave is simply to quote the results of Sect. 9.3.3, with and ; the only change is that is now greater than 1, and so is imaginary. (Obviously, can no longer be interpreted as an*angle*!)- Show that
,
where
and
.
This describes a wave propagating in the
*x*direction (*parallel*to the interface!) and attenuated in the*z*direction. - Noting that
is now imaginary,
use Eqs. (9.109),

to calculate the reflection coefficient for polarization parallel to the plane of incidence. [Notice that you get 100% reflection, which is better than at a conducting surface (see for example Problem**9.21**).] - Do the same for polarization perpendicular to the plane of incidence
(use the results of Problem 9.16):

- In the case of polarization perpendicular to the plane of incidence, show that the (real) evanescent fields are , .
- Check that the fields in the last part satisfy all of MAXWELL'S EQUATIONS (9.67).
- For those same fields, construct the Poynting vector
and show that, on average, no energy is transmitted
in the
*z*direction.

- Show that
,
where
and
.
This describes a wave propagating in the

2006-02-19