As the dierential equations we have been dealing with
are linear and without time-dependent parameters,
Fourier transform from time to frequency results in
equations for which frequency is simply a parameter, and
the question of dispersion, tha
Waves can be guided not only by conductors, but by
dielectrics. Fiber optics cable of silica has n(r) varying
Simplest: core radius a with
n = n1 , surrounded (radius
b) with n = n0 < n1 .
Total internal reection if
Cartesian coordinates ri , i = 1, 2, . . . D for Euclidean
Distance by Pythagoras: (s)2 =
(ri )2 .
Unit vectors ei , displacement r = i ri ei
Fields are functions of position, or of r or of cfw_ri .
Scalar elds (r),
Lecture 13 March 7, 2011
Last time we discussed a small scatterer at origin.
Interesting eects come from many small scatterers
occupying a region of size d large compared to . The
scatterer j at position xj has an Einc with an extra factor
of eik i xj ,
Lecture 12 March 3, 2011
We will rst nish up the = 1 term from the Green
function. This is giving us magnetic dipole and
electric quadripole contributions.
We will briey describe a more consistent
Resonant cavities do not need to be cylindrical, of course.
The surface of the Earth (RE 6400 km) and the
ionosphere (R = RE + h, h 100 km) form concentric
spheres which are suciently good conductors to form a
To review, in our original presentation of Maxwells
equations, all and Jall represented all charges, both free
and bound. Upon separating them, free from
bound, we have (dropping quadripole terms):
For the electric eld
E called electric eld
P called elect
Physics 504, Spring 2011
Electricity and Magnetism
Joel A. Shapiro
January 20, 2011
5-5500 X 3886, shapiro@physics
Wave guide traveling modes in z direction
E, B eikzit
with dispersion relation k 2 = 2 .
Same form as for high-frequencies in dielectrics (Jackson
7.61), with plasma frequency.
How much power is dissipated (per unit area?). 2 ways:
1) Flow of energy into conductor: Energy ow given by
S = E H, for real elds E and H.
so1 S = 1 Re E H , and dPloss /dA = S , so
We begin with waves in a non-conducting uniform linear
medium, so we are discussing solutions of Maxwells
equations without sources. As we are assuming no
time-dependence of the properties of the medium, we will
fourier transform in
We have seen that the issue of how , and n depend on
raises questions about causality: Can signals travel
faster than c, or even backwards in time?
It is very often useful to assume that polarization is linear
and local in space, and the polari
Sources of Electromagnetic Fields
Lecture 11 February 28, 2011
We now start to discuss radiation in free space.
We will reorder the material of Chapter 9, bringing