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## A brief review of field theory concepts

• Gauge Fields, Knots and Gravity'' by John Baez  & Javier P. Muniain , World Scientific Publishing Co. Pty Ltd, 1994, ISBN 9810220340
• Only the first 158 pages: in fact much less, we'll review the most basic stuff only.
• Material covered by the lecture should be sufficient.
• An example of field equations:
 (3.1) (3.2) (3.3) (3.4)

• The operators are vector   differentiation operators, which, in Cartesian coordinates only, look as follows

 (3.5)

• t is time
• E is the electric field
• B is the magnetic field
• is the charge density
• j is the current density
• c and are constants, which depend on the choice of a system of units.

• It is easy to see that c is the   speed of light. Take of equation (3.4), and assume vacuum, i.e., and j = :

hence

 (3.6)

Here is the Laplace   operator, which, in Cartesian coordinates only, is given by:

 (3.7)

• Observe that is multiplied by c2 in the   wave equation. This is necessary, so that both differential operators would divide B by the same thing, i.e., .
• Emacs Calc has   numerous functions for operations on units, which
• simplify units
• convert units
• remove units
• autorange units (i.e., prefixes such as k'' or M'' are automatically added so that the numbers don't get ridiculously large or small)
• let you define your own units.
• example:
'150 km
'3 hr
/
1:  50 km / hr
uc
New units: mi/hr
1:  31.0685596119 mi / hr
uc
New units: m/s
13.8888888889 m / s
'172 cm
uc
New units: ft+in
1:  5 ft + 7.71653543307 in
'69 kg
uc
New units: lb
1:  152.118960908 lb

• If you don't know how to input a particular unit, view calc-units.el in
/afs/ovpit.indiana.edu/common/gnu/share/emacs/site-lisp/calc-2.02f

• Emacs Calc also knows about fundamental physics constants, e.g., the Planck   constant, Gravitational   constant, Avogadro   constant and many others:
'1 Grav
uc
New units: N
1:  6.67259e-17 N m^2 / g^2

• Similarly   we can derive the wave equation for E by taking the time derivative of the Ampère-Maxwell Law (equation (3.3))

hence

 (3.8)

• In order to get these wave equations for B and E we have assumed and j = too. In presence of charges and currents things get complicated.
• waves in conductors  propagate only above the   plasma frequency, i.e., longer waves reflect, shorter waves pass through; cf. colour of gold, reflection of AM waves from the ionosphere;
• waves in semiconductors   propagate only if photon energy is less than the bandgap (this is a quantum effect that cannot be explained using Maxwell equations);
• waves in   dielectrics (glass, water) propagate but with a different phase velocity, and usually with some dispersion, i.e., different wavelengths have different phase velocities.
• nonlinear effects   can sometimes act in the opposite direction to dispersion and form   the so called solitons. These are of increasing importance in physics of optical fibres and laser beams.

Next: Electromagnetic Potentials Up: Fields Previous: Fields
Zdzislaw Meglicki
2001-02-26