Page 154 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Basic Equations of Electrodynamics
These two equations represent four separate scalar wave equations of
identical form, for which one can develop a (see the discussion in [4.1])
time-dependent fundamental solution or Green function. The Green func-
tion is nothing else than the analytical solution of the wave equation to a
unit (read Dirac delta function) right-hand side. We define the Kronecker
delta function positioned at s′ as

 1 s = s′
(
δ s′ – s) =  (4.20)
 0 otherwise
so that it looks like a unit pulse function at s′ . For the wave equation, the
Green function

 − [ r – r′] 
δ t′ – t ----------------- 
+

± () c
,
,,
G ( r t r′ t′) = ----------------------------------------------- (4.21)
r – r′
is called retarded/advanced because it describes the effect of a unit load
t′
at time and position r′ on another point located at , at a later/earlier
r
time . Equations (4.19a) and (4.19b) are linear, and therefore permit
t
superposition of solutions. By superposition of elemental right-hand
sides, the Green functions can be used to build up a complete response to
a right-hand side build–up of a spatially and temporally distributed
charge and/or current density.

4.1.2 Quasi-Static and Static Electric and Magnetic Fields
Many effects of interest to us are dominated either by the electric field or
by the magnetic field, and do not require us to consider the complete cou-
pling of both. We follow the discussion in [4.5], which is highly recom-
mended for additional reading. We start by normalizing the
electromagnetic quantities using unit-carrying scale factors and unit-less
scalar or vector variables (indicated by under-bars). We choose the scal-
ing of length, time and the electric field:

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