Page 160 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Basic Equations of Electrodynamics
chy, we form the multipole expansion coefficients. These are now trans-
lated through the hierarchy so that, at the very worst, it is possible to
perform only of the order of n log( n) vs. evaluations for a system of n
n
charges. Remarkably, further efforts have shown that the number of eval-
uations can be reduced to the order of the number of charges in the
n
system.
Magneto- In analogy with the electrostatic case, we now consider magneto-quasi-
statics static situations where, in addition, ∂B ∂t⁄ = 0 , so that again magnetic
and electrostatic phenomena are completely de-coupled, but stationary
magnetic phenomena dominate. We consider the equation ∇× H = . J
From the equation for the vector potential B = ∇× A , and the constitu-
⁄
tive equation H = B µ , we obtain

∇×  1 A = J (4.38)
---∇×
 µ 
The vector identity ∇× ( ∇× A) = ∇ ( ∇• A) ∇ 2 A , together with
–
∇• A = 0 , transforms (4.38) to

– ∇ 2 A = µJ (4.39)

which now represents a separate Poisson equation for each component of
the vector potential, i.e.

– ∇ 2 A x J x
– ∇ 2 A y = µ J y (4.40)

– ∇ 2 A J
z z
This is very convenient, for we may now use the same methods to solve
(4.39) as for the electrostatic equation (4.35).

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