Page 152 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Basic Equations of Electrodynamics
which is a wave equation for E
type with periodic solutions of the following
•
E = E g ωt –( kr) (4.11)
0
where the second derivatives of g are assumed to exist. Notice that
(4.11) requires to be periodic but does not specify the exact form. The
g
trigonometric sin, cosine and the exponential functions fit the specifica-
tions exactly.

4.1.1 Time-Dependent Potentials
Up to now the formulation for electromagnetics has been in terms of the
electric and magnetic field variables. We can go one step further by intro-
ducing potentials for the field variables. If a vector field is divergence
free, i.e., when ∇• B = 0 holds, then there exists a vector such that
A
B = ∇× A (4.12)

A is the vector potential. Insert this definition for into the Faraday law
B
(4.1a):

∂A  ∂A 
∇× E = – ∇× ⇔ ∇× E + = 0 (4.13)
t ∂  t ∂ 

Whenever the curl of a quantity is zero (we say that it is irrotational or
rotation free), it can be written as the gradient of a scalar potential, say
Φ , which provides us with a definition for the electric field entirely in
terms of the vector and scalar potentials

∂A  ∂A 
E + = – ∇ Φ ⇔ E = – + ∇ Φ (4.14)
t ∂ t ∂  
The two Maxwell equations that we started with are automatically satis-
fied by the potentials, and the remaining two now become (after a num-
ber of manipulation steps)

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