Page 153 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

The Electromagnetic System
2
∇ 2 A µε ∂ A – ∇  ∇• A + µε ∂Φ  = – µJ (4.15a)
–
t ∂ 2  t ∂ 
∂ ∇• ∇ 2 ρ
ε
t ∂ A + Φ = – --- (4.15b)
⁄
We now consider the case for a vacuum; recall that µ ε = 1 c 2 .
0 0
Because of the relation B = ∇× A B
, is unchanged through the addition
of the gradient of an arbitrary scalar function Λ , because
∇× ( A + ∇ Λ) = ∇× A . However, through equation (4.14), which under
A
the same substitution for results in
∂A  ∂Λ 
E = – – ∇ + Φ (4.16)
t ∂ t ∂  

we see that the scalar potential must be transformed according to
⁄
Φ ⇒ Φ ∂Λ ∂t to achieve invariance of the electric field. Now choos-
–
Λ
ing such that
2
∂Λ
∇ 2 Λ + µε = 0 (4.17)
t ∂ 2

implies that

1 ∂Φ
∇• A + ---- = 0 (4.18)
c 2 t ∂

which completely de-couples the remaining two vacuum Maxwell equa-
tions to give

2
1 ∂Φ ρ
∇ 2 Φ – ---- = – --- (4.19a)
2
c ∂ t 2 ε
2
1 ∂ A
∇ 2 A – ---- = – µJ (4.19b)
2
c ∂ t 2

150 Semiconductors for Micro and Nanosystem Technology