Page 186 - Maxwell House

166 Chapter 4

Evidently, the set of equations (4.1) consists of two first-order differential equations relating
two vectors , . Bearing in mind that, in general, each of this vectors can have three coordinate
components the real number of scalar differential equations is up to six. Certainly, it is too many
for any engineering analysis not involving a computer. So we start by transferring (4.1) into the
set of second-order differential equations while each of them includes a single field vector only.
First of all, we apply the ( x) operation to both parts of the first equation in (4.1)

× × = − × � � = − ( × ) (4.3)
0
0

The fact that the time and spatial coordinates are independent variables helped us to change the
order of the differentiation on the right-hand side of (4.3). Using the triple vector product
identity (see (A.16) in Appendix) we obtain
× × = ∙ ( ∘ ) − ∇ (4.4)
2
Again returning to Maxwell’s equations (4.1) and the last equation from (4.2), we have now

Σ 2 + � (4.5)
∙ � � − ∇ = − �
0 Σ

or
2 1
1
2
∇ − − = ∙ + (4.6)
2
2 0 Σ 0
Here = / √ [m/s] is the speed of light in a media with the relative dielectric and

magnetic constant . The last equation in (4.1) and (4.2) highlights the fact that the electric

charges and created by them currents are dependable quantities
Σ Σ Σ
0 = ∘ Σ + = ∘ + ∘ + = ∘ + ∘ + = ∘ +

Σ
Σ + (4.7)

If so, the distribution of total charge (, ) can be found as the solution of the simple first-
Σ
order ordinary linear differential equation
Σ (,)
+ (, ) = − ∘ (, ) (4.8)
Σ
The solution of (4.8) is given by the integral

− − (−)
(, ) = (, 0) − ∫ ∘ (, ) (4.9)
Σ
Σ
0
Here (, 0) = ∘ (, 0) is the initial charge or divergence of electric field distribution
Σ
at the moment of time t = 0 and should be given. Typically, we can put (, 0) = 0. In other
Σ
words, the sum of two terms on the right-hand side of (4.6) can be a quite complicated vector
function

1
(, ) = ∙ Σ + (4.10)
0