Chapter 1
Maxwell’s Equations, Conservation Laws
1.1
Maxwell’s Equations in Materials: D, H, P and M
Our starting point will be the experimentally deduced Maxwell’s Equations These consist of the four differential
equations (in gaussian, cgs units)
u ±
G
+
u
>w
,
@
7
± ²
+
u
> w
,
(1a)
u ±
E
+
u
> w
,
@
3
(1b)
u ²
H
+
u
> w
,
@
³
4
f
C
E
+
u
> w
,
Cw
(1c)
u ²
K
+
u
> w
,
@
4
f
C
G
+
u
> w
,
Cw
.
7
±
f
M
+
u
> w
,
(1d)
In S.I units the equations are (see Physics 411 notes):
u ±
G
+
u
>w
,
@
²
+
u
> w
,
(2a-2d)
u ±
E
+
u
> w
,
@
3
u ²
H
+
u
> w
,
@
³
C
E
+
u
> w
,
Cw
u ²
K
+
u
> w
,
@
C
G
+
u
> w
,
Cw
.
M
+
u
> w
,
for the
displacement
¿
eld
D
+
u
> w
,
>
the
magnetic induction
B
+
u
> w
,
>
the
electric
¿
eld
E
+
u
> w
,
>
and the
magnetic
¿
eld
H
+
u
> w
,
=
The sources of the
¿
elds are the
charge density
²
+
u
> w
,
and the
charge current density
J
+
u
>
w
,
=
Conservation of charge
requires that the charge density and charge current density satisfy
u
±
M
+
u
> w
, .
C
Cw
²
+
u
> w
, @ 3
=
(3)
This set of equations require information concerning the properties and responses of the materials in the region of the
¿
elds.
These are isolated in the constituent equations
G
+
u
> w
,
@
H
+
u
> w
, . 7
±
S
+
u
> w
,
gaussian units
(4a)
K
+
u
> w
,
@
E
+
u
> w
,
³
7
±
P
+
u
> w
,
gaussian units
(4b)
G
+
u
> w
,
@
³
H
+
u
> w
, .
S
+
u
> w
,
S.I units
(5a)
K
+
u
> w
,
@
4
´
E
+
u
> w
,
³
P
+
u
> w
,
S. I. units
(5b)
where
P
+
u
> w
,
is the
polarization
¿
eld and
M
+
u
> w
,
is the
magnetization
¿
eld for the materials.
u
±
S
+
u
>
w
, @
³
²
+
u
> w
,
the
1