1
st
lecture
The Maxwell equations
There are four basic equations, called Maxwell equations, which form the axioms of
electrodynamics. The so called
local
forms of these equations are the following:
rot H
=
j
+
∂
D
/
∂
t
(1)
rot E
= 
∂
B
/
∂
t
(2)
div
B
= 0
(3)
div
D
=
ρ
(4)
Here
rot
(or
curl
in English literature) is the so called vortex density,
H
is vector of
the magnetic field strength,
j
is the current density vector,
∂
D
/
∂
t is the time derivative of the
electric displacement vector
D
,
E
is the electric field strength,
∂
B
/
∂
t is the time derivative of
the magnetic induction vector
B
, div is the so called source density and
ρ
is the charge
density.
While the above local or differential forms are easy to remember and useful in
applications, they are not so easy to understand as they use vector calculus to give spatial
derivatives of vector fields like
rot H
or div
D
. The
global
or integral forms of the Maxwell
equations are somewhat more complicated but at the same time they can be understood
without knowing vector calculus. They are using path, surface, and volume integrals,
however:
∫
G
H
•
dr
= I + I
DISP
(1)
∫
G
E
•
dr
= 
∂Φ
B
/
∂
t
(2)
∫
A
B
•
dA
= 0
(3)
∫
A
D
•
dA
=
∫
V
ρ
dV
(4)
where
I is the electric current
I =
∫
A
j
•
dA
,
I
DISP
is the so called displacement
I
DISP
=
∫
A
(
∂
D
/
∂
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 Spring '08
 Tannebaum
 Magnetic Field

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