MAXWELL’S EQUATIONS
Maxwell’s four equations as studied in this course can be written as
1. Gauss’s law for E:
G G Q
1
= ∫∫∫ ρ dV
w
∫∫S E ⋅ dA = enclosed
εo
εo V
2. Gauss’s law for B:
G G
w
∫∫ B ⋅ dA = 0
S
3. Generalized Ampere’s law
 G G
G G
d  G G 
B
⋅
ds
=
µ
I
+
I
=
µ
J
⋅
dA
+
ε

(
)
 E ⋅ dA 
o
conduction
displacement
o
o
vC∫
 ∫∫
dt  ∫∫
S

S
4. Faraday’s law
G G
d  G G
ε = −  ∫∫ B ⋅ dA = v∫ E ⋅ ds .
dt  S
 C
As these stand they are very powerful equations that summarize all of the fundamental work we
have done this term. But as to their usefulness as integral equations, we have seen that they are
limited to situations where we have sufficient symmetry to allow us to solve for E and B. Here
we will see how to re-write these as differential equations and some very general consequences
of these equations.
First, we can re-write Gauss’s law for E using the divergence theorem:
G G
G
1
E
⋅
dA
=
∇
⋅
EdV
= ∫∫∫ ρ dV .
∫∫S
∫∫∫
w
εo V
V
Then since the volume (bounded by S) is totally arbitrary, we can equate the integrands in the
last two volume integrals to find the differential form of Gauss’s law for E:
G
ρ ( x, y , z )
∇ ⋅ E ( x, y , z ) =
εo
In a similar way we find the differential form of Gauss’s law for B to be:
G
∇ ⋅ B ( x, y , z ) = 0 .
Next, we examine the generalized Ampere’s law and use Stoke’s theorem to re-write it:
G
G G
 G G
G G
 G G
 ∂E G  
d  G G 
vC∫ B ⋅ ds = µo  ∫∫S J ⋅ dA + ε o dt  ∫∫S E ⋅ dA  = ∫∫S ∇ × B ⋅ dA = µo  ∫∫S J ⋅ dA + ε o  ∫∫S ∂t ⋅ dA  .




Here we have used Stoke’s theorem and also interchanged the time and spatial operations in the
last integral. Since the surface (bounded by C) is totally arbitrary, we can equate the integrands
of the surface integrals in the last equality above to find
G
G
G
∂E
∇ × B = µ o J + µ 0ε o
∂t
Lastly, we re-write Faraday’s law, interchanging the order of time differentiation and spatial
integration and using Stoke’s theorem again, to find:
G
G G
G G
d  G G
∂B G
−  ∫∫ B ⋅ dA = − ∫∫
⋅ dA = v∫ E ⋅ ds = ∫∫ ∇ × E ⋅ dA
dt  S
∂t
S
C
S

and on equating the last two surface integrals we have the differential form of Faraday’s law:
G
G
∂B
∇× E = −
∂t
These four equations can now be manipulated and solved for any situation using the mathematics
of differential equations and the power of modern computers.