Chapter 6
Maxwell Equations
6.1
Maxwell equations
So far we have derived two electrostatic equations
ρ
"0
∇×E = 0
(6.1)
∇·E =
(6.2)
and two magnetostatics equations
∇·B = 0
∇ × B = µ0 J
(6.3)
(6.4)
which are to be modified due to Faraday’s observation,
∇·E =
ρ
"0
∂B
= 0
∂t
∇·B = 0
∇ × B = µ0 J.
∇×E+
(6.5)
(6.6)
(6.7)
(6.8)
However, this is not a self consistent set of equation for time varying configurations. Clearly,
∇ · (∇ × B) = µ0 ∇ · J = 0
(6.9)
and the continuity equation (5.4) allow only steady states configurations of
charges
∂ρ
=0
(6.10)
∂t
75
76
CHAPTER 6. MAXWELL EQUATIONS
which cannot be true in general.
The problem was fixed by Maxwell who found a small modification to
(6.8) which makes the new set of (Maxwell) equations self-consistent. By
starting from the continuity equation (5.4) we get
∂ρ
+∇·J =
∂t
∂ ("0 ∇ · E)
=
−
∂t
1
∂ ("0 ∇ · E)
∇ · (∇ × B) −
=
µ0
∂t
1 ∂E
∇×B− 2
= µ0
c ∂t
0
(6.11)
∇·J
(6.12)
∇·J
(6.13)
J+∇×C
(6.14)
where c = (µ0 "0 )−1/2 is the speed of light and C is an arbitrary timeindependent integration constant. If we set ∇ × C = 0, then we obtain
a self-consistent set of microscopic Maxwell equations
ρ
(6.15)
∇·E =
"0
∂B
∇×E+
= 0
(6.16)
∂t
∇·B = 0
(6.17)
1 ∂E
= µ0 J.
(6.18)
∇×B− 2
c ∂t
In a media the macroscopic Maxwell equations are rewritten as
∇·D = ρ
(6.19)
∂B
∇×E+
= 0
(6.20)
∂t
∇·B = 0
(6.21)
∂D
∇×H−
= J.
(6.22)
∂t
where the (partial) fields H and D have the same definition as before
D P
−
"0
"0
B = µ0 H + µ0 M.
E =
(6.23)
(6.24)
Since the divergence equations (6.19) and (6.21) are not unchanged the
boundary conditions are still give by (4.33) and (5.101),
n̂ · (D2 − D1 ) = σ
n̂ · (B2 − B1 ) = 0.
(6.25)
(6.26)
77
CHAPTER 6. MAXWELL EQUATIONS
In contrast the curl equations (6.20) and (6.22) are modified, but it turns out
that the corresponding boundary (4.38) and (5.96) conditions are unchanged,
(6.27)
(6.28)
n̂ × (E2 − E1 ) = 0
n̂ × (H2 − H1 ) = K.
This is due to the fact the integrals of finite
area at the boundary vanish.
6.2
∂
D
∂t
and
∂
B
∂t
over infinitesimal
Gauge transformations
The divergence equation (6.17) suggests that we can still introduce a vector
potential,
B≡∇×A
(6.29)
and from (6.16) the Faraday’s law is given by
"
!
∂A
=0
∇× E+
∂t
(6.30)
which can be rewritten by defining a scalar potential
E+
∂A
≡ −∇Φ
∂t
E = −∇Φ −
(6.31)
∂A
.
∂t
(6.32)
While (6.16) and (6.17) where used to define the (vector and scalar) potentials
the remaining two equations (6.15) and (6.18) can be used to determine the
dynamics of this potentials, i.e.
∂
ρ
∇2 Φ +
(∇ · A) = −
∂t
"0
!
"
1 ∂
1 ∂2
2
= −µ0 J.
∇ A − 2 2A − ∇ ∇ · A + 2 Φ
c ∂t
c ∂t
(6.33)
(6.34)
Note that the observable quantities E and B defined by (6.29) and (6.32)
are unchanged if we replace (or gauge transform)
Anew = Aold + ∇Λ
∂
Φnew = Φold − Λ.
∂t
(6.35)
(6.36)
CHAPTER 6. MAXWELL EQUATIONS
78
Then it is often convenient to choose the potentials such that
∇·A+
1 ∂
Φ=0
c2 ∂t
(6.37)
This choice of gauge is known as Lorentz gauge (to be compared with Coulomb
gauge, i.e ∇ · A = 0) which can be used to simplify (6.33) and (6.34),
1 ∂2
ρ
Φ = −
2
2
c ∂t
"0
2
∂
J
1
∇2 A − 2 2 A = − .
c ∂t
"0
∇2 Φ −
(6.38)
(6.39)
Lorentz gauge can always be constructed with desired gauge transformation (6.35) and (6.36),
0 = ∇ · Anew +
1 ∂
1 ∂2
1 ∂
2
Φ
=
∇
·
A
+
Φ
+
∇
Λ
−
Λ (6.40)
new
old
old
c2 ∂t
c2 ∂t
c2 ∂t2
where Λ is to be chosen to satisfy
∇2 Λ −
!
"
1 ∂
1 ∂2
Λ
=
−
∇
·
A
+
Φ
old
old .
c2 ∂t2
c2 ∂t
(6.41)
However, even in the Lorentz gauge there is still a remaining freedom to
choose potentials (called restricted gauge freedom) with Λ satisfying
∇2 Λ −
1 ∂2
Λ = 0.
c2 ∂t2
(6.42)
We have already discussed the Coulomb gauge, i.e.
∇·A =0
(6.43)
in the context of magnetostatics and in electrodynamics the above choice is
also known as transverse or radiation gauge. In this gauge (6.33) takes the
form of the Poisson equation
∇2 Φ = −
ρ
"0
(6.44)
with solution corresponding to the instantaneous Coulomb potential
#
1
ρ(x" , t) 3 "
dx
(6.45)
Φ(x, t) =
4π"0
|x − x" |
CHAPTER 6. MAXWELL EQUATIONS
79
which is the reason why the gauge has its name. The other dynamical equation (6.34) in Coulomb gauge takes the following form
1 ∂
1 ∂2
A = −µ0 J + 2 ∇ Φ.
(6.46)
2
2
c ∂t
c ∂t
It is now convenient to split the current into longitudinal (or scalar Jl =
∇S) and transverse (or vector Jt ) parts which can be accomplished by first
solving the Poisson equation
∇2 A −
∇2 S = ∇ · J
(6.47)
and then by setting
Jl = ∇S.
Jt = J − ∇S.
such that
∇ × Jl = 0
∇ · Jt = 0.
(6.48)
(6.49)
Then we can also split (6.46) into longitudinal
µ 0 Jl =
1 ∂
∇ Φ.
c2 ∂t
(6.50)
and transverse part
1 ∂2
A = −µ0 Jt
(6.51)
c2 ∂t2
which is the reason why the Coulomb gauge is also known as the transverse
gauge. In a special case when there are no charges (i.e. free electromagnetism) equations (6.45) and (6.51) imply
∇2 A −
Φ = 0
1 ∂
A = 0
c2 ∂t2
and the fields are determined form the vector potential
∇2 A −
(6.52)
2
∂
A
∂t
B = ∇ × A.
E = −
(6.53)
(6.54)
(6.55)
We note that due to symmetry of Maxwell equation one can define alternative (Lorentz and Coulomb) gauges where the magnetic scalar potential
ΦM and electric vector potential AE are used instead to more conventional
electric scalar potential ΦE and magnetic vector potential AM .
CHAPTER 6. MAXWELL EQUATIONS
6.3
80
Green Functions
The dynamics of potential usually involves the so-called wave equations (for
example (6.38), (6.39),(6.51)) of the basic form
∇2 Ψ(x, t) −
1 ∂2
Ψ(x, t) = −4πf (x, t)
c2 ∂t2
(6.56)
where f (x, t) is a known function. The first step in solving this equation is
to expand both functions into Fourier integral over frequencies (conjugate
variable to time)
# ∞
1
Ψ(x, ω) e−iωtdω
(6.57)
Ψ(x, t) =
2π −∞
# ∞
1
f (x, t) =
f (x, ω) e−iωtdω.
(6.58)
2π −∞
and insert the expansion into (6.56),
!
"
# ∞
# ∞
1 ∂2
1
1
2
∇ − 2 2
Ψ(x, ω) e−iωt dω
=
f (x, ω) e−iωt
(6.59)
dω
c ∂t
2π −∞
2π −∞
"
# ∞
# ∞!
ω2
−iωt
2
dω = −4π
f (x, ω) e−iωtdω.
(6.60)
∇ + 2 Ψ(x, ω) e
c
−∞
−∞
By equating each of the Fourier coefficients we get the inhomogeneous Helmholtz
wave equation
$ 2
%
∇ + k 2 Ψ(x, ω) = −4πf (x, ω)
(6.61)
where k = ω/c is the wave number (Note that in the limit k = 0 the
Helmholtz wave equation reduces to the Poisson equation). Equation (6.61)
can be solved using the method of Green functions that obey
$ 2
%
∇ + k 2 Gk (x, x" ) = −4πδ (x − x" )
(6.62)
and appropriate boundary conditions.
For example, if there are no boundaries then the solution must only depend on r = |x − x" | and (6.62) can be rewritten in spherical coordinates
as
1 d2
(rGk ) + k 2 Gk = −4πδ (r)
(6.63)
r dr 2
or for r %= 0
d2
(rGk ) + k 2 (rGk ) = 0
(6.64)
dr 2
81
CHAPTER 6. MAXWELL EQUATIONS
and for r → 0
d2
(rGk ) ≈ −4πδ (r)
(6.65)
dr 2
since in this limit the second term in (6.63) is much smaller than the first
term,
1 d2
Gk
(rGk ) ∼ 2 ) k 2 Gk .
(6.66)
2
r dr
r
It follows that (6.64) has a solution in terms of outgoing ∝ exp(ikr) and
incoming ∝ exp(−ikr) plane waves, i.e.
r
rGk = Aeikr + Be−ikr
(6.67)
and (6.65) is the equation that we already solved in electrostatics, i.e.
1
lim Gk = .
r
(6.68)
kr→0
These two conditions determine the most general solution
(+)
(−)
Gk = AGk (r) + BGk (r)
(6.69)
e±ikr
r
A + B = 1.
(6.70)
where
(±)
Gk (r) =
(6.71)
To obtain a general solution for the Green functions in space-time we
expand the functions in (6.62) using inverse Fourier transformation,
!
"# ∞
# ∞
ω2
!
2
" iω(t! −t)
∇ + 2
Gk (x, x )e
dω = −4π
δ(x − x" )eiω(t −t)
(6.72)
dω
c
−∞
−∞
!
"
#
#
∞
∞
1 ∂2
!
!
∇2 − 2 2
Gk (x, x" )eiω(t −t) dω = −4π
δ(x − x" )eiω(t −t)
(6.73)
dω
c ∂t
−∞
−∞
"
!
1 ∂2
2
∇ − 2 2 G(x, x" ; t, t" ) = −4πδ(x − x" )δ(t − t" ) (6.74)
c ∂t
where
1
G(x, x ; t, t ) ≡
2π
"
"
#
∞
−∞
!
Gk (x, x" )eiω(t −t) dω.
(6.75)
Since we already solved for G(x, x" ) we can write the two wave solutions of
(6.74) as
# ∞ ±ik|x−x! |
1
e
!
G(±) (x, x" ; t, t" ) =
eiω(t −t) dω.
(6.76)
2π −∞ |x − x" |
CHAPTER 6. MAXWELL EQUATIONS
82
In the case of nondispersive media (k = ω/c) the integral of (6.76) yields
& !
!
""'
# ∞
1
1
|x − x" |
(±)
"
"
"
G (x, x ; t, t ) =
exp iω t − t ∓
(6.77)
dω
|x − x" | 2π −∞
c
& !
!
""'
|x − x" |
1
"
δ
iω
t
−
t
∓
(6.78)
=
|x − x" |
c
These are the retarded (or causal) G(+) and advanced G(−) Green functions.
Then the solution of (6.61) in terms of Green functions is given by
# #
(±)
Ψ (x, t) =
G(±) (x, x" ; t, t" )f (x" , t)d3 x" dt"
(6.79)
but arbitrary solutions of the homogeneous wave equations
1 ∂2
Ψin (x, t) = 0
c2 ∂t2
1 ∂2
∇2 Ψout (x, t) − 2 2 Ψout (x, t) = 0.
c ∂t
∇2 Ψin (x, t) −
(6.80)
(6.81)
can also be added to satisfy the incoming wave condition or the outgoing
wave conditions,
Ψ = Ψin (x, t) + Ψ(+) (x, t)
Ψ = Ψout (x, t) + Ψ(−) (x, t).
(6.82)
(6.83)
In other words if either the initial wave configuration or the final wave configurations is known before any sources are turned on, then the combined
solution is given by either (6.82) or (6.83) respectively.