The Continuity Equation and the Maxwell–Ampere Equation
The Continuity Equation
In Physics there are several universal Conservation Laws: the net energy, the net momentum, the net angular momentum, and the net electric charge of a closed system can
never change regardless of what happens to that system. Bodies may collide or blow up,
substances may freeze or boil, there may be all kinds of nuclear reactions, heck, a star may
collapse into a black hole, but the net energy, momentum, angular momentum, and electric
charge will always remain the same as they were when the system in question became closed
(i.e., not interacting with the rest of the Universe).
Moreover, for open systems there are local versions of these conservation laws. The
electric charge and other conserved quantities do not instantaneously jump long distances,
they have to flow at finite rate through all the intermediate locations. For example, consider
the net electric charge Q contained in some volume V. This change may change if there is
an electric current through the surface S of the volume V, but that is the only way: the
charge cannot disappear from V and jump far away without flowing through the surface.
Specifically,
∆Q[inside V] = Inflow[through S] − Outflow[through S],
(1)
and the rate of this change is
d
Q[inside V] = Inet [into V] = I[into V] − I[out from V].
dt
(2)
In terms of electric current density J(x, y, z),
d
Q[inside V] = −
dt
ZZ
J · d2 A
(3)
S
where the minus sign is due to infinitesimal area vector d2 A pointing out from the volume
V rather than into V.
1
Let’s assume a continuous distribution of the electric charge inside V with some volume
density ρ(x, y, z), so the net charge inside V is given by the volume integral
Q[inside V] =
ZZZ
ρ(x, y, z) d3 Vol
(4)
V
The volume V here may have any shape, but we take this shape to be time-independent, so
the time derivative of the charge inside this volume is
d
d
Q[inside V] =
dt
dt
ZZZ
3
ρ(x, y, z) d Vol =
ZZZ
V
∂ρ(x, y, z; t) 3
d Vol.
∂t
(5)
V
Consequently, eq. (3) gives us an integral relation between the time derivative of the electric
charge density and the electric current,
ZZZ
∂ρ(x, y, z; t) 3
d Vol = −
∂t
V
ZZ
J · d2 A.
(6)
S
On the right hand side here, we may use the Gauss Theorem to trade the surface integral of
the current density J to a volume integral of its divergence ∇ · J, thus
ZZZ
V
∂ρ(x, y, z; t) 3
d Vol = −
∂t
ZZ
2
J·d A = −
S
ZZZ
∇ · J(x, y, z; t) d3Vol.
(7)
V
Note: this equation must hold for any volume V, so the integrands on the LHS and on the
RHS must be identically equal to each other. Thus, for every point in space and time, we
must have
∂
ρ(x, y, z; t) = −∇ · J(x, y, z; t).
∂t
(8)
This continuity equation is the differential form of the local conservation law for the electric
charge.
The local conservation laws of energy, momentum, and angular momentum also have
differential forms, but they involve tensors, so I won’t write them down in these notes.
2
Maxwell–Ampere Equation
Consider the differential form of the Ampere’s Law
∇ × B(x, y, z) = µ0 J(x, y, z).
(9)
Let’s take the divergence of the vector fields on both sides of this equation. On the LHS,
the divergence of a curl is automatically zero,
∇ · (∇ × B) ≡ 0,
(10)
so on the RHS of eq. (9) we should also have identical zero, thus
∇ · J(x, y, z) ≡ 0.
(11)
But alas, this condition of divergence-less current density does not quite agree with the
continuity equation (8).
There is no disagreement for steady, time-independent currents. For such currents, there
is no temporary accumulation of electric charges, so that the net charge density ρ(x, y, z)
remains time-independent and hence the continuity equation reduces to the ∇ · J ≡ 0 condition. Consequently, for steady currents the Ampere’s Law may be used as it is. But for
time-dependent currents — which may be accompanied by a time-dependent ρ(x, y, z; t) —
the Ampere’s Law must be modified in order to be consistent with the continuity equation.
But before we modify the Ampere’s Law, let’s look at its integral form
I
B · d~ℓ = µ0 × I net [through loop L].
(12)
loop L
To be precise, the net current through a closed loop L is defined as the net current through
some surface S spanning the loop L,
I
net
[through L] =
ZZ
J · d2 A.
(13)
S
As long as the current density J is divergence-less, Gauss’s theorem assures us that the
integral here is the same for any surface spanning the same loop L, so the net current
3
through L is well defined. But that’s no longer true for non-zero ∇ · J. Indeed, let S1 and
S2 be two surfaces spanning the same loop L, and let V be the volume enclosed between the
S1 and the S2 . In this case, Gauss Theorem gives us
I
net
[through S1 ] − I
net
[through S2 ] =
ZZ
2
J·d A −
ZZ
S
=
1
ZZZ
J · d2 A
S2
∇ · J d3 V
(14)
V
= −
ZZZ
∂ρ 3
d V
∂t
V
d
= − Q[inside V].
dt
Thus, if the charge enclosed between the surfaces S1 and S2 changes with time, then the
currents through these surfaces differ by − the time derivative of that charge. Consequently,
the net current through the loop L spanned by each of these surfaces becomes ill-defined
— we may no longer talk about the current through the loop but only about the current
through a particular surface which spans it.
As an example, consider AC current flowing through a long wire interrupted by a parallelplate capacitor. Let the Ampere Loop L go around the capacitor. We may span this loop
with a surface S1 which crosses the wire, or with the surface S2 which goes between the
capacitor’s plates and avoids the wire, as illustrated on the following picture:
4
S1
S2
loop L
The net current I1 through the surface S1 is the AC current I(t) in the wire, while the net
current I2 through the surface S2 is zero. So which of these two currents should we compare
to the integral of the magnetic field along the Ampere loop L?
To resolve this quandary, James Clerk Maxwell introduced the displacement current ID
due to time-dependent electric field. Specifically, the displacement current density is
JD = κǫ0
∂E
∂t
(15)
where κ is the dielectric constant. In the capacitor example, the current I(t) through the
capacitor results in the time-dependent charges ±Q(t) on the capacitor plates, and hence
the time-dependent electric field between the plates,
E(t) =
Q(t)
κǫ0 A
(16)
where A is the plates’ area. The time dependence of this field gives rise to the displacement
current density
JD (t) = κǫ0
1
dQ
1
∂E
=
×
=
× Iwire (t) ,
∂t
A
dt
A
(17)
so the net displacement current between the plates is the same as the electric current in the
5
wire,
ID (t) = A × JD (t) = Iwire (t).
(18)
Therefore, if we add the displacement current ID to the ordinary electric current through
the wire, then the net current Iwire + ID through the surface S2 which passes between the
capacitor’s plates is the same as the net current through the surface S1 which crosses a
wire connected to the capacitor. This allows us to unambiguously define the net electric +
displacement current through the Ampere’s loop L, so we may compare that net current to
the loop integral of the magnetic field.
In general, the combined electric + displacement current density
∂E
∂t
(19)
∂E
= ∇ · κǫ0
∂t
∂
∇ · κǫ0 E
=
∂t
∂
=
ρ
∂t
(20)
Jnet = J + JD = J + κǫ0
is always divergence-less. Indeed,
∇ · JD
hh interchanging derivatives ii
hh by Gauss Law ii
while by the continuity equation
∇·J = −
∂
ρ,
∂t
(21)
so together
∇ · Jnet = ∇ · JD + ∇ · J =
∂ρ
∂ρ
−
= 0.
∂t
∂t
(22)
Consequently, the net electric+displacement current through a closed Ampere Loop is always
well-defined and does not depend on a particular surface spanning the loop. And it is this
observation which allowed James Clerk Maxwell to repair the Ampere’s Law by adding the
displacement current to the ordinary electric current.
6
In the integral form the Ampere–Maxwell Law says
I
ZZ
combined
~
B · dℓ = µ0 (I + ID )
[through L] = µ0 ×
J + JD · d2 A.
L
(23)
S
while in the differential form it becomes
∇ × B = µ 0 × J + JD .
(24)
To be precise, both of these form apply in the non-magnetic media only. IN a magnetic
medium, in addition to the ordinary current and the displacement current, there is also
an effective surface current due to magnetization of the medium. Macroscopically, this
magnetization can be handled by introducing the magnetic field strength H, which is related
to the magnetic induction B according to
H =
B
magnetic moment
B
−
=
.
µ0
Volume
µrel µ0
In terms of the H field, the Ampere–Maxwell Law becomes
I
ZZ
combined
~
H · dℓ = (I + ID )
[through L] =
J + JD · d 2 A
L
(25)
(26)
S
in the integral form, or
∇ × H = J + JD = J + κǫ0
∂E
∂t
(27)
in the differential form. Finally, in dielectric media it is often convenient to rewrite the
displacement current in terms of the electric displacement field
D = ǫ0 E +
dipole moment
= κǫ0 × E
Volume
(28)
as
JD =
∂D
.
∂t
(29)
Consequently, the Ampere–Maxwell equation becomes
∇×H = J +
7
∂D
.
∂t
(30)
Maxwell Equations
And God said:
∇ · D = ρ,
(31.a)
∇ · B = 0,
(31.b)
∂B
,
∂t
∂D
∇×H = J +
,
∂t
∇×E = −
(31.c)
(31.d)
and there was light!
Collectively, equations (31.a–d) — which comprise the Gauss Laws for the electric and
magnetic fields, the induction equation, and the Ampere-Maxwell Law — are known as the
Maxwell equations after James Clerk Maxwell, although Maxwell himself wrote them in a
quite different form using quaternion algebra instead of vectors. The vector form of Maxwell
equations — and also their name — is due to Oliver Heaviside and Heinrich Hertz. Maxwell
equations explain the electromagnetic waves, which were predicted by Maxwell himself and
experimentally produced and detected by Heinrich Hertz. Maxwell also argued that light is
nothing but an electromagnetic wave with a rather short wavelength (400 nm to 700 nm for
visible light). Thus, the four Maxwell equations unify Electricity, Magnetism, and Optics —
which used to be separate branches of Physics — into a common field of Electromagnetism.
Equations (31.a–d) are written in MKSA units. In Gaussian units, they become
∇ · D = 4πρ,
(32.a)
∇ · B = 0,
(31.b)
1 ∂B
,
c ∂t
1 ∂D
4π
J +
,
∇×H =
c
c ∂t
∇×E = −
(32.c)
(32.d)
where
D = κE and H =
8
1
B.
µrel
(33)