It is useful to recall the background of Maxwell’s Equations. We consider this for each of the four equations separately.
E ( r ) =
1
4 πε o
ρ ( r ) r − r
| r − r | 3 d 3 r (2.1)
This is essentially Coulomb’s Law; giving the field E ( r ) at r due to charge density ρ ( r ) at r . It follows from this that
∇ · E =
ρ
ε o
(2.2)
In polarizable materials, it is useful to consider induced electric dipoles. In non-polar materials, in the linear regime, the average dipole moment of a molecule is proportional to the applied field E , that is, p = ε o
χ M OL e
E (2.3) where α is the molecular polarizability. The electric polarization P = N p = ε o
N χ is the dipole moment per volume, N is the number density. The electric po-
M OL e tential V , defined by E = − ∇ V , due to this polarization is
E
V ( r ) =
1
4 πε o
P ( r ) · r − r
| r − r | 3 d 3 r (2.4)
13

14 CHAPTER 2. REVIEW OF ELECTRICITY AND MAGNETISM which may be written as
V ( r ) =
1
4 πε o surf ace
1
| r − r |
P ( r ) · d Ω −
1
4 πε o
∇ · P ( r )
| r − r |
· d 3 r
We note that −∇· P ( r ) is a charge density; it is the density of bound charges
ρ b
. Writing ρ = ρ b
+ ρ f
, where ρ f is the density of free charges, Eq. 15.2
becomes
∇ · E =
ρ f
ε o
−
∇ · P
ε o
(2.5) which may be rearranged to give
∇ · ( ε o
E + P ) = ρ f
This leads to the definition of electric displacement D
(2.6)
D = ε o
E + P (2.7) which, since P = N χ M OL e
E , may be written as
D = ε o
(1+ N χ M OL e
) E = ε o
(1+ χ e
) E = ε E (2.8) where ε = ε o
ε r is the dielectric permittivity.
dielectric permittivity of free space, ε r
ε o
= 8 .
85 × 10 − 12 F/m is the is the dielectric constant, and χ is the dielectric susceptibility.
Gauss’ Law is then simply
∇ · D = ρ f
(2.9) and in the absence of free charges it becomes
∇ · D = 0 (2.10)
Not that the above arguments assumed that the charges were stationary , if there are time varying magnetic fields, then the electric field is not simply the Coulomb contribution given by Eq. 15.1
The Biot-Savart law gives
B ( r ) =
µ o
4 π
J ( r ) × r − r
| r − r | 3 d 3 r (2.11)

2.2. GAUSS’ LAW FOR THE MAGNETIC FIELD: 15 where B ( r ) is the magnetic field at r due to the current density J ( r ) at r .
It follows from this that
∇ · B =0 (2.12)
Note that this also implies that there is no magnetic ’free charge’ - i.e. that there are no magnetic monopoles. Taking the curl of both sides of Eq. 15.13
gives the original Ampère’s law:
(2.13)
In magnetizable materials, it is useful to consider the induced magnetization
M , the magnetic dipole moment per volume. The magnetic vector potential
A , defined by B = ∇ × A , due to this magnetization is
A ( r ) =
µ o
4 π
M ( r ) × r − r
| r − r | 3 d 3 r (2.14) which may be written as
1
µ o
∇ × B = J
A ( r ) =
µ o
4 π surf ace
M ( r ) × d Ω
| r − r |
+
µ o
4 π
1
H =
µ o
B − M
∇ × M ( r ) d 3 r
| r − r |
(2.15)
We note that ∇ × M is a current density; it is the current density due to bound charges J b
. Writing J = J b
+ J f
, where J f is the current density due to free charges, Eq. 2.13 becomes
1
µ o
∇ × B = J f
+ ∇ × M which may be rearranged to give
1
∇ × (
µ o
B − M ) = J f
This leads to the definition of the magnetic intensity H
(2.16)
(2.17)
(2.18) or
∇ × H = J f
(2.19)

16 CHAPTER 2. REVIEW OF ELECTRICITY AND MAGNETISM
Thus H really plays a role similar to D (cf. Eq. 15.12). In materials without permanent magnetization, in the linear regime, the average magnetic moment m of a molecule is proportional to the applied field H , that is, m = χ MOL m
H (2.20) where χ MOL m is the molecular magnetic susceptibility. The magnetization
M = N m = N χ MOL m
H is the magnetic dipole moment per volume, N is the number density.Since
M = N χ MOL m
H , Eq. 2.18 may be written as
B = µ o
(1 + N χ M OL m
) H = µ o
(1 + χ m
) H = µ H (2.21) where µ = µ o
µ r is the magnetic permeability.
µ o permeability of free space, µ r
= 4 π × 10 − 7 is the relative permeability and
H/m
χ m is the is the magnetic susceptibility.
Again, note that the above arguments assumed that the currents were steady , if there are time varying electric fields, then the magnetic induction is not simply the Biot-Savart contribution given by
Eq. 15.13.
Maxwell corrected the original form of Ampère’s Law (Eq.2.13) to include displacement current to give
∇ × H = J f
+
∂ D
∂t
(2.22)
This is essentially the statement that, in the general time-varying case, in addition to the Biot-Savart contribution, time-varying electric fields also act as sources of the magnetic field. It is interesting to note that even with this contribution, ∇ · B = 0 .
Finally, we have Faraday’s Law of induction, which, in differential form, is
∇ × E = −
∂ B
∂t
(2.23)

2.4. FARADAY’S LAW: 17
This is essentially the statement that, in the general time-varying case, in addition to the Coulomb contribution, time-varying magnetic fields act as sources of the electric field. It is interesting to note that even with this contribution, ∇ · D = ρ f
.

18 CHAPTER 2. REVIEW OF ELECTRICITY AND MAGNETISM