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Classical Electromagnetism
Richard Fitzpatrick
Professor of Physics
The University of Texas at Austin
- 1 Maxwell’s Equations Contents
- 1.1 Introduction
- 1.2 Maxwell’s Equations
- 1.3 Scalar and Vector Potentials
- 1.4 Dirac Delta Function
- 1.5 Three-Dimensional Dirac Delta Function
- 1.6 Solution of Inhomogeneous Wave Equation
- 1.7 Retarded Potentials
- 1.8 Retarded Fields
- 1.9 Electromagnetic Energy Conservation
- 1.10 Electromagnetic Momentum Conservation
- 1.11 Exercises
- 2 Electrostatic Fields
- 2.1 Introduction
- 2.2 Laplace’s Equation
- 2.3 Poisson’s Equation
- 2.4 Coulomb’s Law
- 2.5 Electric Scalar Potential
- 2.6 Electrostatic Energy
- 2.7 Electric Dipoles
- 2.8 Charge Sheets and Dipole Sheets
- 2.9 Green’s Theorem
- 2.10 Boundary Value Problems
- 2.11 Dirichlet Green’s Function for Spherical Surface
- 2.12 Exercises
- 3 Potential Theory
- 3.1 Introduction
- 3.2 Associated Legendre Functions 2 CLASSICAL ELECTROMAGNETISM
- 3.3 Spherical Harmonics
- 3.4 Laplace’s Equation in Spherical Coordinates
- 3.5 Poisson’s Equation in Spherical Coordinates
- 3.6 Multipole Expansion
- 3.7 Axisymmetric Charge Distributions
- 3.8 Dirichlet Problem in Spherical Coordinates
- 3.9 Newmann Problem in Spherical Coordinates
- 3.10 Laplace’s Equation in Cylindrical Coordinates
- 3.11 Poisson’s Equation in Cylindrical Coordinates
- 3.12 Exercises
- 4 Electrostatics in Dielectric Media
- 4.1 Polarization
- 4.2 Boundary Conditions for E and D
- 4.3 Boundary Value Problems with Dielectrics
- 4.4 Energy Density Within Dielectric Medium
- 4.5 Force Density Within Dielectric Medium
- 4.6 Clausius-Mossotti Relation
- 4.7 Dielectric Liquids in Electrostatic Fields
- 4.8 Exercises
- 5 Magnetostatic Fields
- 5.1 Introduction
- 5.2 Biot-Savart Law
- 5.3 Continuous Current Distribution
- 5.4 Circular Current Loop
- 5.5 Localized Current Distribution
- 5.6 Exercises
- 6 Magnetostatics in Magnetic Media
- 6.1 Magnetization
- 6.2 Magnetic Susceptibility and Permeability
- 6.3 Ferromagnetism
- 6.4 Boundary Conditions for B and H
- 6.5 Permanent Ferromagnets
- 6.6 Uniformly Magnetized Sphere
- 6.7 Soft Iron Sphere in Uniform Magnetic Field
- 6.8 Magnetic Shielding
- 6.9 Magnetic Energy
- 6.10 Exercises
- 7 Wave Propagation in Uniform Dielectric Media
- CONTENTS
- 7.1 Introduction
- 7.2 Form of Dielectric Constant
- 7.3 Anomalous Dispersion and Resonant Absorption
- 7.4 Wave Propagation in Conducting Media
- 7.5 High Frequency Limit
- 7.6 Polarization of Electromagnetic Waves
- 7.7 Faraday Rotation
- 7.8 Wave Propagation in Magnetized Plasmas
- 7.9 Wave Propagation in Dispersive Media
- 7.10 Wave-Front Propagation
- 7.11 Sommerfeld Precursor
- 7.12 Method of Stationary Phase
- 7.13 Group Velocity
- 7.14 Brillouin Precursor
- 7.15 Signal Arrival
- 7.16 Exercises
- 8 Wave Propagation in Inhomogeneous Dielectric Media
- 8.1 Introduction
- 8.2 Laws of Geometric Optics
- 8.3 Fresnel Relations
- 8.4 Total Internal Reflection
- 8.5 Reflection by Conducting Surfaces
- 8.6 Ionospheric Radio Wave Propagation
- 8.7 WKB Approximation
- 8.8 Reflection Coefficient
- 8.9 Extension to Oblique Incidence
- 8.10 Ionospheric Pulse Propagation
- 8.11 Measurement of Ionospheric Electron Density Profile
- 8.12 Ionospheric Ray Tracing
- 8.13 Asymptotic Series
- 8.14 WKB Solution as Asymptotic Series
- 8.15 Stokes Constants
- 8.16 WKB Reflection Coefficient
- 8.17 Jeffries Connection Formula
- 8.18 Exercises
- 9 Radiation and Scattering
- 9.1 Introduction
- 9.2 Basic Antenna Theory
- 9.3 Antenna Directivity and Effective Area
- 9.4 Antenna Arrays
- 9.5 Thomson Scattering
- 9.6 Rayleigh Scattering 4 CLASSICAL ELECTROMAGNETISM
- 9.7 Exercises
- 10 Resonant Cavities and Waveguides
- 10.1 Introduction
- 10.2 Boundary Conditions
- 10.3 Cavities with Rectangular Boundaries
- 10.4 Quality Factor of a Resonant Cavity
- 10.5 Axially Symmetric Cavities
- 10.6 Cylindrical Cavities
- 10.7 Waveguides
- 10.8 Dielectric Waveguides
- 10.9 Exercises
- 11 Multipole Expansion
- 11.1 Introduction
- 11.2 Multipole Expansion of Scalar Wave Equation
- 11.3 Angular Momentum Operators
- 11.4 Multipole Expansion of Vector Wave Equation
- 11.5 Properties of Multipole Fields
- 11.6 Solution of Inhomogeneous Helmholtz Equation
- 11.7 Sources of Multipole Radiation
- 11.8 Radiation from Linear Centre-Fed Antenna
- 11.9 Spherical Wave Expansion of Vector Plane Wave
- 11.10 Mie Scattering
- 11.11 Exercises
- 12 Relativity and Electromagnetism
- 12.1 Introduction
- 12.2 Relativity Principle
- 12.3 Lorentz Transformation
- 12.4 Transformation of Velocities
- 12.5 Tensors
- 12.6 Physical Significance of Tensors
- 12.7 Space-Time
- 12.8 Proper Time
- 12.9 4-Velocity and 4-Acceleration
- 12.10 Current Density 4-Vector
- 12.11 Potential 4-Vector
- 12.12 Gauge Invariance
- 12.13 Retarded Potentials
- 12.14 Tensors and Pseudo-Tensors
- 12.15 Electromagnetic Field Tensor
- CONTENTS
- 12.16 Dual Electromagnetic Field Tensor
- 12.17 Transformation of Fields
- 12.18 Potential Due to a Moving Charge
- 12.19 Field Due to a Moving Charge
- 12.20 Relativistic Particle Dynamics
- 12.21 Force on a Moving Charge
- 12.22 Electromagnetic Energy Tensor
- 12.23 Accelerated Charges
- 12.24 Larmor Formula
- 12.25 Radiation Losses
- 12.26 Angular Distribution of Radiation
- 12.27 Synchrotron Radiation
- 12.28 Exercises
Maxwell’s Equations 7
1 Maxwell’s Equations
1.1 Introduction
This chapter gives a general overview of Maxwell’s equations.
1.2 Maxwell’s Equations
All classical (i.e., non-quantum) electromagnetic phenomena are governed by Maxwell’s equa- tions, which take the form
∇ · E =
ρ 0
∇ · B = 0 , (1.2)
∇ × E = −
∂B
∂t
∇ × B = μ 0 j + μ 0 0
∂E
∂t
Here, E(r, t), B(r, t), ρ(r, t), and j(r, t) represent the electric field-strength, the magnetic field- strength, the electric charge density, and the electric current density, respectively. Moreover,
0 = 8. 8542 × 10 −^12 C 2 N −^1 m−^2 (1.5)
is the electric permittivity of free space, whereas
μ 0 = 4 π × 10 −^7 N A −^2 (1.6)
is the magnetic permeability of free space. As is well known, Equation (1.1) is equivalent to Coulomb’s law (for the electric fields generated by point charges), Equation (1.2) is equivalent to the statement that magnetic monopoles do not exist (which implies that magnetic field-lines can never begin or end), Equation (1.3) is equivalent to Faraday’s law of electromagnetic induction, and Equation (1.4) is equivalent to the Biot-Savart law (for the magnetic fields generated by line currents) augmented by the induction of magnetic fields by changing electric fields. Maxwell’s equations are linear in nature. In other words, if ρ → α ρ and j → α j, where α is an arbitrary (spatial and temporal) constant, then it is clear from Equations (1.1)–(1.4) that E → α E and B → α B. The linearity of Maxwell’s equations accounts for the well-known fact that the electric fields generated by point charges, as well as the magnetic fields generated by line currents, are superposable. Taking the divergence of Equation (1.4), and combining the resulting expression with Equa- tion (1.1), we obtain ∂ρ ∂t
8 CLASSICAL ELECTROMAGNETISM
In integral form, making use of the divergence theorem, this equation becomes
d dt
V
ρ dV +
S
j · dS = 0 , (1.8)
where V is a fixed volume bounded by a surface S. The volume integral represents the net electric charge contained within the volume, whereas the surface integral represents the outward flux of charge across the bounding surface. The previous equation, which states that the net rate of change of the charge contained within the volume V is equal to minus the net flux of charge across the bounding surface S , is clearly a statement of the conservation of electric charge. Thus, Equa- tion (1.7) is the differential form of this conservation equation. As is well known, a point electric charge q moving with velocity v in the presence of an electric field E and a magnetic field B experiences a force
F = q (E + v × B). (1.9)
Likewise, a distributed charge distribution of charge density ρ and current density j experiences a force density f = ρ E + j × B. (1.10)
1.3 Scalar and Vector Potentials
We can automatically satisfy Equation (1.2) by writing
B = ∇ × A, (1.11)
where A(r, t) is termed the vector potential. Furthermore, we can automatically satisfy Equa- tion (1.3) by writing
E = −∇φ −
∂A
∂t
where φ(r, t) is termed the scalar potential. The previous prescription for expressing electric and magnetic fields in terms of the scalar and vector potentials does not uniquely define the potentials. Indeed, it can be seen that if A → A − ∇ψ and φ → φ + ∂ψ/∂t, where ψ(r, t) is an arbitrary scalar field, then the associated electric and magnetic fields are unaffected. The root of the problem lies in the fact that Equation (1.11) specifies the curl of the vector potential, but leaves the divergence of this vector field completely unspecified. We can make our prescription unique by adopting a convention that specifies the divergence of the vector potential—such a convention is usually called a gauge condition. It turns out that Maxwell’s equations are Lorentz invariant. (See Chapter 12.) In other words, they take the same form in all inertial frames. Thus, it makes sense to adopt a gauge condition that is also Lorentz invariant. This leads us to the so-called Lorenz gauge condition (see Section 12.12),
0 μ 0
∂φ ∂t
+ ∇ · A = 0. (1.13)
10 CLASSICAL ELECTROMAGNETISM
where f (r) is an arbitrary function that is well behaved at r = r′. It is also easy to see that
δ(r′^ − r) = δ(r − r′). (1.24)
We can show that ∇ 2
|r − r′^ |
= − 4 π δ(r − r′). (1.25)
(Here, ∇ 2 is a Laplacian operator expressed in terms of the components of r, but independent of the components of r′.) We must first prove that
|r − r′^ |
= 0 for r r′, (1.26)
in accordance with Equation (1.21). If R = |r − r′^ | then this is equivalent to showing that
1 R 2
d dR
[
R 2
d dR
R
)]
for R > 0, which is indeed the case. (Here, R is treated as a radial spherical coordinate.) Next, we must show that (^) ∫
V
|r − r′^ |
dV = − 4 π, (1.28)
in accordance with Equations (1.22) and (1.25). Suppose that S is a spherical surface, of radius R, centered on r = r′. Making use of the definition ∇ 2 φ ≡ ∇ · ∇φ, as well as the divergence theorem, we can write ∫
V
|r − r′^ |
dV =
V
|r − r′^ |
dV =
S
|r − r′^ |
· dS
= 4 π R 2
d dR
R
= − 4 π. (1.29)
(Here, ∇ is a gradient operator expressed in terms of the components of r, but independent of the components of r′. Likewise, dS is a surface element involving the components of r, but indepen- dent of the components of r′.) Finally, if S is deformed into a general surface (without crossing the point r = r′) then the value of the volume integral is unchanged, as a consequence of Equa- tion (1.26). Hence, we have demonstrated the validity of Equation (1.25).
1.6 Solution of Inhomogeneous Wave Equation
Equation (1.14), as well as the three Cartesian components of Equation (1.15), are inhomogeneous three-dimensional wave equations of the general form ( 1 c 2
∂t 2
u = v, (1.30)
Maxwell’s Equations 11
where u(r, t) is an unknown potential, and v(r, t) a known source function. Let us investigate whether it is possible to find a unique solution of this type of equation. Let us assume that the source function v(r, t) can be expressed as a Fourier integral,
v(r, t) =
−∞
vω(r) e−i^ ω^ t^ dω. (1.31)
The inverse transform is
vω(r) =
2 π
−∞
v(r, t) e+i^ ω^ t^ dt. (1.32)
Similarly, we can write the general potential u(r, t) as a Fourier integral,
u(r, t) =
−∞
uω(r) e−i^ ω^ t^ dω, (1.33)
with the corresponding inverse
uω(r) =
2 π
−∞
u(r, t) e+i^ ω^ t^ dt. (1.34)
Fourier transformation of Equation (1.30) yields
(∇ 2 + k 2 ) uω = −vω , (1.35)
where k = ω/c. Equation (1.35), which reduces to Poisson’s equation (see Section 2.3),
∇ 2 uω = −vω , (1.36)
in the limit k → 0, is known as Helmholtz’s equation. Because Helmholtz’s equation is linear, it is appropriate to attempt a Green’s function method of solution. Let us try to find a Green’s function, Gω(r, r′), such that (∇ 2 + k 2 ) Gω(r, r′) = −δ(r − r′). (1.37)
The general solution to Equation (1.35) is then [cf., Equation (2.16)]
uω(r) =
vω(r′) Gω(r, r′) dV′. (1.38)
Let us adopt the spatial boundary condition Gω(r, r′) → 0 as |r − r′^ | → ∞, so as to ensure that the potential goes to zero a long way from the source. Because Equation (1.37) is spherically symmetric about the point r′, it is plausible that the Green’s function itself is spherically symmetric: that is, Gω(r − r′) = Gω(|r − r′^ |). In this case, Equation (1.37) reduces to
1 R
d 2 (R Gω) dR 2
Maxwell’s Equations 13
The real-space Green’s function specifies the response of the system to a point source located at position r′^ that appears momentarily at time t′. According to the retarded Green’s function, G(+)^ , this response consists of a spherical wave, centered on the point r′, that propagates forward in time. In order for the wave to reach position r at time t, it must have been emitted from the source at r′^ at the retarded time t (^) r = t − |r − r′^ |/c. According to the advanced Green’s function, G(−)^ , the response consists of a spherical wave, centered on the point r′, that propagates backward in time. Clearly, the advanced potential is not consistent with our ideas about causality, which demand that an effect can never precede its cause in time. Thus, the Green’s function that is consistent with our experience is
G(r, r′; t, t′) = G(+)^ (r, r′; t, t′) =
δ(t′^ − [t − |r − r′^ |/c]) 4 π |r − r′^ |
Incidentally, we are able to find solutions of the inhomogeneous wave equation, (1.30), that prop- agate backward in time because this equation is time symmetric (i.e., it is invariant under the transformation t → −t). In conclusion, the most general solution of the inhomogeneous wave equation, (1.30), that satisfies sensible boundary conditions at infinity, and is consistent with causality, is
u(r, t) =
v(r′, t − |r − r′^ |/c) 4 π |r − r′^ |
dV′. (1.50)
This expression is sometimes written
u(r, t) =
[v(r′)] 4 π |r − r′^ |
dV′, (1.51)
where the rectangular bracket symbol [ ] denotes that the terms inside the bracket are to be eval- uated at the retarded time t − |r − r′^ |/c. Note, in particular, from Equation (1.50), that if there is no source [i.e., if v(r, t) = 0] then there is no field [i.e., u(r, t) = 0]. But, is expression (1.50) really the only solution of Equation (1.30) that satisfies sensible boundary conditions at infinity? In other words, is this solution really unique? Unfortunately, there is a weak link in our derivation— between Equations (1.38) and (1.39)—where we assumed, without proof, that the Green’s function for Helmholtz’s equation, subject to the boundary condition Gω(r, r′) → 0 as |r−r′^ | → ∞, is spher- ically symmetric. Let us try to fix this problem. With the benefit of hindsight, we can see that the Fourier-space Green’s function
Gω =
e+i^ k R 4 π R
corresponds to the retarded solution in real space, and is, therefore, the correct physical Green’s function in Fourier space. The Fourier-space Green’s function
Gω =
e−i^ k R 4 π R
corresponds to the advanced solution in real space, and must, therefore, be rejected. We can select the retarded Green’s function in Fourier space by imposing the following boundary condition at
14 CLASSICAL ELECTROMAGNETISM
infinity
Rlim→∞ R
∂Gω ∂R
− i k Gω
This is called the Sommerfeld radiation condition, and basically ensures that infinity is an absorber of radiation, but not a source. But, does this boundary condition uniquely select the spherically symmetric Green’s function (1.52) as the solution of
(∇ 2 + k 2 ) Gω(R, θ, ϕ) = −δ(R)? (1.55)
Here, (R, θ, ϕ) are spherical polar coordinates. If it does then we can be sure that Equation (1.50) represents the unique solution of the inhomogeneous wave equation, (1.30), that is consistent with causality. Let us suppose that there are two different solutions of Equation (1.55), both of which satisfy the boundary condition (1.54), and revert to the unique (see Section 2.3) Green’s function for Poisson’s equation, (1.42), in the limit R → 0. Let us call these solutions u 1 and u 2 , and let us form the difference w = u 1 − u 2. Consider a surface Σ 0 which is a sphere of arbitrarily small radius centred on the origin. Consider a second surface Σ∞ which is a sphere of arbitrarily large radius centred on the origin. Let V denote the volume enclosed by these surfaces. The difference function w satisfies the homogeneous Helmholtz equation,
(∇ 2 + k 2 ) w = 0 , (1.56)
throughout V. According to the generalized (to deal with complex potentials) Green’s theorem (see Section 2.9),
∫
V
(w ∇ 2 w∗^ − w∗^ ∇ 2 w) dV =
Σ 0
Σ∞
w
∂w∗ ∂n
− w∗^
∂w ∂n
dS , (1.57)
where ∂/∂n denotes a derivative normal to the surface in question. It is clear from Equation (1.56) that the volume integral is zero. It is also clear that the first surface integral is zero, because both u 1 and u 2 must revert to the Green’s function for Poisson’s equation in the limit R → 0. Thus,
∫
Σ∞
w
∂w∗ ∂n
− w∗^
∂w ∂n
dS = 0. (1.58)
Equation (1.56) can be written
∂ 2 (R w) ∂R 2
D (R w) R 2
where D is the spherical harmonic operator
D =
sin θ
∂θ
sin θ
∂θ
sin 2 θ
∂ϕ 2
16 CLASSICAL ELECTROMAGNETISM
It follows that if f 0 = 0 then all of the f (^) n are equal to zero. Let us now consider the surface integral (1.58). Because we are interested in the limit R → ∞, we can replace w by the first term of its expansion in (1.66), so that
∫
Σ∞
w
∂w∗ ∂n
− w∗^
∂w ∂n
dS = −2 i k
| f 0 | 2 dΩ = 0 , (1.70)
where dΩ is an element of solid angle. It is clear that f 0 = 0. This implies that f 1 = f 2 = · · · = 0, and, hence, that w = 0. Thus, there is only one solution of Equation (1.55) that is consistent with the Sommerfeld radiation condition, and this is given by Equation (1.52). We can now be sure that Equation (1.50) is the unique solution of Equation (1.30), subject to the boundary condition (1.54). This boundary condition ensures that infinity is an absorber of electromagnetic radiation, but not an emitter, which seems entirely reasonable.
1.7 Retarded Potentials
We are now in a position to solve Maxwell’s equations. Recall, from Section 1.3, that Maxwell equations reduce to
( 1 c 2
∂t 2
φ =
ρ 0
c 2
∂t 2
A = μ 0 j. (1.72)
We can solve these inhomogeneous three-dimensional waves equations using the appropriate Green’s function, (1.49). In fact, making use of Equation (1.46), we find that
φ(r, t) =
4 π 0
ρ(r′, t − |r − r′^ |/c) |r − r′^ |
dV′^ (1.73)
A(r, t) =
μ 0 4 π
j(r′, t − |r − r′^ |/c) |r − r′^ |
dV′. (1.74)
Alternatively, we can write
φ(r, t) =
4 π 0
∫ [
ρ(r′)
]
|r − r′^ |
dV′, (1.75)
A(r, t) =
μ 0 4 π
∫ [
j(r′)
]
|r − r′^ |
dV′. (1.76)
The above potentials are termed retarded potentials (because the integrands are evaluated at the retarded time). Finally, according to the discussion in the previous section, we can be sure that Equations (1.75) and (1.76) are the unique solutions to Equations (1.71) and (1.72), respectively, subject to sensible boundary conditions at infinity.
Maxwell’s Equations 17
1.8 Retarded Fields
We have found the solution to Maxwell’s equations in terms of retarded potentials. Let us now construct the associated retarded electric and magnetic fields using (see Section 1.3)
E = −∇φ −
∂A
∂t
B = ∇ × A. (1.78)
It is helpful to write R = r − r′, (1.79)
where R = |r − r′^ |. The retarded time becomes t (^) r = t − R/c, and a general retarded quantity is written [F(r′, t)] ≡ F(r′, t (^) r). Thus, we can express the retarded potential solutions of Maxwell’s equations in the particularly compact form
φ(r, t) =
4 π 0
[ρ] R
dV′, (1.80)
A(r, t) =
μ 0 4 π
[j] R
dV′. (1.81)
It is easily seen that
∇φ =
4 π 0
[ρ] ∇
R −^1
[∂ρ/∂t] R
∇t (^) r
dV′
4 π 0
[ρ] R 3
R +
[∂ρ/∂t] c R 2
R
dV′, (1.82)
where use has been made of
∇R =
R
R
R −^1
R
R 3
, ∇t (^) r = −
R
c R
Likewise,
∇ × A =
μ 0 4 π
R −^1
× [j] +
∇t (^) r × [∂j/∂t] R
dV′
μ 0 4 π
R × [j] R 3
R × [∂j/∂t] c R 2
dV′. (1.84)
Equations (1.77), (1.78), (1.82), and (1.84) can be combined to give
E =
4 π 0
[ρ]
R
R 3
[
∂ρ ∂t
]
R
c R 2
[∂j/∂t] c 2 R
dV′, (1.85)
and
B =
μ 0 4 π
[j] × R R 3
[∂j/∂t] × R c R 2
dV′. (1.86)
Maxwell’s Equations 19
et cetera. Thus, the electric field reduces to
E(r, t) −
4 π 0
[∫
∂j⊥/∂t dV′
]
c 2 r
whereas the magnetic field is given by
B(r, t)
4 π 0
[∫
∂j⊥/∂t dV′
]
× r c 3 r 2
Here, [· · · ] merely denotes evaluation at the retarded time t − r/c. Note that
E B
= c, (1.98)
and E · B = 0. (1.99)
This configuration of electric and magnetic fields is characteristic of an electromagnetic wave. In fact, Equations (1.96) and (1.97) describe an electromagnetic wave propagating radially away from the charge and current containing region. The wave is clearly driven by time-varying electric currents. Now, charges moving with a constant velocity constitute a steady current, so a nonsteady current is associated with accelerating charges. We conclude that accelerating electric charges emit electromagnetic waves. The wave fields, (1.96) and (1.97), fall off like the inverse of the distance from the wave source. This behavior should be contrasted with that of Coulomb or Biot-Savart fields, which fall off like the inverse square of the distance from the source. In conclusion, electric and magnetic fields look simple in the near field region (they are just Coulomb fields, etc.), and also in the far field region (they are just electromagnetic waves). Only in the intermediate region, R ∼ R 0 , do the fields get really complicated.
1.9 Electromagnetic Energy Conservation
Consider the fourth Maxwell equation:
∇ × B = μ 0 j + 0 μ 0
∂E
∂t
Forming the scalar product with the electric field, and rearranging, we obtain
−E · j = −
E · ∇ × B
μ 0
+ 0 E ·
∂E
∂t
which can be rewritten
−E · j = −
E · ∇ × B
μ 0
∂t
0 E 2
Now, ∇ · (E × B) ≡ B · ∇ × E − E · ∇ × B, (1.103)
20 CLASSICAL ELECTROMAGNETISM
so
−E · j = ∇ ·
E × B
μ 0
B · ∇ × E
μ 0
∂t
0 E 2
Making use of third Maxwell equation,
∇ × E = −
∂B
∂t
we obtain
−E · j = ∇ ·
E × B
μ 0
∂B
∂t
∂t
0 E 2
which can be rewritten
−E · j = ∇ ·
E × B
μ 0
∂t
0 E 2
B 2
2 μ 0
Thus, we get ∂U ∂t
where U and u are specified in Equations (1.109) and (1.110), respectively. By comparison with Equation (1.7), we can recognize the previous expression as some sort of conservation equation. Here, U is the density of the conserved quantity, u is the flux of the conserved quantity, and −E·j is the rate at which the conserved quantity is created per unit volume. However, E · j is the rate per unit volume at which electric charges gain energy via interaction with electromagnetic fields. Hence, −E·j is the rate per unit volume at which electromagnetic fields gain energy via interaction with charges. It follows that Equation (1.108) is a conservation equation for electromagnetic energy. Thus.
U =
0 E 2
B 2
2 μ 0
can be interpreted as the electromagnetic energy density, and
u =
E × B
μ 0
as the electromagnetic energy flux. The latter quantity is usually called the Poynting flux, after its discoverer.
1.10 Electromagnetic Momentum Conservation
Let g(i)^ be the density of electromagnetic momentum directed parallel to the ith Cartesian axis. (Here, i = 1 corresponds to the x-axis, i = 2 to the y-axis, and i = 3 to the z-axis.) Furthermore, let G(i)^ be the flux of such momentum. We would expect the conservation equation for electromagnetic momentum directed parallel to the ith Cartesian axis to take the form
∂g(i) ∂t
- ∇ · G(i)^ = −(ρ E + j × B)i, (1.111)
