20 Maxwell’s Equations II:Electromagnetic Waves 20.1 The Electromagnetic Energy Theorem; Poynting Vector The Poynting vector which is defined as has the meaning of “energy current density
Trang 1160 19 Maxwell’s Equations I: Faraday’s and Maxwell’s Laws
where i2=−1, we thus have
U G (t) = Re( U Geiωt) and
I(t) = Re( J e iωt
) , where
U ≡ U(0)
G and
J ≡ (I(0)e−iα )
By analogy with Ohm’s law we then define the complex quantityR, where
U G=R · I
The quantityR is the complex a.c resistance or simply impedance.
The total impedance of a circuit is calculated from an appropriate com-bination of three types of standard elements in series or parallel, etc
1 Ohmic resistances (positive and real) are represented by the well-known rectangular symbol and the letter R The corresponding
com-plex resistance is
R R = R
2 Capacitive resistances (negatively imaginary) correspond to a pair
of capacitor plates, together with the letter C The corresponding
impedance is given by
R C= 1
iωC . (A short justification: U C (t) = Q(t) C , i.e., ˙U C (t) = I(t) C Thus with the
ansatz U C (t) ∝ e iωtone obtains ˙U C (t) ≡ iωU C (t)).
3 Inductive resistances (positively imaginary) are represented by a solenoid symbol, together with the letter L The corresponding impedance is
R L = iωL
(The induced voltage drop in the load results from building-up the
magnetic field, according to the relation U L (t) = L · dI(t)
dt , i.e., U L (t) =
L · ˙I(t) But with the ansatz I(t) ∝ e iωtwe obtain ˙I(t) ≡ iωI(t).) One can use the same methods for mutual inductances (i.e., transformers;
see exercises)5
5
The input (load) voltage of the transformer is given by the relation U(1)
Tr =
iωL 1,2 · J2, while the output (generator) voltage is given byU(2)
=−iωL2,1 · J1
Trang 219.4 Applications: Complex Resistances etc 161
d) An a.c resonance circuit
The following is well-known as example of resonance phenomena For a se-ries RLC circuit connected as a load to an alternating-voltage generator
U G (t) = U G(0)· cos(ωt) ,
one has
J
U =
1
R=
R + i
ωL − 1 ωC
−1
.
Thus we obtain
I(t) = I(0)· cos(ωt − α) , with
I(0)
U G(0)
=
1
R
R2+ (ωL − 1
ωC)2 and tan α = ωL − 1
ωC
For sufficiently small R (see below) this yields a sharp resonance at the
resonance frequency
ω0:=√ 1
L · C . For this frequency the current and the voltage are exactly in phase, whereas for higher frequencies the current is delayed with respect to the
voltage (inductive behavior) while for lower frequencies the voltage is de-layed with respect to the current (capacitive behavior) At the resonance
frequency ω0 the current has a very sharp maximum of height U G(0)/R, and for weak damping (i.e., for sufficiently small values of R) it decays very quickly as a function of ω, for very small deviations from ω = ω0;
i.e., for
ω ± := ω0± ε , where ε = R
2L is |ω0| ,
the current has already decreased to 70% of the maximum (more precisely: from 1×I(0) down to √1
2× I(0))
The ratio
Q := ω0 R/L
is called the quality factor of the resonance; it characterizes the sharpness
of the phenomenon In fact, Q often reaches values of the order of 103 or more.
(Here the reader could try solving exercises 11 and 12 (file 6) from the summer term of 2002, which can be found on the internet, [2] This can simultaneously serve as an introduction to MAPLE In fact, it may be helpful to illustrate resonance phenomena using mathematical computer tools such as MAPLE or MATHEMATICA See for example [12] as a rec-ommendable presentation.)
Trang 3162 19 Maxwell’s Equations I: Faraday’s and Maxwell’s Laws
Now consider the power loss in an a.c circuit By forming the derivative
of the energy, we may write
(dE/dt)(t) = U G (t) · I(t) = U(0)
G · I(0)cos ωt · cos(ωt − α)
≡ U(0)
G · I(0)·(cos α · (cos ωt)2+ sin α · (cos ωt · sin ωt)) .
Averaged over a complete cycle the first term gives
U G(0)· I(0)· 1
2· cos α
This is the resistive part, and represents the energy dissipated The second term, however, vanishes when averaged over a complete cycle, and is called the reactive part Using complex quantities one must explicitly take into account the factor 12 The resistive part may be written
{(dE/dt)(t)} = Re
1
2U G J ∗
, while the reactive part (which vanishes on average) is given by
Im
1
2U G J ∗
.
Further details on alternating-current theory can be found in many standard textbooks on applied electromagnetism
Trang 420 Maxwell’s Equations II:
Electromagnetic Waves
20.1 The Electromagnetic Energy Theorem;
Poynting Vector
The Poynting vector which is defined as
has the meaning of “energy current density”:
Firstly we have the mathematical identity
which can be proved using the relation
div[E × H] = ∂ i e i,j,k E j H k =
With the Maxwell equations
∂t
one then obtains the continuity equation corresponding to the conservation
of field energy:
∂t +H · ∂ B
i.e.,
divjenergy+∂wenergy
where
∂wenergy
∂t :=E · ∂ D
∂t +H · ∂ B
∂t
Trang 5164 20 Maxwell’s Equations II: Electromagnetic Waves
is the (formal)1 time-derivative of the energy density and −j · E describes
the Joule losses, i.e., one expects
j · E ≥ 0
(The losses are essentially sources of heat production, since the Ohmic be-havior,j = σ ·E (where σ is the specific conductivity) arises due to frictional
processes leading to energy dissipation and heat production due to scattering
of the carriers of the current, e.g., by impurities The case of vanishing losses
is called ballistic.)
For Ohmic behavior one has
κ E ,
where
κ = 1 σ
is the specific resistivity2, a constant property of the Ohmic material; i.e.,
−j · E = − E2
κ .
In the absence of electric currents the electromagnetic field energy is con-served If the material considered shows Ohmic behavior, the field energy decreases due to Joule losses.
At this point we shall just mention two further aspects: (i) the role of the Poynting vector for a battery with attached Ohmic resistance The Poynting vector flows radially out of the battery into the vacuum and from there into the Ohmic load Thus the wire from the battery to the Ohmic resistance
is not involved at all, and (ii) Drude’s theory of electric conductivity This theory culminates in the well-known formula
j(ω) = σ(ω)E(ω) ,
where the alternating-current specific conductivity is given by
σ(ω) ≡ σ(0)/(1 + iωτ) , with σ(0) = n V e2τ /me.
Here n V is the volume density of the carriers, i.e., typically electrons; meis the
electron mass, e the electron charge, and τ is a phenomenological relaxation
time corresponding to scattering processes
1
In the case of a linear relation (e.g., betweenE and D or B and H) the derivative
is non-formal; otherwise (e.g., for the general relationD = ε0E + P ) this is only
formally a time-derivative
2 The Ohmic resistance R of a wire of length l and cross-section F made from material of specific resistivity κ is thus R = κ · ( l
)
Trang 620.2 Retarded Scalar and Vector Potentials I: D’Alembert’s Equation 165
20.2 Retarded Scalar and Vector Potentials I:
D’Alembert’s Equation
For given electromagnetic fieldsE(r, t) and B(r, t) one can satisfy the second
and third Maxwell equations, i.e., Gauss’s magnetic law, divB = 0, and
Faraday’s law of induction,
∂t , with the ansatz
∂t . (20.4) The scalar potential φ( r, t) and vector potential A(r, t) must now be
calculated simultaneously However, they are not unique but can be “gauged”
(i.e., changed according to a gauge transformation without any change of the
fields) as follows:
A(r, t) → A (r, t) := A (r, t) + gradf(r, t),
φ( r, t) → φ (r, t) := φ (r, t) − ∂f ( r, t)
Here the gauge function f ( r, t) in (20.5) is arbitrary (it must only be
differentiable) (The proof that such gauge transformations neither change
E(r, t) nor B(r, t) is again based on the fact that differentiations can be
permuted, e.g., ∂t ∂ ∂f ∂x =∂x ∂ ∂f ∂t.)
In the following we use this “gauge freedom” by choosing the so-called
Lorentz gauge :
divA + 1
c2
∂φ
After a short calculation, see below, one obtains from the two remaining
∂t, the so-called
d’Alembert-Poisson equations:
−
∇2− ∂2
c2∂t2
−
∇2− ∂2
c2∂t2
E andj B are the effective charge and current density, respectively These deviate from the true charge and true current density by polarization contributions:
and
j B(r, t) := j(r, t) + J(r, t)
∂ P
Trang 7166 20 Maxwell’s Equations II: Electromagnetic Waves
A derivation of the two d’Alembert-Poisson equations now follows a)
∂t E /ε0.
With
div∂ A
∂t =
∂
∂tdivA
and with the Lorentz gauge (20.6) we obtain the first d’Alembert-Poisson equation
b)
∂t → curlB − curlJ = μ0·
j + ε0
∂ E
∂t +
∂ P
∂t
;
→ curl curlA = μ0
μ0
+∂ P
∂t
+ μ0 ε0
∂ E
∂t .
One now inserts
curl curlA ≡ grad(divA) − ∇2A and E = −gradφ − ∂ A
∂t
and obtains with
ε0μ0= 1
c2 the gradient of an expression that vanishes in the Lorentz gauge The remaining terms yield the second d’Alembert-Poisson equation
In the next section (as for the harmonic oscillator in Part I) we discuss
“free” and “fundamental” solutions of the d’Alembert equations, i.e., with vanishing r.h.s of the equation (and ∝ δ(r)) Of special importance among
these solutions are planar electromagnetic waves and spherical waves.
20.3 Planar Electromagnetic Waves; Spherical Waves
The operator in the d’Alembert-Poisson equations (20.7)
:=
∇2− ∂2
c2∂t2
(20.10)
is called the d’Alembert operator or “quabla” operator.
Amongst the general solutions of the free d’Alembert equation
φ(r, t) ≡ 0
Trang 820.3 Planar Electromagnetic Waves; Spherical Waves 167 (also known simply as the wave equation) are right-moving planar waves of the kind
Here g(x) is a general function, defined on the whole x-interval, which must
be continuously differentiable twice g(x) describes the profile of the traveling wave, which moves to the right here (positive x-direction) with a velocity c.
A wave traveling to the left is described by
φ −(r, t) ≡ g(x + ct)
Choosing x as the direction of propagation is of course arbitrary; in general
we could replace x by k · r, where
ˆ
k := k/|k|
is the direction of propagation of the planar wave
All these relations can be evaluated directly from Maxwell’s equations In particular, it is necessary to look for the polarization direction, especially for
the so-called transversality The first two Maxwell equations,
imply (if the fields depend only on x and t) that the x-components E x and
B x must be constant (i.e., = 0, without lack of generality) Thus, only the! equations
∂ E
∂t
remain (i.e.,≡ 1
c2
∂ E
∂t), which can be satisfied by
with one and the same arbitrary profile function g(x) (For an electromagnetic wave traveling to the left one obtains analogously E ≡ g(x + ct)e y and
For electromagnetic waves traveling to the right (or left, respectively) the
propagation direction ˆ k and the vectors E and cB thus form a right-handed
rectangular trihedron (in both cases!), analogous to the three vectors ±e x ,
e y and ±e z In particular, the amplitude functions of E and cB (in the cgs
system: those of E and B ) are always identical.
The densities of the electromagnetic field energy are also identical:
w E= ε0
2 E
2≡ w B:= B
2
2μ0 .
The Poynting vector
S := [E × H]
Trang 9168 20 Maxwell’s Equations II: Electromagnetic Waves
is related to the total field-energy density of the wave,
wtotal:= w E + w B ,
as follows:
S ≡ cˆk · wtotal,
as expected
Spherical waves traveling outwards,
Φ+(r, t) := g(t − r
c)
for r > 0 are also solutions of the free d’Alembert equation This can easily
be seen from the identity
∇2f (r) = r −1d2(r · f(r))
dr2 .
If the singular behavior of the function 1/r at r = 0 is again taken into
account by exclusion of a small sphere around the singularity, one obtains from the standard definition
V →0
1
V
∂V
v · nd2A
for sufficiently reasonable behavior of the double-derivative ¨g(t) the following
identity:
g(t −
r
c)
∇2− 1
c2
∂2
∂t2
g(t − r
c)
r ≡ −4πδ(r)gt − r
c
. (20.12)
This corresponds to the analogous equation in electrostatics:
∇21
r ≡ −4πδ(r)
As a consequence we keep in mind that spherical waves traveling outwards,
Φ+(r, t) := g
t − r c
are so-called fundamental solutions of the d’Alembert-Poisson equations (The corresponding incoming spherical waves, Φ − (r, t) := g(t+
r)
r , are
also fundamental solutions, but in general they are non-physical unless one
is dealing with very special initial conditions, e.g., with a pellet bombarded from all sides by intense laser irradiation, which is performed in order to initiate a thermonuclear fusion reaction.)
Trang 1020.4 Retarded Scalar and Vector Potentials II 169
20.4 Retarded Scalar and Vector Potentials II:
The Superposition Principle with Retardation
With equation (20.12) we are now in a position to write down the explicit
solutions of the d’Alembert-Poisson equations (20.7), viz,
φ( r, t) =
dV E
r , t − |r−r c | 4πε0 |r − r | ,
A(r, t) =
dV μ0j B
r , t − |r−r c |
In principle these rigorous results are very clear For example, they tell
us that the fields of single charges and currents
a) on the one hand, can be simply superimposed, as in the static case with
Coulomb’s law, while
b) on the other hand, the retardation between cause and effect has to be taken into account, i.e., instead of t = t (instantaneous reaction) one has
to write
t = t − |r − r |
i.e., the reaction is retarded, since electromagnetic signals betweenr and
r propagate with the velocity c.
c) Huygens’s principle3 of the superposition of spherical waves is also con-tained explicitly and quantitatively in equations (20.13)
d) According to the rigorous result (20.13) the mutual influences propagate
at the vacuum light velocity, even in polarizable matter.
(This does not contradict the fact that stationary electromagnetic waves in
dielectric and permeable matter propagate with a reduced velocity (c2 →
c2/(εμ)) As with the driven harmonic oscillator (see Part I) these stationary waves develop only after a finite transient time The calculation of the
tran-sition is one of the fundamental problems solved by Sommerfeld, who was Heisenberg’s supervisor.)
The explicit material properties enter the retarded potentials (20.13) only through the deviation between the “true charges” (and “true currents”) and the corresponding “effective charges” (and “effective currents”) One should
remember that in the rigorous equations (20.13) the effective quantities enter Only in vacuo do they agree with the true quantities.
3
See section on optics