Chapter 1 Maxwell’s Equations, Conservation Laws
1.1 Maxwell’s Equations in Materials: D, H, P and M
Our starting point will be the experimentally deduced Maxwell’s Equations These consist of the four differential equations (in gaussian, cgs units)
uG +u>w,@7 3 +u>w, (1a) uE +u>w,@3 (1b) 4 CE +u>w, uH +u>w,@ f Cw (1c) 4 CG +u>w, 7 uK +u>w,@ . M +u>w, f Cw f 3 (1d) In S.I units the equations are (see Physics 411 notes):
uG +u>w,@ 3 +u>w, (2a-2d) uE +u>w,@3 CE +u>w, uH +u>w,@ Cw CG +u>w, uK +u>w,@ . M +u>w, Cw 3 for the displacement ¿eld D+u>w, > the magnetic induction B+u>w, > the electric ¿eld E+u>w, > and the magnetic ¿eld H+u>w, =
The sources of the ¿elds are the charge density 3 +u>w, and the charge current density J+u> w, = Conservation of charge requires that the charge density and charge current density satisfy C u M +u>w,. +u>w,@3= 3 Cw 3 (3) This set of equations require information concerning the properties and responses of the materials in the region of the ¿elds. These are isolated in the constituent equations G +u>w,@H +u>w,.7 S +u>w, gaussian units (4a) K +u>w,@E +u>w, 7 P +u>w, gaussian units (4b)
G +u>w,@rH +u>w,.S +u>w, S.I units (5a) 4 K +u>w,@ E +u>w, P +u>w, S. I. units (5b) r where P+u>w, is the polarization ¿eld and M+u>w, is the magnetization ¿eld for the materials. u S +u> w,@ s +u>w, the 1
Section 1.1 Maxwell’s Equations in Materials: D, H, P and M
Now uMpdw . C pdw @Cw @3therefore u^Mpdw Ms`@3= Since its divergence vanishes the current density ^Mpdw Ms` can be written as the curl of a vector ^Mpdw Ms`@fuP +u> w, = P +u>w, is identi¿ed as the magnetization of the materials. In terms of P and S the Ampere-Maxwell law is in gaussian units:
4 CH +u>w, 7 7 CS +u>w, uE +u>w,@ . M +u>w,. .7 uP +u>w, f Cw f 3 f Cw (9) 4 C^H +u>w,.7 S +u>w,` 7 u^E +u>w, 7 P +u>w,` @ . M +u>w, f Cw f 3 becomes 4 CG +u>w, 7 uK +u>w,@ . M +u>w, f Cw f 3 or Eq. (1d )
which is consistent with the constituent equations (Equations 4 and 5).
In gaussian (cgs) units the Lorentz force law is
y I @ t H . E f gaussian (10) I @ t +H . y E, SI units which indicates that E, B, D, and H have dimensions of charge per length squared and that (charge/length)5 has the dimensions of force (dyne).
Outline of topics : a) We will ¿rst investigate some general properties of the equations. Symmetry properties and conservation laws for the ¿elds will be examined. The solutions of the equations in a uniform isotropic medium will be developed using scalar and vector potentials. And ¿nally the wave equation in a uniform, isotropic medium will be obtained. (Chapter 6) b) Next we will investigate the propagation of electromagnetic waves in various systems. This will include the propagation of waves in dissipative and non-uniform systems. We will also consider the conditions placed by causality on the response of a system to applied electromagnetic ¿elds. (Chapter 7 and 8) c) The interaction of the electromagnetic waves and matter will be investigated through the scattering of the waves and through radiation by simple systems, oscillating electric and magnetic dipoles. (Chapter 9) d) The properties of charge plasmas and electromagnetic waves in plasmas will be studied. (Chapter 10) e) Finally we will consider in more detail phenomenon involving moving charged particles. This will encompass collisions between charged particles, the energy loss and scattering (Chapter 13), and the potentials due to moving charges. The latter will involve both uniform motion and accelerated motion. (Chapter 14).
Unless stated otherwise, we shall be using gaussian units. 3 Section 1.2 Field Energies 1.2 Field Energies
From our experience with mechanical systems we deduce that the power density supplied to the charged particles of a charge current J3 +u>w, in an electric ¿eld E+u>w, is s +u>w,@H +u>w, M3 +u>w, = From Equations 1a and 1d and using the following identity
u ^H +u>w, K +u>w,` @ H +u>w, u K +u>w,.K +u>w, u H +u>w, (11) we obtain, f 4 C C u ^H +u>w, K +u>w,` . K +u>w, E +u>w,.H +u>w, G +u> w, . H +u>w, M +u>w,@3 7 7 Cw Cw 3 (12)
This has the form of a conservation law, . C u V +u>w,. x +u>w,.H +u>w, M +u>w,@3 Cw 3 (13) where the energy current density can be identi¿ed as f V +u>w,@ ^H +u>w, K +u>w,` 7 (14) and the time rate of change of the energy density stored in the ¿elds and the medium in the region of the ¿elds is C 4 C C x +u>w,@ K +u>w, E +u=w,.H +u>w, G +u> w, = Cw 7 Cw Cw (15)
It is interesting to examine in detail the rate of change of the energy density. We ¿rst consider the energy density associated with the electric ¿eld, using G +u>w,@H +u>w,.7 S +u>w, . C 4 C x +u>w,@ H +u>w, G +u>w, Cw H 7 Cw (16a) 4 C C @ H +u>w, H +u>w,.H +u>w, S +u>w, 7 Cw Cw (16b)
C As previously noted u S +u>w,@ s +u>w, >the polarization charge density and CwS +u>w, has been identi¿ed as the
polarization charge current density Js +u>w, = The second term in Eq. 16b is then the power going into the ‘mechanical’ energy of the system. The relationship between the electric ¿eld and the polarization is generally very interesting. In the linear response approximation [6 ]]]] +H, 3 3 3 3 6 3 3 Sl +u>w,@ "lm +u u >w w , Hm +u >w, g u gw (17) m@4 where "+H, +u>w, is the electric susceptibility tensor for the system. The energy density associated with the magnetic ¿eld is C 4 C x +u>w,@ K +u>w, E +u>w, = Cw P 7 Cw (18) There are two points of interest. First, the magnetic force ‘does no work’ (in the Lorentz force law B is perpendicular to gu@gw) and, second, the time derivative operates on the source of the force, the magnetic induction vector B. Using Faraday’s 4 Section 1.2 Field Energies law and the relationship K +u>w,@E +u>w, 7 P +u>w,, C 4 C C x +u>w,@ E +u>w, E +u>w,.P +u>w, E +u>w, Cw P 7 Cw Cw (19) 4 C @ E +u>w, E +u>w,.P +u>w, +fuH +u>w,, 7 Cw 4 C @ E +u>w, E +u>w,.fu^H +u>w, P +u>w,` . fH +u>w, uP +u>w, 7 Cw 4 C @ E +u>w, E +u>w,.fu^H +u>w, P +u>w,` . H +u>w, ^M M ` 7 Cw pdw s (19) The ¿rst term in Eq. 19 is the power transfer to the magnetic induction, the second term is the divergence of an energy Àux associated with the magnetization of the material, and the last term is the energy supplied to the rotational current density by the electric ¿eld generated by the time varying magnetic induction vector. In terms of the total charge current density, externally controlled (measured) and material response, the time rate of change of the total energy density change is C 4 C C 4 C x +u>w,@ H +u>w, H +u>w,.H +u>w, S +u>w,. E +u>w, E +u>w,. Cw 7 Cw Cw 7 Cw (20) fu^H +u>w, P +u>w,` . H +u>w, ^Mpdw Ms` 4 C @ ^H +u>w, H +u>w,.E +u>w, E +u>w,` . ; Cw C H +u>w, S +u>w,.^M M ` . fu^H +u>w, P +u>w,` Cw pdw s 4 C k l @ mE +u>w,m5 . mH +u>w,m5 . H +u>w, ^M .^M M `` . ; Cw s pdw s fu^H +u>w, P +u>w,`
and the conservation of energy equation is written as f 4 C k l 3@ u ^H +u>w, +E +u>w, 7 P +u>w,,` . mE +u>w,m5 . mH +u>w,m5 . 7 ; Cw (21) fu^H +u>w, P +u>w,` . H +u>w, ^Mpdw . M3`
f 4 C k l u ^H +u>w, E +u>w,` . mE +u>w,m5 . mH +u>w,m5 . H +u>w, M +u>w,@3 7 ; Cw wrwdo (22) This is the microscopic equation for the conservation of energy known as Poynting’s theorem. The transfer of energy between the ¿elds and the mechanical system is carried out by the interaction of the electric ¿eld and the charge current density. The energy transferred to the current can reside in the translational kinetic energy of the system, the potential energy of interaction of the particles of the system, or thermal energy. We will return to this equation as we consider models of systems.
Exercise 1: Using the following table, convert Poynting’s theorem into SI units: formals conversions formal conversionss BVL s7 @3 @Bjdxvv BVL @Bjdxvv s3@7 EVLs7 3 @Ejdxvv EVL @Ejdxvv@s7 3 HVLs7 3 @Hjdxvv HVL @Hjdxvvs@ 7 3 DVL s7 @3 @Djdxvv DVL @Djdxvv s3@7 JVL@^7 7 @3`@Jjdxvv JVL @Jjdxvv 7 7 @3
5 Section 1.3 Vector and scalar potentials 1.3 Vector and scalar potentials
In the following we will restrict our investigation to the microscopic form of Maxwell’s equations. We start with the two source free equations. First we consider gauss’ law for the magnetic induction, uE @3. In this case a vector potential, D +u>w, > can be de¿ned such that the magnetic induction vector is given by
E +u>w,@uD +u>w, = (23)
The vector potential is then used in Faraday’s law yielding 4 C u H +u>w,. D +u>w, @ 3 f Cw (24)
It follows that we can ¿nd a scalar potential, ! +u>w, > such that 4 C H +u>w,. D +u>w,@u! +u>w, = f Cw (25)
Gauss’ law for the electric ¿eld and Ampere-Maxwell law now provide the equations which the vector and scalar potential must satisfy. First we consider Gauss’ law and obtain 4 C u5! +u>w,. uD +u>w,@7 +u>w, f Cw wrwdo (26) while the Ampere-Maxwell law 4 C 7 u^uD +u>w,` @ H +u>w,. M +u>w, f Cw f wrwdo 4 C 4 C 7 u^uD +u>w,` . D +u>w,.u! +u>w, @ M +u>w, f Cw f Cw f wrwdo ((27)) 4 C 4 C 7 u+uD +u>w,, u5D +u>w,. D +u>w,.u! +u>w, @ M +u>w, f Cw f Cw f wrwdo 4 C 4 C5 7 u5D +u>w,.u^uD +u>w,. ! +u>w,` . D +u>w,@ M +u>w, f Cw f5 Cw5 f wrwdo (27b)
Equations (26) and (27b )are coupled second order partial differential equations. Since the ¿elds only depend on derivatives of the potentials there is freedom to chose the potentials so as to provide a convenient form for the equations for the potentials. The addition of a term u +u>w, to the vector potential does not change the magnetic induction ¿eld. In this case we obtain the new vector potential
D3 +u>w,@D +u>w,.u +u>w, (28)
Consider now Eq. (26). 4 C 4 C H +u>w,. D3 +u>w, u +u>w,@u! +u>w, f Cw f Cw (29) Changing the vector potential will change the electric ¿eld unless the scalar potential is also changed 4 C !3 +u>w,@! +u>w, +u>w, f Cw (30) 6 Section 1.4 Lorentz and Coulomb gauges
Equations (28) and (30) constitute a gauge transformation of the potentials. Fields which are invariant under the transformation speci¿ed in Equations (28) and (30) are said to be gauge invariant. One possible requirement on the new potentials is that 4 C uD3 +u>w,. !3 +u>w,@3 f Cw (31)
Given potentials A and ! which do not satisfy this condition we chose to be a solution of the equation 4 C5 4 C u5 +u>w, +u>w,@u D +u>w, ! +u>w, > f5 Cw5 f Cw (32) i.e., satis¿es the wave equation with the source determined by the potentials we wish to modify. In the case that the potentials are related by Eq. (31) the Equations (26) and (27) are uncoupled and have the form 4 C5 u5!3 +u>w, !3 +u>w,@7 +u>w, f5 Cw5 wrwdo (33a) 4 C5 u5D3 +u>w, D3 +u>w,@7 M +u>w, f5 Cw5 wrwdo (33b)
Both the scalar and vector potentials satisfy a wave equation with the respective sources the charge density and the charge current density.
1.4 Lorentz and Coulomb gauges
3 Equation (31) is called the Lorentz condition.IfD3 and ! satisfy the Lorentz condition and is a solution of the homogeneous wave equation 4 C5 u5 +u>w, +u>w,@3 f Cw5 (34) then
D33 +u>w,@D3 +u>w,.u +u>w, (35a) 4 C !33 +u>w,@!3 +u>w, +u>w, f Cw (35b) de¿nes new vector and scalar potentials which satisfy the Lorentz condition and give the same ¿elds. The class of all such combinations of vector and scalar potentials related by this restricted gauge transformation belong to the Lorentz gauge. Choosing potentials which satisfy the Lorentz condition not only uncouples the equations for the potentials but also yields equations which are invariant under the Poincaré transformations., i.e., the set of space-time translations and Lorentz transformations. Another gauge which is often is in radiation theory is the Coulomb, transverse, or radiation gauge. The gauge condition is
uD +u>w,@3 (36)
In this gauge the potentials satisfy the equations
u5! +u>w,@7 +u>w, (37a) wrwdo 4 C5 7 4 C u5D +u>w, D +u>w,@ M +u>w,. u ! +u>w, f5 Cw5 f wrwdo f Cw (37b) 7 Section 1.4 Lorentz and Coulomb gauges
Taking the divergence of Eq. (36) and using
5 l m 5 u u D +u>w,@ l{alu mu umD +u>w,@u u D +u>w,@3 (38) we ¿nd that
4 C5 +u5 ,uD +u>w,@3 f5 Cw5 (39) and
4 C u M +u>w,. u5! +u>w, @3= wrwdo 7 Cw
If we separate Mwrwdo into a transverse and longitudinal part, Mwrwdo @ Mw . Mo>
4 C u ^M . M ^u! +u>w,` @ 3= w o 7 Cw
with
4 C M @ u ! +u>w, uM +u>w,@3 o 7 Cw so that o (40)
we have
uMw +u>w,@3 (41b)
8 Section 1.4 Lorentz and Coulomb gauges
Mathematical aside: Let J+u>w,@Mw +u>w,.Mo +u>w, with uMw +u>w,@3and uMo +u>w,@3= The vector identity
u+uM +u>w,, @ u +uM +u>w,, u5M +u>w, (42a) u5M +u>w,@u +uM +u>w,, u+uM +u>w,, @ I+u>w, together with the Green’s function for Laplace’s equation
5 u J +u> u3>w,@+u u3, (42b) 4 4 J +u> u3>w,@ 7 mu u3m (42c)
with boundaries at in¿nity gives
]]] M +u>w,@ I+u>w,J +u> u3>w, g6u3 (43a) ]]] ]]] 4 u3 u3 M +u3>w, 4 u3 u3 M +u3>w, @ g6u3 . g6u3 7 mu u3m 7 mu u3m ]]] ]]] 4 u3 M +u3>w, 4 u3 M +u3>w, @ u g6u3 . u g6u3 7 mu u3m 7 mu u3m (43b)
3 3 One can obtain Eq. 43b directly from the Helmholtz theorem by noting that u M +u3>w,@i and u M +u3>w,@z provide the trivial sources for
]]] ]]] 4 i +u3>w, 4 z+u3>w, @ g6u3 N @ g6u3 7 mu u3m and 7 mu u3m (43c) where M @ u .uN (43d)
The result holds if the integrals are over all space and olpu$4 i @3and olpu$4 mzm @3= From Eq. 43b the longitudinal and transverse parts of the current density, respectively, are identi¿ed as
]]] 4 u3 M +u3>w, M +u>w,@ u g6u3 o 7 mu u3m (44a) ]]] 4 u3 M +u3>w, M +u>w,@ u g6u3 w 7 mu u3m (44b)
9 Section 1.4 Lorentz and Coulomb gauges
It follows that
Mw +u>w,@Mwrwdo +u>w, Mo +u>w, (45)
and Eq. (37b) becomes
4 C5 7 u5D +u>w, D +u>w,@ M +u>w, f5 Cw5 f w (46)
As in the case of the Lorentz gauge there is a continuum of potentials which yield the same ¿elds. These potentials are connected by "’s satisfying u5" +u>w,@3= As suggested by the name ‘radiation gauge’, potentials satisfying this gauge condition are useful in describing the propagation of radiation in the absence of sources. In that case ! +u>w,@3and
4 C H +u>w,@ D +u>w, > f Cw (47a) E +u>w,@uD +u>w, = (47b)
In the case that the charge density has non-zero values the solution to the scalar potential equation, Eq. (37), in the Coulomb gauge is
]]] +u3>w, ! +u>w,@ wrwdo g6u3 mu u3m (48)
We note that the scalar potential in this gauge responds instantaneously, at each point in space, to any temporal variation in the charge density. This can be compared with the equivalent description furnished by working in the Lorentz gauge. In the Lorentz gauge we will ¿nd that the effects of a temporal variation in the sources propagate through the potentials with a ¿nite speed. 10 Section 1.4 Lorentz and Coulomb gauges
...... Assignment 1: Obtain the transverse and longitudinal components of the current densities:
q k d dl r d, m +u>w,@Uhhà l$w l } . } +{, +|, n 3 5 5
à l$w a e, m +u>w,@ Uhh l3 +}, + e, !> cylindrical coordinates
(c) For each current density obtain the related charge density (charge conservation) and the potential which vanishes at points in¿nitely far from the charges. (d) Verify that Eq. (40) is correct.
...... 11 Section 1.5 TheGreen’sfunctionforthewaveequation 1.5 The Green’s function for the wave equation The wave equation, with sources, has the general form 4 C5 u5 +u>w, +u>w,@7 I +u>w, f5 Cw5 (49) One approach to the solution of this equation is to work with the space-time Fourier transforms of the ¿eld and source. The space-time functions are related to the wave vector - angular frequency functions by ]]]] 4 +u>w,@ # +n>$,h{s+l ^n u $w`, g6ng$ (50a) +5 ,7 ]]]] 4 I +u>w,@ i +n>$,h{s+l ^n u $w`, g6ng$ (50b) +5 ,7
]]]] # +n>$,@ +u>w,h{s+ l ^n u $w`, g6ugw (51a) ]]]] i +n>$,@ I +u>w,h{s+ l ^n u $w`, g6ugw (51b)
...... whereweusetherelationship ] 4 4 hÃlxygy @ +x, (51c) 5 Ã4 ...... Assignment 2: Calculate the fourier transform of the Coulomb potential t +u> u3,@ mu u3m ......
The Fourier transform of Eq. (49) converts it into an algebraic equation. ]]]] 4 C5 4 ^u5 ` # +n>$,h{s+l ^n u $w`, g6ng$ f5 Cw5 +5 ,7 ]]]] 4 @ 7 i +n>$,h{s+l ^n u $w`, g6ng$ +5 ,7 or ]]]] 4 C5 ^# +n>$,+u5 ,.7 i +n>$,` h{s + l ^n u $w`, g6ng$ @3 f5 Cw5 ]]]] $5 ^# +n>$,+n n . ,.7 i +n>$,` h{s + l ^n u $w`, g6ng$ @3 f5
Since each h{s + l ^n u $w`, is linearly independent, the coef¿cients must all be zero. The solution to the algebraic equation is i +n>$, # +n>$,@7 = (52) n5 +$@f,5
12 Section 1.5 TheGreen’sfunctionforthewaveequation
The problem is now reduced to taking the inverse transform. There is at least one dif¿culty. The integrand will have singularities at $ @ f mnm = To understand the source of this dif¿culty and to determine how it is handled we consider the Green’s function for the wave equation.. The Green’s function will solve Eq. (49) with the source term given by ]]]] 4 I +u>w,@+6, +u u3, +w w3,@ h{s l n +u u3, $+w w3, g6ng$ ^5 `7 (53)
One can ¿nd the j +n>$, using the same method as above: ]]]] 4 C5 ^j +n>$,+u5 ,.7 h{s l n u3 $w3 `h{s+l ^n u $w`, g6ng$ @3 f5 Cw5 ]]]] $5 ^j +n>$,+n n . ,.7 h{s l n u3 $w3 `h{s+l ^n u $w`, g6ng$ @3 f5
h{s l n u3 $w3 j +n>$,@7 (54) n5 +$@f,5
In terms of space-time coordinates, then the Green’s function for the wave equation is ]]]] 4 4 h{s + l ^n +u u3, $ +w w3,`, J +u u3>w w3,@ g$g6n 6 5 5 (55) 7 Ã4 n +$@f,
The integrand has singularities at $ @ fn (or n @ $@f,. How these singularities are handle depends on the boundary conditions imposed on the system. The basic approach involves the Cauchy’s Residue Theorem.
Mathematical aside: Cauchy’s Residue Theorem wk wk Assume an analytic function I +},, } @ { . l|> has an m order pole at } @ }3. A function, I +}, > has an p order pole at } @ }3 if in the neighborhood of }3 it has an expansion [4 qÃp I +},@ dq +} }3, (56) q@3
Then if I +}, is integrated counter-clockwise around a contour enclosing }3 we obtain the residue of I +}, at } @ }3> L
I +}, g} @5 l uhv +}3, (57) counter clockwise In the case of an mwk order pole 4 gpÃ4 i+} } ,p I +},j uhv +} ,@ 3 3 +p 4,$ g}pÃ4 (58) }@}3
Note that the residue is always the coef¿cient, dpÃ4+}3,, of the simple pole term in the expansion of I+}, d +} , d +} , uhv+} , d +} , I+},@ r 3 . 4 3 . === 3 . p 3 . ======p pÃ4 4 3 (59) +} }3, +} }3, +} }3, +} }3, No other term contributes to the contour integral.
The $ integral’s integrand has two ¿rst order poles but it is an integral along the real $ axis. 13 Section 1.5 TheGreen’sfunctionforthewaveequation
]]] 4 h{s + l n +u u3,, J +u u3>w w3,@ g6n 7 6 5n (60) ] .4 h{s+l$ +w w3,, h{s++l$ +w w3,, f g$ Ã4 $ . fn $ fn
In order to apply Cauchy’s theorem we must have a closed path, so we write J as follows,
]]] L ] h{s+l$ +w w3,, h{s+l$ +w w3,, J @ i+n, g6n g$ g$` [ 5 5 5 5 . ((61)) forvhg $ +fn, vhplÃflufoh $ +fn,
We will try to close the path with a semi-circle, with in¿nite radius, in either the upper half complex $ plane or the lower Ã5 complex $ half plane. The denominator will cause the integrand to vanish, on either semi-circle, as m$m for large m$m if the numerator is well behaved. It suf¿ces to check the numerator on the imaginary $ axis, i.e., for $ @ l$l== The value is determined by the exponential
3 3 h{s ^l$+w w ,`@h{s^$l +w w ,` = (62)
3 3 3 This diverges for m$lm$4if $l +w w , A 3 and vanishes exponentially with $l for $l +w w , ? 3= For w?w we will close the path in the upper half complex $ plane while for wAw3 the path will be closed in the lower half complex $ plane. We now know how to close the path of integration but because the poles lie on the real axis the integral is not well-de¿ned. That is, it is an improper integral. The ambiguity provides the freedom to impose temporal boundary conditions on the Green’s function... One approach distorts the path so that it avoids the poles. It that case there are four possible paths. The pertinent paths for our problems are shown in Figure 1.
.Figure 1. Location and direction of the contours CU and CD relative to the real $ axis (center line) Contour CU is completed below the real axis and contour CD is completed above the real axis so that in each case the poles at fn are inside the closed contour.
It suf¿ces to ¿rst consider the $ or angular frequency part of the integrand
L h{s+l$ +w w3,, h{s++l$ +w w3,, L +F> n,@ g$ (63) F $ . fn $ fn where F is the path of integration. If the path FU inFigure1isusedweobtain 14 Section 1.5 TheGreen’sfunctionforthewaveequation
3 L +FU>n,@5 l+w w ,^uhv+fn, uhv+.fn,` (64) @5 l+w w3,^h{s+l+fn,+w w3,, h{s++lfn +w w3,,` @ 7 +w w3,vlq+fn+w w3,,= (64b)
The Green’s function obtained with this path choice will be labelled JU=
]]] f h{s + l n +u u3,, J +u u3>w w3,@+w w3, vlq+fn+w w3,,g6n U 5 5 n (65)
Now we do the k integration in spherical n space, letting +u u3, be along the a} direction: ]]] f h{s + lnmu u3m frv , J @+w w3, n vlq+fn+w w3,,n5gng! g+ frv , U 5 5 n n n (66) ]] f5 4 h{s + lnmu u3mx, @+w w3, vlq+fn+w w3,,n5gngx 5 5 n ] Ã4 f 4 h{s + lnmu u3m, h{s + lnmu u3m, @+w w3, vlq+fn+w w3,,n5gn 5 3 3 ln mu u m ] 4 3 f 3 3 @+w w , 3 h{s + lnmu u m,vlq+fn+w w ,,gn lmu u m Ã4 ] 4 3 f 4 3 3 3 @+w w , 3 h{s + lnmu u m, ^h{s+lfn+w w ,, h{s+lfn+w w ,,`gn mu u m Ã4 5 ] 4 3 f 3 3 3 3 @+w w , 3 ^h{s + ln^mu u m . f+w w ,`, h{s + ln^mu u mf+w w ,`,`gn 5 mu u m Ã4
Retarded Green’s function: Finally, we have (using Eq. 51c):
f J +u u3>w w3,@ f+w w3,^+mu u3m . f+w w3,, +mu u3mf+w w3,,` U mu u3m (67a) f @ +w w3,+mu u3mf+w w3,, mu u3m (67b)
In the last step the additional term involving +mu u3m . f +w w3,, was dropped since the argument of this delta 3 function could not vanish for wAw. The Green’s function JU is called the retarded Green’s function. It gives a disturbance (created at time w3) propagating outward from a source point u3. The condition +w w3, A 3 is often interpreted as a causality condition. That is, the disturbance is detected at a time, w, which must be larger than the creation time, w3, at the source. The Green’s function is also called a propagator as it ”propagates” the disturbance from point +u3>w3, to the point +u>w, within the integral expression for the solution to the wave equation. Advanced Green’s Function:
3 The Green’s function obtained using the path FD is similar, but corresponds to w?w. Note that one must repeat all the steps leading to Eq. (67b), but use the contour FD, rather than FU= 15 Section 1.5 TheGreen’sfunctionforthewaveequation
f J +u u3>w w3,@+w3 w, +mu u3m . f +w w3,, D mu u3m (68)
This is the advanced Green’s function and corresponds to the time reversed retarded Green’s function. The interpretation of this Green’s function is that the disturbance is ‘converging on its source’ Solution to the Wave Equation:
The Green’s function solution of Eq. (49) is thus given by
]]]] 3 3 3 3 6 3 3 +u>w,@ k +u>w,. JU +u u >w w , I +u >w, g u gw (69a) ]]]] f @ +u>w,. +w3 w, +mu u3mf +w w3,, I +u3>w3, g6u3gw3 k mu u3m ]]]] f 4 mu u3m @ +u>w,. +w3 w, w3 ^w ` I +u3>w3, g6u3gw3 k mu u3m f f ]]] mu u3m 4 @ +u>w,. I u3>w g6u3 k f mu u3m (69b)
where k +u>w, is a solution of the homogeneous wave equation.
Normally the Green’s function solution would have ”surface integral terms” evaluated at u3 @ 4 and at w3 @ 4. They would be of the form:
] 4 ]] ]]] 3 3 3 3 3 C C g+fw , ^ u J Ju ` gV . gY ^ J J ` 3 = 3 3 w $Ã4 (70) Ã4 u3$4 Cfw Cfw
The surface integrals do not contribute since both the solution and the Green’s function vanish as u3 $4.The time derivatives at w3 @ 4. do not appear because of the +w w3, in the Green’s function and at w3 @ 4 one assumes that there is no time derivative of the source signal in the Green’s function, and no time derivative of the ”wave” function, .The w3 @ 4 boundary condition has been taken care of by our choice of contour. In most applications of the Green’s function the disturbance is assumed to take place near w3 @3 andto”turnoff”atw3 ?? 4 In this case one can see that the boundary conditions in time are automatically taken care of.
16 Section 1.5 TheGreen’sfunctionforthewaveequation
...... Assignment 3: The ‘displacement’ of a damped, harmonic oscillator satis¿es the equation
5 { . {b . $3{ @ i +w, =
(a) Obtain the Green’s function for this equation. Note
3 3 5 3 3 j +w w ,.j +w w ,.$3j +w w ,@ +w w ,
(b) In the case that the ‘force’ is given by
i +w,@i3 ^ +w,.+w,h{s+w@ ,` and { +4,@3> {b +4,@3evaluate { +w, = (c) Let @3=4$3 and evaluate the energy (per unit mass) supplied by the force in the time interval from 4 to .4= Plot this energy, divided by the ¿nal energy stored by the oscillator, as a function of oq +$3 , from $3 @3=34 to $3 @433=
Assignment 4: Jackson Problem 6.20 ...... 17 Section 1.6 Macroscopic equations measurements 1.6 Macroscopic equations measurements The macroscopic equations involve expressing all or a subset of the response of the system to external ¿elds in terms of observed electric and magnetic susceptibilities .[An interesting development of the macroscopic equations is given by Mazur and Nijboer, Physica 19, 971 (1953).] The basic problem is the determination of the proper equations to use to describe a given phenomenon. Experience tells us that the answer generally depends on the relationships between the characteristic lengths of the system. Fields which have wavelengths approximating the characteristic lengths of the system will probe the detailed structure of the material. The principal example would be the x-ray diffraction patterns generated by the scattering of x-rays by the electron density of the system. If the ¿elds have wavelengths which are large compared to the characteristic lengths of the system we expect to use the macroscopic equations and tabulated electric and magnetic susceptibilities. Of course, if we are interested in the ¿elds on the molecular level it would not make sense to use the bulk susceptibilities.. However, we often use tabulated atomic susceptibilities to describe the interaction of an atom with an applied ¿eld. We can qualitatively investigate the transition from microscopic to macroscopic by considering an atomic gas. Take an atomic gas at a ¿xed number density, q3, and temperature, T3> such that its properties are approximately those of 4@6 an ideal gas. The average atomic spacing will be g @ q3 = In the following discussion we will neglect resonance effects. An electromagnetic wave of a given (angular) frequency, $> is propagated through this gas. At high frequencies (high energy x-rays), very short wavelengths, the wave will propagate as if in a vacuum except for scattering by the electrons of the atoms. As the frequency decreases and the wavelength becomes long compared to the atomic size but short compared to the average interatomic spacing, the wave will still be propagating as in a vacuum. In this case the scattering will be atomic scattering and the atomic susceptibility can be used to obtain the scattered waves. As long as the wavelength of the waves is much shorter than the average interatomic distance, g> the electromagnetic interaction between the atoms should average to zero. For these wavelengths we expect that the phases of the scattered waves at any point, not in the forward direction, will be randomly distributed. We will consider the forward scattering below. For wavelengths of the order of the interatomic distance we obtain some coherence between the scattered waves. At these wavelengths the electromagnetic interaction between the atoms will start to contribute to the polarization of the atoms. In the long wavelength limit the wavelength of the wave is long compared to the interatomic distances. In this limit the susceptibility of the gas is approximated by the treating the ¿elds as if instantaneously static. The dominant scattering mechanism for the long wavelengths are Àuctuations in the atomic density which give rise to a spatial variation in the index of refraction of the gas. Macroscopic treatment of E and B: wavelengths : 100 Angstroms E/eV 1.2443Ã7 1.2443Ã5 1.24 12.4 124 1.24436 1.24437 1.24438 1.24439 1cm 10Ã5fp 10Ã7fp 1000 Å 100 Å 10 Å 1 Å 10Ã The waves scattered in the forward direction, the direction of propagation of the incident wave, are coherent with the incident wave. Conservation of energy requires that the forward scattered wave interfere destructively with the incident wave so as to account for the energy lost to the scattered wave (This is the optical theorem of scattering theory.). This results in an exponentially decreasing wave amplitude in the forward direction. In addition the wavelength of the superposition of the incident wave and the forward scattered wave will be less than the vacuum wavelength of the wave. Derivation of Macroscopic D and H ¿elds from Microscopic Properties In this approach we consider material with 4356 atoms (or molecules) per cm6 and make the plausible assumption that a region of the material to be treated macroscopically is of the order of 433 Å U in dimension. In a volume U6 there are approximately 438 atoms and the charge constituents (electrons, nuclei) can be treated as point charges, t. (If one prefers to view this quantum mechanically, the averages would be substituted for the charge.) There will be free charges, tm,atum and charges associated with the qwk molecule (or atom), tmq at umq= We shall be interested in the averages of the microscopic electric ¿eld, h+u>w,, and of the microscopic magnetic induction ¿eld, e+u>w, at the point u at time, w= The u is understood to represent a position vector in a much larger region of space (of the order of fp6 )at which the macroscopic electric ¿eld H+u>w,, and the macroscopic magnetic induction ¿eld, E+u>w, are set equal to the average of the microscopic ¿elds over the volume U6: 18 Section 1.6 Macroscopic equations measurements ]]] H+u>w, ? h+u>w, A@ i+u3,h+u u3>w,g6{3 (71a) ]]] E+u>w, ? e+u>w, A@ i+u3,e+u u3>w,g6{3 (71b) Time averaging is (as Jackson argues) not meaningful since the period associated with a @433Å light wave (43Ã4:v .) is close to the period of the electronic charge vibrations. Also, unless the system is being driven by the external ¿eld, the atomic electronic (and nuclear) charge oscillations are not correlated and provide no net effect. The space averaging described in Eq. 71 is done instantaneously. The exact form of the distribution function, i+u3,, is not crucial and it is convenient to use 4 u5 i+u3,@ h{s^ ` ^ U6`6@5 U5 where (72) ]]] 4@ i+u3,g6{3 This function has the advantage that the distribution falls off smoothly at the edge of the volume. It is clear that ]]] u ? h+u>w, A@ i+u3,uh+u u3>w,g6{3 @? uh+u>w, A (73) ]]] C C C ? h+u>w, A@ i+u3, h+u u3>w,g6{3 @? h+u>w, A Cw Cw Cw The (averaged) microscopic Maxwell’s equations are: ? urh+u>w, A@? mtm+u um, A . q m ?tmq+u uq umq, A (74a) ? ue+u>w, A@3 (74b) C ? u{h+u>w, A . ? e+u>w, A@3 Cw (74c) C gu g+u u , ? u{e+u>w, A ? h+u>w, A@ ^? t m +u u , A . ?t q mq +u u u , A` f5Cw r m m gw m q m mq gw q mq (74d) The Displacement Field, D(r,t): The total charge density can be written: ? +u>w, A@? t +u u , A . ? t +u u u , A ]]] m m m q m mq q mq (75) 3 3 3 6 3 @ i+u ,^ mtm+u um u ,. q m ?tmq+u uq umq u ,`g { @ mtmi+u um,. q mtmqi+u uq umq, Next, we expand the i+u uq umq, in a Taylor series about the point u uq: 19 Section 1.6 Macroscopic equations measurements mtmqi+u uq umq,@ mtmq h{s^umq u`i+u uq, (76) 4 @ t ^4 +u u,. +u u,+u u,.===`i+u u , m mq mq 5$ mq mq q @ mtmqi+u uq, mtmq+umq u,i+u uq,. 4 C5i mtmq {{ . ==== 5 C{C{ We identify these terms with the multipole moments of the qwk molecule, the ¿rst three as follows: mtmqi+u uq,@tq totali+u uq, (77a) mtmq+umq u,i+u uq,@sq ui+u uq, (77b) 5 5 4 C i 4 q C i+u uq, mtmq {{ @ T (77c) 5 C{C{ 9 mq C{C{ Thus, Eq. 74a can be written : ? urh+u>w, A@ mtmi+u um,. qtqi+u uq, u qsqi+u uq, (78) 5 4 q C i+u uq, . q T . === 9 mq C{C{ +4,p @ +u,u ^ s i+u u , s 4 +u u,pÃ4i+u u ,` iuhh q q q q m mq p@5 p$ mq q +4,p @ +u, u^S+u>w, s ^ 4 +u u,pÃ4`i+u u ,` iuhh q m mq p@5 p$ mq q . (78b) The functions i+u, fall off rapidly for uAAUand provide the correct density units. This Maxwell equation can be written: +4,p u^ H+u>w,.S+u>w, s 4 +u u,pÃ4i+u u ,` @ uG+u>w,@ +u, r q m mq p@5 p$ mq q iuhh (79) where G+u>w, can be written as: +4,p G+u>w,@ H+u>w,.S+u>w, s ^ 4 +u u,pÃ4`i+u u ,======r q m mq p@5 p$ mq q (80) 5 Note that theu operator only acts on i+u uq,. The terms in the summation are generally small (of the order of mumqm or smaller) and in most cases one can assume G+u>w,@rH+u>w,.S+u>w,. The Magnetic Field, H(r,t): 20 Section 1.6 Macroscopic equations measurements 5 4 gumq guq For Eq. (79d) we ¿nd in a similar fashion (using f @ and ymq @ yq @ ): rr gw gw 4 C ? u{e+u>w, A ?r h+u>w, A@? m+u>w, A (81) r Cw gu @ ? t m +u u , A . ? t +y . y ,+u u u , A m m gw m q m mq mq q q mq gu @ t m i+u u ,. t +y . y ,i+u u u , i+u u u ,= m m gw m q m mq mq q q mq Expand the q mq gu +4,p @ t m i+u u ,. t +y . y ,^4 . 4 ^u u`p `i+u u , m m gw m q m mq mq q p@4 p$ mq q +4,p @ t y i+u u ,. t y i+u u , t y u u^4 4 +u u,pÃ4`i+u u , m m m m q q q q q m mq q mq p@5 p$ mq q +4,p t y u u^4 4 +u u,pÃ4=`i+u u , q m mq mq mq p@5 p$ mq q whereweassumethat q mtmq+ymq . yq, qtqyq.Lettheqwk molecule magnetic moment be de¿ned as follows 4 p @ t u y , q 5 m mq+ mq mq , (83) and M+u>w,@? mtmym+u um, A . ? qtqyq+u uq, A (84) @ mtmymi+u um,. qtqyqi+u uq,. Using summation notation to rewrite the following (noteu acts only on i+u um, ): opn u^ur yr`@ {aoupn^{a ur`^{a yr` (85) @^op op`{aoup^{a ur`^{a yr` @ ur+yr u, yr+ur u, where u does not act on uror yr so that +ymq . yq,umq u@ umq+ymq . yq, uu^umq +ymq . yq,` (85b) we can write 4 C ? u{e+u>w, A ?r h+u>w, A@ M+u>w,. (86) r Cw . q mtmq^u^umq +ymq . yq,` umq+ymq . yq, u,`^i+u uq, +4,p 4 +u u,pÃ4i+u u ,` p@5 p$ mq q 4 C ? u{e+u>w, A ?r h+u>w, A@ M+u>w,. (86b) r Cw . m q^u5pqm . u+sqm yq, smq+ymq . yq, u,`^i+u uq, +4,p 4 +u u,pÃ4i+u u ,` p@5 p$ mq q 21 Section 1.6 Macroscopic equations measurements C Using +ymq . yq, u@ +ymq . yq, uu .u @ (note that ui+u uq, uuÃu Ãu i+u uq umq,muÃu ): q qm Cw q mq q +4,p @ M+u>w,.u^ 5p i+u u , 5p 4 +u u,pÃ4i+u u ,` q q q q m qm p@5 p$ mq q (86c) C +4,p . ^S+u>w, s 4 +u u,pÃ4i+u u ,` . Cw q m mq p@5 p$ mq q +4,p .u^ +s y ,i+u u , +s y , 4 +u u,pÃ4i+u u ,` q q q q q m mq q p@5 p$ mq q Using P+u>w,@? q5pq+u uq, A@ q5pqi (87) for the macroscopic magnetization, Eq. 86 becomes 4 C C ? u{e+u>w, A ?r h+u>w, A@ M+u>w,. ^G+u>w, rH+u>w,` . (88) r Cw Cw +4,p .u^P+u>w, 5p 4 +u u,pÃ4i+u u ,` . q m qm p@5 p$ mq q +4,p .u^ +s y ,i+u u , +s y , 4 +u u,pÃ4i+u u ,` q q q q q m mq q p@5 p$ mq q After making use Eqs. 71 and the following de¿nition for the macroscopic magnetic ¿eld, K+u>w, 4 K+u>w, E+u>w, P+u>w, q+sq yq,i+u uq,. (89) r +4,p . +s +y . y ,, 4 +u u,pÃ4i+u u , q m qm mq q p@5 p$ mq q (see Jackson page 256) C u{K+u>w,` @ G+u>w,.M+u>w, Cw (90) Note that when the solid moves as a whole (each yq @ y),the dipole moments of the molecules, sq, can produce a contribution to the magnetic ¿eld (see third term on right hand side of Eq. 89). 22 Section 1.6 Macroscopic equations measurements Example: a scalar wave travelling through a medium. Consider a harmonic plane, scalar wave propagating through a medium of identical scatterers/absorbers, say in the .} direction. We will investigate the change in the phase of the wave between the plane } and the plane } .#}= . Figure 2. Phase shift diagram The (angular) frequency of the wave is $3 and the vacuum wave number is n3= The geometry of the example is given in Fig. 2. In the plane } the wave is given by +}> w,@#3 h{s ^l +n3} $3w . ! +},,` (91) where ! +}, is the phase ‘shift’ in the wave function because the wave is travelling in a medium. If there were no medium between the plane } and the plane } .#} the wave transmitted through the slab of thickness #} would be r +} .#}>w,@#3 h{s ^l +n3 +} .#}, $3w . ! +},,` = (92) The total wave at } .#} is +} .#}> w,@ r +} .#}> w,. +}>w, where the forward scattered wave from the scatterers in the slab between } and } .#} is l +}> w,@#3 h{s ^l +n3 +} .#}, $3w . ! +},,` h #} (93) The h{s +l, depends on the density of scattering centers and the interactions between the wave and the scatterers. On a microscopic scale the ”scattered wave” is generated as follows. The incident wave perturbs the electronic structure of the atoms and molecules and this rearrangement produces an emitted electromagnetic wave which adds to the transmitted wave. This assumes that there is a suf¿cient density of scatterers to generate a plane wave. We assume that the scattered wave is due to a response of the scatterer to the incident wave. In this case we expect that the scattered wave will temporally lag the incident wave so is expected to be positive. In addition, in order to provide destructive interference, @5 ? = (Why 23 Section 1.6 Macroscopic equations measurements this is the case will become clear later.) The resultant wave at } .#} is +} .#}>w,@# h{s ^l +n +} .#}, $ w . ! +} .#},,` 3 3 3 (94) g! +}, # h{s l +n +} .#}, $ w . ! +},, . l #} 3 3 3 g} l @ #3 h{s ^l +n3 +} .#}, $3w . ! +},,` 4.h #} so that g! +}, h{s l #} @ 4.h l#} g} or g! +}, l #} @oq 4.hl#} h l#} g} g! +}, @ +l frv vlq , g} giving ! +},@ +l frv vlq , } (95) Up to an arbitrary phase constant the wave function for the scalar wave in the medium can be written +}>w,@#3 h{s ^l +n3} $3w . vlq +, },` h{s +} frv +,, (96) @ #3 h{s ^l ^n3 . vlq +,` } l$3w`h{s+} frv +,, Thewaveisdamped,withadampingconstant frv +, > as it propagates through the medium. The speed of the wave in the medium is $3 yydf yphg @ @ ?yydf (97) n3 . vlq +, 4.+@n3,vlq+, Note that the range @5 ? now provides an exponential damping [frv +, ? 3] and a reduced speed [vlq +, A 3 giving yphg ?yydf]. This example for a scalar wave indicates how several of the features of the electromagnetic wave in a medium can be explained. 1.6.1 Measurements The above discussion indicates how the wavelength dependence of the propagation and scattering of the electromagnetic wave can provide information concerning the structure of a system. In particular x-ray diffraction is commonly used to determine the structure parameters of liquids and solids. These measurements are particularly sensitive to the electron density function. In our discussion we have assumed that the frequency of the waves will not drive the system at a resonance. This would have complicated the discussion of the macroscopic equations.. However the frequency dependence of the propagation, scattering, and absorption of the electromagnetic waves provides information concerning the dynamics of a system (the frequency response function for the system). These measurements of the wavelength and frequency dependence of the interaction of the electromagnetic waves and a system are generally done with harmonic, continuous waves (cw). Another class of measurements which have become more common with the advent of lasers are those in which an electromagnetic wave excites a system to an initial state and the evolution of the system is probed using a second wave. The availability of pulsed lasers, with pulses less than a picosecond (even of order 10 femtoseconds), provides the opportunity to follow the evolution of a system on a time scale of order a picosecond. 24 Section 1.7 Harmonic waves 1.7 Harmonic waves We’ve noted that response of the system to the applied ¿elds is generally a convolution of the ¿elds and a response function of the system. In a large class of experiments, for which the wavelengths of the electromagnetic ¿elds are large compared to the characteristic lengths of the system, the spatial dependence of the response function is approximated by a delta function of u u3. For these systems it is suf¿cienttoworkin‘frequency space’. Maxwell’s equations are $ uh +u>$,@l e +u>$, f (98a) $ 7 uk +u>$,@l g +u>$,. m +u>$, f f (98b) ug +u>$,@7 +u>$, (98c) ue +u>$,@3 (98d) g H +u>w,@h +u>$, hl$w > l $ Note that one can obtain the above by setting etc in Eqs. (6a)-(6d) so that fgw is replaced by f . The constituent equations, in the linear response approximation and for cubic solids or isotropic systems, are g +u>$,@% +$, h +u>$, > (99) Ã4 k +u>$,@ +$, e +u>$, > (100) where if % +$, and +$, are complex functions (have an imaginary part) the medium is dispersive. The charge conservation law is um +u>$, l$ +u>$,@3= (101) The condition that the time dependent ¿elds and sources be real places constraints on the fourier transformed ¿elds and sources. To examine the constraint assume I +w, is a real function and i +$, is its fourier transform then ] 4 i +$,@ I +w,h{s+l$w, gw (102) Ã4 The fourier transform, i +$, > will generally be a complex function. Its complex conjugate is ] 4 Æ i +$, @ I +w,h{s+l$w, gw (103) Ã4 @ i +$, Æ This constraint, that i +$, @ i +$, > requires that the real part of i +$, be a symmetric function of $ and the imaginary part be an odd function of $: i +$,@Uhi+$,.l Lp i+$, i +$,Æ @Uhi+$, l Lp i+$, 25 Section 1.7 Harmonic waves 4 4 Uh i+$,@ ^i+$,.i +$,Æ`@ ^i+$,.i +$,` 5 5 4 4 Lp i+$,@ ^i+$, i +$,Æ`@ ^i+$, i +$,` 5l 5l Another relationship which is useful is the relationship between the frequency and temporal widths of the ¿elds. At this point we assume, without proof, that a relationship similar to that obtained in ordinary quantum mechanics will hold. In this case if #w is the temporal width of the ‘pulse’ and G H 5 #$5 $5 km$ml (104) then #w #$ ) 4@5. (105) 1.7.1 Conservation of energy in ‘frequency space’ Our observations of a system and its interaction with an electromagnetic ¿eld are made over time intervals. The observation time interval measured in the characteristic times of the ¿elds and the system can range from very small to very large. At this point we will consider the time interval for measurement to be large compared with the characteristic times of the system. The electromagnetic ¿elds can either be cw or pulse. In the case of cw operation we generally assume that the system is in the ‘steady state’ and our measurements are time averages of the system and ¿eld properties. In the pulse mode we measure the average properties of the system and the ¿elds per pulse. 1.7.1.1 Pulse mode We ¿rst consider the case of the pulse mode. The energy density delivered to the current by the ¿elds is ] 4 z +u,@ H +u>w, M +u>w, gw (106) Ã4 ]] ] 4 4 @ h +u>$, m +u>$3, h{s ^l +$ . $3, w` gw g$g$ 3 +5 ,5 ]] Ã4 4 @ h +u>$, m +u>$3,5 +$ . $3, g$g$ 3 +5 ,5 ] 4 4 @ h +u>$,Æ m +u>$, g$> +5 , Ã4 the energy density going to the ‘¿elds’ is ] 4 4 C C x +u,@ K +u>w, E +u=w,.H +u>w, G +u> w, gw 7 Cw Cw (107) ]Ã4 l 4 @ $ k +u>$,Æ e +u>$, h +u>$, g +u>$,Æ g$> 5 ; Ã4 and the energy density carried by the Poynting’s vector is ] f 4 v +u,@ ^H +u>w, K +u>w,` gw 7 (108) ]Ã4 f 4 @ h +u>$, k +u>$,Æ g$ 5 ; Ã4 26 Section 1.7 Harmonic waves Poynting’s theorem, the conservation of energy, is ]] ]]] v +u, gV. ^x +u,.z +u,` g6u @3 (109) erxqgdu| yroxph We now consider under what conditions is there a net transfer to or from the ¿elds during a pulse. For this we use to the constituent equations to rewrite x +u, in terms of h +u>$,and k +u>$, ] l 4 k l x +u,@ $ +$, mk +u>$,m5 % +$,Æ mh +u>$,m5 g$ 5 (110) ; Ã4 Because the ¿elds are real, the % +$, and +$, > like the fourier transforms of the ¿elds, have the property that their real parts are even functions of $ while their imaginary parts are odd functions of $= It follows then that ] 4 4 k l x +u,@ $ Lp + +$,, mk +u>$,m5 .Lp+% +$,, mh +u>$,m5 g$ 5 (111) ; Ã4 and x +u, is non-zero only if the imaginary parts of and % do not both vanish. In this case energy will be dissipated in the response of the mechanical system to the applied ¿elds. If +$, and % +$, are both real the system is not dispersive and no energy is lost doing mechanical work on the sources. 1.7.1.2 CW mode In the cw mode the average power delivered to the currents by the ¿elds is ] 4 W@5 z +u,@ H +u>w, M +u>w, gw (112) W ÃW@5 ]] ] 4 4 4 W@5 @ h +u>$, m +u>$33, h{s ^l +$ . $33, w` gw g$g$ 33 Let $3 @+$ . $33, +5 ,5 W ]] Ã4 ÃW@5 4 4 4 4 W W @ h +u>$, m +u>$3 $, ^h{s+l$3 , h{s+l$3 ,` g$g$ 3 +5 ,5 W l$3 5 5 ]] Ã4 4 4 vlq ^$3 W@5` @ h +u>$, m +u>$3 $, g$g$ 3> 5 3 +5 , Ã4 $ W@5 In the cw mode the frequency dependent ¿elds and sources should be sharply peaked around the selected frequency, say $3= The time average will be taken over a time which is short compared to the #$Ã4 where #$ is the frequency spread of the ¿elds and sources, $ @ $3 #$ In the integral over $ the value of $ will be restricted to be near $3 by the ¿eld h +u>$, = In that case $3 will be restricted to be less than #$ by the current density m +u>$3 $,. It follows that $3 W 4 (so that vlq^$3 W@5` $3W@5 4)and ] 4 ] 4 4 3 Æ 3 z+u,@ 5 h +u>$, g$ m +u>$ , g$ . f=f= (113) +5 , 3 3 where the ”cc” indicates the complex conjugate. Similarly the average power density delivered to the ¿elds is ] 4 ] 4 ] 4 ] 4 l$3 3 Æ 3 3 3 Æ si +u,@ 6 k +u>$ , g$ e +u>$, g$ h +u>$ , g$ g +u>$, g$ . f=f= (114) 5+5 , 3 3 3 3 Ã4 This result neglects a term which is smaller by a factor of +$3W , = If we assume the constituent equations, Equations (99),1.1)and (100,1.1), with the added assumption that the % +$, and +$, vary slowly in the frequency range for which the ¿elds are non-zero we obtain % ] ] & l$ 4 5 4 5 s +u,@ 3 +$ ,Ã4 e +u>$, g$ . % +$ , h +u>$3, g$3 f=f= = i 6 3 3 (115) 5+5 , 3 3 27 Section 1.7 Harmonic waves Considering these expression we can understand how the calculations assuming a single frequency can provide reasonable agreement with observations. As long as the experiment is not sensitive to the frequency spread we expect that we can use ¿elds and sources such as H +u>w,@3=8 h +u,h{s+l$3w,.ff G +u>w,@3=8 g +u,h{s+l$3w,.ff E +u>w,@3=8 e +u,h{s+l$3w,.ff K +u>w,@3=8 k +u,h{s+l$3w,.ff M +u>w,@3=8 m +u,h{s+l$3w,.ff +u>w,@3=8 +u,h{s+l$3w,.ff U Ã4 4 where we can identify h +u,@ 3 h +u>$, g$>etc. This provides a physical basis and a connection with reality for the harmonic waves we use in a number of models. With these identi¿cations the time averaged Poynting’s vector is f v +u,@ h +u, k +u,Æ . f=f= 49 (116) and time averaged Poynting’s theorem becomes 4 f 4 l$ 3@ u h +u, k +u,Æ . h +u, m +u,Æ . 3 h +u, g +u,Æ e +u, k +u,Æ . f=f= 5 ; 5 ; (117) The time averaged Poynting’s theorem involves only the real part of f 4 l$ &+u,@u h +u, k +u,Æ . h +u, m +u,Æ . 3 h +u, g +u,Æ e +u, k +u,Æ ; 5 ; (118) This is the dissipated or resistive part of the average power densities. The imaginary part of &+u, corresponds to the stored or reactive power. This is best seen by repeating the development of Poynting’s theorem starting with the positive frequency waves. The result is f 4 l$ &+u,@u h +u, k +u,Æ . h +u, m +u,Æ . 3 h +u, g +u,Æ e +u, k +u,Æ @3 ; 5 ; (119) The imaginary part of each term will give the average power which is received and returned without loss by each part of the system. In this case 4 x +u,@ Uh h +u, g +u,Æ e +u, k +u,Æ ; (120) 4 k l @ Uh % +$ , mh +u,m5 Uh +$ , mk +u,m5 ; 3 3 (120b) is the time average of the energy stored by the ¿elds. This is readily shown if we assume that the energy densities stored by the ¿elds are 3=8 H +u>w, G +u>w, and 3=8 K +u>w, E +u>w, = The real parts of % +$, and +$, give the stored energies and the imaginary parts of % +$, and +$, give the dissipated energies. 1.7.1.3 Recap, harmonic waves Quite generally the real parts of % +$, and +$, (and the imaginary part of the electrical conductivity) are associated with the energy stored by the system while the imaginary parts of % +$, and +$, (and the real part of the electrical conductivity) give the dissipated energies. In the case of non-conductors the imaginary parts of % +3, and +3, vanish and only the real parts are non-zero. It follows that, for static ¿elds, the energy stored by the ¿elds is 4 k l x +u,@ % +3, mH +u,m5 . +3, mK +u,m5 3 7 (121) We will ¿nd that the harmonic oscillator provides a model for the dielectric constant of non-conductors where the ¿elds provide the forces. The results obtained in Assignment 3 indicate that if the ¿elds are turned on ‘adiabatically’($3 4) 28 Section 1.7 Harmonic waves Eq. (121) 1.0121 will give the instantaneous energy density stored by the electric ¿eld and probably the the magnetic ¿eld. The above discussion of pulse mode measurements and cw measurements cover only two types of general measurements on systems. The variety of electromagnetic measurements which can be made to analyze a system is being expanded continually by improving technology and the imagination of the researchers. There is also a rapid expansion in the applications of electromagnetic waves. These range from the cd players, optical scanners, and communications to medical and manufacturing. Much of the impact has come from the development of lasers over the last 35 years. Lasers have provided the light beams which are a close approximation to the light rays of geometrical optics and nearly monochromatic waves which approximate the pure harmonic waves of our electromagnetic models. 29 Section 1.7 Harmonic waves ...... Assignment 5: For a given system the electric ¿eld and current density are given by % & +} fw,5 H +u>w,@H h{s frv +n +} fw,, h 3 5O5 3 4 % & +} fw,5 M +u>w,@M h{s frv +n +} fw,.! , h 3 5O5 3 3 4 (a) Calculate the magnetic Àux density, E +u>w, > for the magnetic part of the electromagnetic wave whose electric part is H +u>w, = (b) Calculate the frequency dependent ¿elds and current densities h +u>$, > e +u>$, > and m +u>$, = (c) What are the units of these three quantities in the gaussian units used by Jackson , in the SI (MKSC) units? (d) Starting with the results of 45 part (b) evaluate h +u, > e +u, > and m +u, which would appear in Equation (119) ??.(e)Let!3 @ @9 and n3O @5 43 and calculate the time average of H +u>w, M +u>w, over the period from w @ W@5 to w @.W@5 at } @ O and at } @3= (Note, a tabulated integral). (f) Plot the results of part (e) for 3 ?fW@O?4= ...... 30