Finding the fields produced by time-varying electric dipoles has been historically impor- tant and has served as a prototypical example for radiation problems.
14.5. Fields of Electric and Magnetic Dipoles 581 We consider a point dipole located at the origin, in vacuum, with electric dipole momentp. Assuming harmonic time dependenceejωt, the corresponding polarization (dipole moment per unit volume) will be: P(r)=pδ(3)(r). We saw in Eq. (1.3.18) that the corresponding polarization current and charge densities are:
J=∂P
∂t =jωP, ρ= −∇∇∇ ãP (14.5.1) Therefore,
J(r)=jωpδ(3)(r) , ρ(r)= −pã ∇∇∇δ(3)(r) (14.5.2) Because of the presence of the delta functions, the integrals in Eq. (14.3.3) can be done trivially, resulting in the vector and scalar potentials:
A(r)=μ0
jωpδ(3)(r)G(r−r) dV=jωμ0pG(r) ϕ(r)= −1
0
pã ∇∇∇δ(3)(r)
G(r−r) dV= −1 0
pã ∇∇∇G(r)
(14.5.3)
where the integral forϕwas done by parts. Alternatively,ϕcould have been determined from the Lorenz-gauge condition∇∇∇ ãA+jωμ00ϕ=0.
TheE,H fields are computed from Eq. (14.3.6), or from (14.3.7), or away from the origin from (14.3.9). We find, wherek2=ω2/c20=ω2μ00:
E(r)= 1 0∇∇∇ ×
∇∇∇G(r)×p
= 1 0
k2p+(pã ∇∇∇)∇∇∇ G(r) H(r)=jω∇∇∇G(r)×p
(14.5.4)
forr=0. The Green’s functionG(r)and its gradient are:
G(r)=e−jkr
4πr , ∇∇∇G(r)= −ˆr jk+1
r
G(r)= −ˆr jk+1
r e−jkr
4πr
wherer= |r|and ˆris the radial unit vector ˆr=r/r. Inserting these into Eq. (14.5.4), we obtain the more explicit expressions:
E(r)= 1 0
jk+1 r
3ˆr(ˆrãp)−p r
G(r)+k2 0
ˆr×(p׈r)G(r) H(r)=jω
jk+1 r
(p׈r)G(r)
(14.5.5)
If the dipole is moved to locationr0, so thatP(r)=pδ(3)(r−r0), then the fields are still given by Eqs. (14.5.4) and (14.5.5), with the replacementG(r)→G(R)and ˆr→ˆR, whereR=r−r0.
Eqs. (14.5.5) describe both the near fields and the radiated fields. The limitω=0 (or k=0) gives rise to the usual electrostatic dipole electric field, decreasing like 1/r3. On the other hand, as we discuss in Sec. 14.7, the radiated fields correspond to the terms decreasing like 1/r. These are (withη0= μ0/0):
582 14. Radiation Fields
Erad(r)=k2 0
ˆr×(p׈r)G(r)=k2 0
ˆr×(p׈r)e−jkr 4πr Hrad(r)=jω jk(p׈r)G(r)= k2
η00
(ˆr×p)e−jkr 4πr
(14.5.6)
They are related by η0Hrad =ˆr×Erad, which is a general relationship for radia- tion fields. The same expressions can also be obtained quickly from Eq. (14.5.4) by the substitution rule∇∇∇ → −jkˆr, discussed in Sec. 14.10.
The near-field, non-radiating, terms in (14.5.5) that drop faster than 1/rare im- portant in the new area ofnear-field optics[517–537]. Nanometer-sized dielectric tips (constructed from a tapered fiber) act as tiny dipoles that can probe the evanescent fields from objects, resulting in a dramatic increase (by factors of ten) of the resolution of optical microscopy beyond the Rayleigh diffraction limit and down to atomic scales.
A magnetic dipole at the origin, with magnetic dipole momentm, will be described by the magnetization vectorM=mδ(3)(r). According to Sec. 1.3, the corresponding magnetization current will beJ= ∇∇∇ ìM= ∇∇∇δ(3)(r)ìm. Because∇∇∇ ãJ=0, there is no magnetic charge density, and hence, no scalar potentialϕ. The vector potential will be:
A(r)=μ0
∇
∇
∇δ(3)(r)×mG(r−r) dV=μ0∇∇∇G(r)×m (14.5.7) It then follows from Eq. (14.3.6) that:
E(r)= −jωμ0∇∇∇G(r)×m H(r)= ∇∇∇ ×
∇
∇∇G(r)×m
=
k2m+(mã ∇∇∇)∇∇∇ G(r)
(14.5.8)
which become explicitly, E(r)=jωμ0
jk+1 r
(ˆr×m)G(r)
H(r)= jk+1
r
3ˆr(ˆrãm)−m r
G(r)+k2ˆr×(m׈r)G(r)
(14.5.9)
The corresponding radiation fields are:
Erad(r)=jωμ0jk(ˆr×m)G(r)=η0k2(m׈r)e−jkr 4πr Hrad(r)=k2ˆr×(m׈r)G(r)=k2ˆr×(m׈r)e−jkr 4πr
(14.5.10)
We note that the fields of the magnetic dipole are obtained from those of the electric dipole by thedualitytransformationsE→H,H→ −E,0→μ0,μ0→0,η0→1/η0, and p→μ0m, that latter following by comparing the termsPandμ0Min the constitutive relations (1.3.16). Duality is discussed in more detail in Sec. 17.2.
The electric and magnetic dipoles are essentially equivalent to the linear and loop Hertzian dipole antennas, respectively, which are discussed in sections 16.2 and 16.8.
Problem 14.4 establishes the usual resultsp=Qd for a pair of charges±Qseparated by a distanced, andm=ˆzISfor a current loop of areaS.
14.5. Fields of Electric and Magnetic Dipoles 583 Example 14.5.1: We derive explicit expressions for the real-valued electric and magnetic fields of an oscillatingz-directed dipolep(t)=pˆzcosωt. And also derive and plot the electric field lines at several time instants. This problem has an important history, having been considered first by Hertz in 1889 in a paper reprinted in [58].
Restoring theejωtfactor in Eq. (14.5.5) and taking real parts, we obtain the fields:
EE E(r)=p
ksin(kr−ωt)+cos(kr−ωt) r
3ˆr(ˆrãˆz)−ˆz
4π0r2 +pk2ˆr×(ˆz׈r)
4π0r cos(kr−ωt) H
HH(r)=pω
−kcos(kr−ωt)+sin(kr−ωt) r
ˆz׈r 4πr
In spherical coordinates, we have ˆz=ˆrcosθ−θθθˆsinθ. This gives 3 ˆr(ˆrãˆz)−ˆz=2 ˆrcosθ+
θˆ
θθsinθ, ˆr×(ˆz׈r)= −θθθˆsinθ, and ˆz׈r=φφφˆsinθ. Therefore, the non-zero components ofEEEandHHHareEr,EφandHφ:
Er(r)=p
ksin(kr−ωt)+cos(kr−ωt) r
2 cosθ 4π0r2
Eθ(r)=p
ksin(kr−ωt)+cos(kr−ωt) r
sinθ 4π0r2
− pk2sinθ
4π0r cos(kr−ωt) Hφ(r)=pω
−kcos(kr−ωt)+sin(kr−ωt) r
sinθ 4πr
By definition, the electric field is tangential to its field lines. A small displacementdralong the tangent to a line will be parallel toEEEat that point. This implies thatdr×EEE =0, which can be used to determine the lines. Because of the azimuthal symmetry in theφvariable, we may look at the field lines that lie on thexz-plane (that is,φ=0). Then, we have:
dr× EEE =(ˆrdr+θθθ r dθ)×(ˆ ˆrEr+θθθˆEθ)=φφφ(drˆ Eθ−r dθEr)=0 ⇒ dr dθ=rEr
Eθ
This determinesras a function ofθ, giving the polar representation of the line curve. To solve this equation, we rewrite the electric field in terms of the dimensionless variables u=krandδ=ωt, definingE0=pk3/4π0:
Er=E0
2 cosθ u2
sin(u−δ)+cos(u−δ) u
Eθ= −E0
sinθ u
cos(u−δ)−cos(u−δ)
u2 −sin(u−δ) u
We note that the factors within the square brackets are related by differentiation:
Q(u)=sin(u−δ)+cos(u−δ) u Q(u)=dQ(u)
du =cos(u−δ)−cos(u−δ)
u2 −sin(u−δ) u Therefore, the fields are:
Er=E0
2 cosθ
u2 Q(u) , Eθ= −E0
sinθ u Q(u)
584 14. Radiation Fields
It follows that the equation for the lines in the variableuwill be:
du dθ=uEr
Eθ = −2 cotθ Q(u)
Q(u)
⇒ d dθ
lnQ(u)
= −2 cotθ= −d dθ
ln sin2θ
which gives:
d dθln
Q(u)sin2θ
=0 ⇒ Q(u)sin2θ=C whereCis a constant. Thus, the electric field lines are given implicitly by:
sin(u−δ)+cos(u−δ) u
sin2θ=
sin(kr−ωt)+cos(kr−ωt) kr
sin2θ=C
−1.5 −1 −0.5 0 0.5 1 1.5
−1.5
−1
−0.5 0 0.5 1 1.5 t = 0
−1.5 −1 −0.5 0 0.5 1 1.5
−1.5
−1
−0.5 0 0.5 1 1.5 t = T/ 8
−1.5 −1 −0.5 0 0.5 1 1.5
−1.5
−1
−0.5 0 0.5 1 1.5 t = T/ 4
−1.5 −1 −0.5 0 0.5 1 1.5
−1.5
−1
−0.5 0 0.5 1 1.5 t = 3T/ 8
Fig. 14.5.1 Electric field lines of oscillating dipole at successive time instants.
Ideally, one should solve forrin terms ofθ. Because this is not possible in closed form, we prefer to think of the lines as a contour plot at different values of the constantC. The resulting graphs are shown in Fig. 14.5.1. They were generated at the four time instants t=0, T/8, T/4, and 3T/8, whereTis the period of oscillation,T=2π/ω. Thex, z distances are in units ofλand extend to 1.5λ. The dipole is depicted as a tinyz-directed line at the origin. The following MATLAB code illustrates the generation of these plots:
rmin = 1/8; rmax = 1.6; % plot limits in wavelengthsλ
Nr = 61; Nth = 61; N = 6; % meshpoints and number of contour levels t = 1/8; d = 2*pi*t; % time instantt=T/8
[r,th] = meshgrid(linspace(rmin,rmax,Nr), linspace(0,pi,Nth));