Problems
Application 4.1 Monolayer Electric Dipole Drops
4.3 Electric Dipole Layers 101
Up to an additive constant, integration of (4.43) overzimplies that 0
d
dzϕ(rS, z)=τz(rS)δ(z). (4.44)
The constant disappears when we integrate (4.44) fromz=0−toz=0+. What remains is a matching condition for the potential:
ϕ(rS, z=0+)−ϕ(rS, z=0−)=τz(rS)/0. (4.45) The condition (4.45) shows that the electrostatic potential suffers a jump discontinuity at a dipole-layer surface. The variation inϕ(z) sketched as the dashed curve in Figure 4.8 makes this entirely plausible.
The potential rises from left to right across this microscopic dipole layer because the electric field inside the layer points from the positive charge on the right to the negative charge on the left.
The discontinuity (4.45) implies that the tangential component of the electric field is not continuous if the dipole-layer density varies along the surface. In the language of Section 2.3.3, the reader can show that
ˆ
n2×[E1−E2]= ∇ × {ˆn2( ˆn2·τ)}/0. (4.46) Under certain conditions, (4.46) generates corrections to the Fresnel formulae (Section 17.3.2) which describe the reflection and transmission of electromagnetic waves from interfaces.7
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implies that the electrostatic interaction energy of two such rows separated by a distancey comes entirely from the first term in the curly brackets in (4.31):
U(y)= 1 4π 0
L 0
dx1
L 0
dx2
τ2a2
[(x1−x2)2+y2]3/2 = τ2a2 4π 0
2 y2
&
L2+y2−y '
. (4.48) SinceLy, a valid approximation to (4.48) isU(y)τ2a2L/2π 0y2. The total electrostatic energy follows by summing over every pair of lines. There are w/a−1 lines separated by a distance a, w a−2 lines separated by a distance 2a, etc. Therefore, whenwa,
UE= τ2L 2π 0
w/a n=1
(w/a−n) n2 ≈ τ2L
2π 0
w/a 1
dn(w/a−n) n2 ≈ τ2L
2π 0
&w a −lnw
a '
. (4.49) The total energy is the sum of (4.49) and the chemical perimeter energy (4.47):
UT =UC+UE =2A w
λ+ τ2
4π 0
"w a −lnw
a
#
. (4.50)
KeepingAfixed and minimizingUT with respect towgives the equilibrium width we seek,
w0≈aexp(4π 0λ/τ2). (4.51)
The widthw0is independent of the drop lengthL, as claimed in the statement of the problem. The exponential dependence of w0 on material parameters is characteristic of two-dimensional dipole systems. Experiments show that the addition of more molecules does not increase the width of any
drop beyondw0. Instead, new drops form with a widthw0.
4.4 The Electric Quadrupole
The third (electric quadrupole) term dominates the multipole expansion (4.7) when the charge distri- bution of interest has zero net charge and zero dipole moment (Q=p=0). All atomic nuclei have this property,8as do all homonuclear diatomic molecules (like N2). The asymptotic (r→ ∞)
ϕ(r)= 1 4π 0
Qij
3rirj −δijr2
r5 . (4.52)
Repeating (4.6), the components of the electric quadrupole tensor are Qij = 12
d3r ρ(r)rirj. (4.53) Mimicking (4.9), it is not difficult to show that these scalars are uniquely defined and do not depend on the choice of the origin of coordinates ifQ=p=0. Note also that the explicit appearance of Cartesian factors likexyorz2in the integrand does not compel us toevaluatethe integral (4.53) using Cartesian coordinates.
The name “quadrupole” comes from a geometrical construction similar to the dipole case of Fig- ure 4.5. This time, we place two oppositely oriented point dipoles±p/s (thusfourcharges in total) at opposite ends of a vectorsc(Figure 4.10).9In the limits→0, a calculation similar to (4.13) gives
8 See the box which follows Example 4.1.
9 The multiple derivative structure of (4.3) guarantees that each term in the multipole expansion can be represented by point charge configurations constructed from suitable pairings in space of the configurations which describe the previous term. This guarantees that each configuration is electrically neutral and that only 2n-poles appear.
4.4 The Electric Quadrupole 103
sc s –p s
p
Figure 4.10: A quadrupole composed of two oppositely directed dipoles.
the electrostatic potential of a “point quadrupole” located at the origin:
ϕ(r)= 1 4π 0
(c· ∇)(p· ∇)1
r. (4.54)
This is identical to (4.52) with
Qij = 12(cipj+cjpi). (4.55)
The manifestly symmetric definition (4.55) is needed becauseQij =Qj i is explicit in (4.53). By direct construction, or by mimicking (4.14), the reader can confirm that the corresponding singular charge density for a point quadrupole at the origin is
ρQ(r)=Qij∇i∇jδ(r). (4.56) The symmetryQij=Qj i implies that only six (rather than nine) independent numbers are needed to characterize a general quadrupole tensor:10
Q=
⎡
⎣Qxx Qxy Qxz
Qxy Qyy Qyz
Qxz Qyz Qzz
⎤
⎦. (4.57)
This information can be packaged more efficiently if we adopt theprincipal axiscoordinate system (x, y, z) to evaluate the integrals in (4.53). By definition, the quadrupole tensorQisdiagonalin this system, with components
Qij = 12δij
d3rρ(r)rirj. (4.58)
In the general case, familiar theorems from linear algebra tell us that
Q=U−1QU, (4.59)
whereU is the unitary matrix formed from the eigenvectors ofQ. These eigenvectors can always be found becauseQis a symmetric matrix. We conclude that the three numbersQxx,Qyy, andQzz are sufficient to completely characterize the quadrupole potential (4.52) in the principal axis system.
The reader will recognize the direct analogy between the electric quadrupole moment tensor and the moment of inertia tensor used in classical mechanics.
4.4.1 The Traceless Quadrupole Tensor
It is usually not convenient (or even possible) to measure the components of Q directly in the principal axis system defined by (4.59). Luckily, a strategy exists that economizes the description
10 Recall from Section 1.8 that we use boldface symbols for tensor quantities likeQ. Context should prevent confusion with the total chargeQ.
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of the quadrupole potential even if we are forced to work in an arbitrary coordinate system. The calculation begins by using (4.53) to write out (4.52):
ϕ(r)= 1 4π 0
1 2
d3rρ(r)rirj
+3rirj −r2δij
r5 . (4.60)
The key step uses the identity
rirjr2δij =r2r2=rirjr2δij (4.61) to write (4.60) in the form
ϕ(r)= 1 4π 0
1 2
d3rρ(r).
3rirj−δijr2/+rirj
r5 . (4.62)
Therefore, if we define a symmetric,traceless quadrupole tensor, ij= 1
2
d3rρ(r)(3rirj−r2δij)=3Qij−Qkkδij, (4.63) the quadrupole potential (4.60) simplifies to
ϕ(r)= 1 4π 0
ij
rirj
r5 . (4.64)
As its name implies, the point of this algebraic manipulation is that the tensorhas zero trace:
Tr[]=δijij = 1 2
d3r ρ(r)(3riri−r2δii)=0. (4.65) This is important because the constraint
Tr[]=xx+yy+zz=0 (4.66)
reduces the number of independent components ofby one. Hence, a complete characterization of the quadrupole potential in an arbitrary coordinate system requires specification of onlyfivenumbers (the independent components of thetracelessCartesian quadrupole tensor) rather than six numbers (the components of theprimitiveCartesian quadrupole tensorQ). Equivalently, it always possible to choose the vectorscandpin (4.54) so|c| = |p|. This similarly reduces the number of independent Cartesian components from six to five.
4.4.2 Force and Torque on a Quadrupole
The force and torque exerted on a point quadrupole in an external electric field can be calculated using the method of Section 4.2.3. The results are
F=Qij∇i∇jE (4.67)
and
N=2(Q· ∇)×E+r×F. (4.68)
The corresponding electrostatic interaction energy is
VE(r)= −Qij∇iEj(r)= −13ij∇iEj(r). (4.69) A glance back at (4.63) shows that the two expressions in (4.69) differ by a term proportional to δij∂iEj = ∇ ·E. This iszerobecause the source charge for the external fieldEis assumed to be far from the quadrupole. Following the logic of Section 4.2.3, it not difficult to show that (4.67), (4.68), and (4.69) are valid for any neutral charge distribution withp=0 if the electric field does not vary too rapidly over the volume of the distribution.
4.4 The Electric Quadrupole 105