Problems
Application 4.2 Nuclear Quadrupole Moments
4.6 Spherical and Azimuthal Multipoles
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110 ELECTRIC MULTIPOLES: APPROXIMATE ELECTROSTATICS FOR LOCALIZED CHARGE
r R
Figure 4.15: A charge distributionρ(r) (shaded) lies completely outside an origin-centered sphere of radiusR.
An interior multipole expansion is valid at observation pointsrinside the sphere.
4.6.1 The Exterior Spherical Expansion
When the charge density is localized and confined to the sphere of radiusRin Figure 4.1, the spherical analog of the Cartesian expansion (4.3) which converges for observation pointsexteriorto the sphere is
ϕ(r)= 1 4π 0
∞ =0
m=−
Am
Y m()
r+1 r > R. (4.86) The complex numbersA mare theexterior spherical multipole momentsof the charge distribution:
A m= 4π 2+1
d3rρ(r)rY m∗ (). (4.87) The order-of-magnitude estimateA m∼QR shows that, like the Cartesian multipole expansion, (4.86) is an expansion in the small parameterR/r. Moreover, there is a simple relation betweenA m
andA−mbecause the spherical harmonics obey
Y−m(θ, φ)=(−1)mY m∗ (θ, φ). (4.88) The coefficient of 1/r+1in (4.86) must be identical to the corresponding coefficient in the Cartesian expansion (4.7). This means that the information encoded by the three spherical momentsA10, A11, andA1−1is exactly the same as the information encoded by the three components of the electric dipole moment vectorp. Put another way, eachA1mcan be written as a linear combination of thepkand vice versa. Similarly, the five spherical momentsA2mcarry the same information as the five components of the traceless quadrupole tensor.13When higher moments are needed (as in some nuclear, atomic, and molecular physics problems), the relatively simple analytic properties of the spherical harmonics usually make the spherical multipole expansion the method of choice for approximate electrostatic potential calculations.
4.6.2 The Interior Spherical Expansion
The expansion (4.86) is only valid for observation points that lieoutsidea bounded distributionρ(r).
Another common situation is depicted in Figure 4.15. This shows a region of distributed chargeρ(r) which completely surrounds a charge-free, origin-centered sphere of radiusR. To studyϕ(r) inside the sphere, we choose the version of (4.84) that applies whenr < rand substitute this into (4.85).
13 It is not obvious that the traceless tensoris relevant here rather than the primitive tensorQ. This point is explored in Section 4.7.
4.6 Spherical and Azimuthal Multipoles 111
The result is a formula for the potential that is valid at observation points in theinteriorof the sphere in Figure 4.15:
ϕ(r)= 1 4π 0
∞ =0
m=−
B mrY m∗ () r < R. (4.89)
The complex numbersB m in (4.89) are the interior spherical multipole moments of the charge distribution:
B m= 4π 2+1
d3r ρ(r)
r+1Y m(). (4.90)
The Cartesian analog of the interior expansion (4.89) is a power expansion ofϕ(x, y, z) around the origin of coordinates.
Example 4.4 Use the multipole expansion method to findϕ(r) for a spherically symmetric but otherwise arbitrary charge distribution.
Solution: Figure 4.16 shows an observation pointrat a distancerfrom the center of the distribution.
We use an exterior expansion to find the contribution toϕ(r) from the charge in the volumeVand an exterior expansion to find the contribution toϕ(r) from the charge in the volumeV.
V V′ r
Figure 4.16: A spherically symmetric charge distribution of infinite extent.
SinceY00=1/√
4π is a constant, there is no loss of generality if we write ρ(r)=ρ(r)Y00= ρ(r)Y00∗. Then, the exterior and interior multipole moments defined in (4.87) and (4.90) are
Am(r)= 4π 2+1
dY00()Ym∗ () r 0
drr2ρ(r)=4π δ,0δm,0
r 0
drr2ρ(r)
Bm(r)= 4π 2+1
dY00∗()Ym() ∞ r
drρ(r)
r−1 =4π δ,0δm,0
∞ r
drrρ(r).
Since only theA00andB00terms survive we drop the constant factorY00from the definition of ρ(r) and add the potentials (4.86) and (4.89) to get the desired final result,
ϕ(r)= 1 4π 0
A00(r)
r +B00(r) +
= 1 0
⎧⎨
⎩ 1 r r 0
ds s2ρ(s)+ ∞
r
ds sρ(s)
⎫⎬
⎭.
This agrees with the formula derived for this quantity in Example 3.4.
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112 ELECTRIC MULTIPOLES: APPROXIMATE ELECTROSTATICS FOR LOCALIZED CHARGE
4.6.3 Azimuthal Multipoles
Specified charge densities that areazimuthally symmetricwith respect to a fixed axis in space occur quite frequently. For that reason, it is worth learning how (4.86) and (4.89) simplify whenρ(r, θ, φ)= ρ(r, θ). We will need two properties of the spherical harmonics:
Y0(θ, φ)=
)2+1
4π P(cosθ) (4.91)
and
1 2π
2π 0
dφ Y m(θ, φ)=
)2+1
4π P(cosθ)δm,0. (4.92)
Using (4.92), the expansion coefficients (4.87) and (4.90) becomeA m=Aδm0
√4π/(2+1) and B m=Bδm0
√4π/(2+1), where14
A=
d3rrρ(r, θ)P(cosθ) (4.93) and
B =
d3r 1
r+1ρ(r, θ)P(cosθ). (4.94) SubstitutingA mandB minto (4.86) and (4.89), respectively, and using (4.91) yields exterior and interiorazimuthal multipole expansionsin the form
ϕ(r, θ)= 1 4π 0
∞ =0
A
r+1P(cosθ) r > R, (4.95) ϕ(r, θ)= 1
4π 0
∞ =0
BrP(cosθ) r < R. (4.96)
Application 4.3 The Potential Produced byσ(θ)=σ0cosθ
A situation which occurs frequently in electrostatics is an origin-centered spherical shell with radius Rand a specified surface charge density
σ(θ)=σ0cosθ. (4.97)
We can represent the potential produced by (4.97) using the exterior azimuthal multipole expansion (4.95) for the space outside the shell and the interior azimuthal multipole expansion (4.96) for the space inside the shell. The fact thatP1(x)=xand the orthogonality relation (4.75) for the Legendre polynomials show that the expansion coefficients (4.93) and (4.94) are
A=
dS RP(cosθ)σ(θ)=σ0R+2 2π
0
dφ π 0
dθsinθ P(cosθ) cosθ= 4π
3 R3σ0δ ,1
(4.98) B=
dS 1
R+1P(cosθ)σ(θ)=σ0R1−
2π 0
dφ π 0
dθsinθ P(cosθ) cosθ = 4π 3 σ0δ ,1.
14 Notice that (4.93) and (4.94) include an integration over the azimuthal angleφ.
4.6 Spherical and Azimuthal Multipoles 113
The corresponding potential is
ϕ(r, θ)= σ0
30
⎧⎨
⎩
rcosθ r < R, R3
r2 cosθ r > R. (4.99) Note that the two expansions agree atr=R. This reflects the continuity of the electrostatic potential as the observation point passes through a layer of charge. The fact that only the=1 term survives tells us that the electric field outside the shell is exactly dipolar. The interior “dipole” potential is proportional torcosθ =z. This gives a constant electric fieldEˆzinside the shell becauseE= −∇ϕ.
4.6.4 Preview: the Connection to Potential Theory
The multipole calculation outlined in Application 4.3 exploited the fact that all the source charge lay on the surface of a spherical shell. Another approach to this problem, calledpotential theory, uses the same information to infer thatϕ(r) satisfies Laplace’s equation both inside the shell and outside the shell:
∇2ϕ(r)=0 r=R. (4.100)
From this point of view, (4.95) and (4.96) are linear combinations of elementary solutions of Laplace’s equation with azimuthal symmetry. The coefficientsAandB are chosen so the potentials inside and outside the sphere satisfy the required matching conditions (Section 3.3.2) atr=R. If the source charge on the shell is not azimuthally symmetric, similar remarks apply to a potential given by the simultaneous use of (4.86) and (4.89). We will explore potential theory in detail in Chapters 7 and 8.
Example 4.5 Find the electrostatic potential produced by a uniformly charged line bent into an origin-centered circular ring in thex-yplane. The charge per unit length of the line isλ=Q/2π R.
Solution: The potentialϕ(r, θ) is determined completely by an interior and exterior azimuthal multipole expansion with respect to an origin-centered sphere of radiusR. In spherical coordinates, the volume charge density of the ring is
ρ(r)=(λ r)δ(r−R)δ(cosθ).
Note that the right side of this equation is dimensionally correct and satisfies the necessary condition Q=
d3rρ(r)=2π Rλ. Using (4.93) and (4.94), the expansion coefficients areA=QRP(0) andB =QR−(+1)P(0). Therefore,
ϕ(r, θ)= Q 4π 0r
∞ =0
R r
P(0)P(cosθ) r≥R, ϕ(r, θ)= Q
4π 0R ∞ =0
"r R
#
P(0)P(cosθ) r≤R.
As it must be, the potential is continuous as the observation point passes through the ring.