UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede LECTURE NOTES 8

POTENTIAL APPROXIMATION TECHNIQUES: THE ELECTRIC AND MOMENTS OF THE ELECTRIC CHARGE DISTRIBUTION

There are often situations that arise where an “observer” is far away from a localized charge distribution ρ ()r and wants to know what the potential Vr( ) and / or the electric intensity Er() are far from the localized charge distribution.

If the localized charge distribution has a net electric charge Q , then far away from this net localized charge distribution, the potential Vr( ) to a good approximation will behave very much like that of a point charge,

1 Qnet 1 Qnet Vrfar () and Erfar()=−∇ Vr far () − 2 4πε o r 4πε o r when the field point – source charge separation distance, r d, the characteristic size of the charge distribution.

However, as the “observer” moves in closer and closer to the localized charge distribution ρ ()r′ , he/she will discover that increasingly Vr( ) (and hence Er( ) ) may deviate more and more from pure point charge behavior, because ρ (r′) is an extended source charge distribution.

Furthermore, ρ ()r′ may be such that Qnet ≡ 0 , but that does NOT necessarily imply that Vr()= 0 (and Er()=0)!

Example:

A pure, physical electric is a spatially-extended, simple charge distribution where Qnet = 0 but Vr()≠ 0 and Er()=−∇ Vr () ≠0 , as shown in the figure below: +q r+ P (field point)

r A pure physical electric dipole is d θ composed of two opposite electric r− charges separated by a distance d:

−q

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 1 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede The Potential Vr() and Er( ) of a Pure Physical Electric Dipole

“Pure” → Qnet = 0 “Physical” → Spatially extended electric eipole d ≠ 0 , d > 0 {n.b. ∃ “point” electric with d = 0, e.g. neutral atoms & molecules…}

First, let us be very careful / wise as to our choice of . A wrong choice of coordinate system will unnecessarily complicate the and obscure the physics we are attempting to learn about the nature / behavior of this system.

Examples of BAD choices of coordinate systems: zˆ′ q+ a.) q+ zˆ b.) zˆ Ο′ θ′ yˆ′ rdipole Ο Ο ϕ ' yˆ yˆ ϕ q− xˆ′

xˆ q− xˆ

Dipole lying in x – y plane has Even more mathematically complicated!! ϕ -dependence, but (at least it) Origin is not conveniently chosen (arbitrary?) is centered at the origin. the dipole axis makes with respect to zˆ & xˆ axes must be described by two - θ and ϕ .

Smart / wise choice of coordinate system: Exploit intrinsic of problem.

Physical electric dipole has axial symmetry – choose zˆ axis to be along line separating q+ and q−. Choose x-y plane to lie mid-way between q+ and q−:

zˆ P (field point) r+ n.b. This problem +q r now has no Mathematical expressions obtained for θ ϕ -dependence Vr( ), Er( )()=−∇ Vr for this choice d r− yˆ Ο of coordinate system for the physical electric dipole can be explicitly and xˆ −q rigorously related to more complicated / tedious mathematical expressions for a.) and b.) above – via coordinate translations & rotations!

2 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede Pure, Physical Electric Dipole:

zˆ P (field point) +q r + r+′ r Source Charge Locations 1 11 2 d rdzrd++′′=+22ˆ, = 11 Ο θ r− yˆ rdzrd−−′′=−22ˆ, = d π −θ 1 d 2 xˆ r−′ −q

Law of Cosines: c2 = a2 + b2 – 2ab cos θ q+ +q c P −q π −θ P b a b Ο c θ a Ο −q

2 2 22⎛⎞dd ⎛⎞ 22⎛⎞dd ⎛⎞ r + =+−⎜⎟rr2cos ⎜⎟ θ r − = ⎜⎟+−rr2cos ⎜⎟ ()π −θ ⎝⎠22 ⎝⎠ ⎝⎠22 ⎝⎠ 2 2 ⎛⎞d 2 ⎛⎞d 2 =+−⎜⎟ rdr cosθ =++⎜⎟ rdr cosθ ⎝⎠2 ⎝⎠2 2 2 2 ⎛⎞d 2 ⎛⎞d =+rrd⎜⎟ − cosθ =+rrd⎜⎟ + cosθ ⎝⎠2 ⎝⎠2

Use Principle of Linear Superposition for Total Potential:

VrVrVrVTOT()=+≡+− q( ) q( ) dipole ( r)

11++qqq 1 Vr+q ()== = 44πε πε2222 4 πε oor + r+−() d2cos rdθ o r +−() d 2cos rd θ 11−−−qqq 1 Vr−q ()== = 44πε πε2222 4 πε oor − rd++()2cos rdθ o rd ++() 2cos rdθ

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 3 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede 11+qq ∴ VrVrVrdipole()=+=+− q () q () − 44πεoorr +− πε 11qq =− 44πε2222 πε oord+−()2 rd cosθ rd ++() 2 rd cosθ ⎡⎤ q 11 =−⎢⎥ 4πε ⎢⎥2222 o ⎣⎦rd+−()2 rd cosθθ rd ++() 2 rd cos

This is an exact analytic mathematical expression for the potential associated with a pure

()Qnet = 0 physical electric dipole with charges +q and –q separated from each other by a distance d. Note further that, because of the judicious choice of coordinate system and the intrinsic (azimuthal) symmetry, Vrdipole ( ) has no ϕ -dependence.

The exact analytic expression for potential associated with pure physical electric dipole: ⎧⎫ q ⎪⎪11 Vrdipole ()=−⎨⎬ 4πε 2222 o ⎩⎭⎪⎪rd+−()2 rd cosθ rd ++() 2 rd cosθ

As mentioned earlier, often we are / will be interested only in knowing (approximately) Vrdipole () when rd . For example, many kinds of neutral molecules have permanent electric dipole moments p ≡ qd (Coulomb-meters) and (obviously) for such molecules, the dipole’s separation distance d is (typically) on the order of ~ few Ångstroms, i.e. d ~ Ο (5Å) {1 Å ≡ 10−10 m = 10 nm (1 nm = 10−9 m)}. So even if the field point P is e.g. rm==110μ −6 m away from such a molecular dipole, rmdnm=1~5μ , since dr 0.005 !

In such situations, when rd an approximate solution for Vrdipole ( ) which has the benefit of reduced mathematical complexity, will suffice to give a good / reasonable physical description of the intrinsic physics, accurate e.g. to 1% (or better) when compared directly to the exact analytical expression over the range of distance scales rd that are of interest to us.

Thus for rd> , the exact expressions for the r+ and r− separation distances are:

2 2 2 2 r + =+rd()2cos − rdθ r − =+rd()2cos + rdθ

2 2 ⎛⎞d ⎛⎞d ⎛⎞d ⎛⎞d =+rr 1⎜⎟ −⎜⎟ cosθ =+rr 1⎜⎟ +⎜⎟ cosθ ⎝⎠2 ⎝⎠r ⎝⎠2 ⎝⎠r

2 2 1 ⎛⎞dd ⎛⎞ 1 ⎛⎞dd ⎛⎞ =+r 1⎜⎟ − ⎜⎟ cosθ =+r 1⎜⎟ + ⎜⎟ cosθ 4 ⎝⎠rr ⎝⎠ 4 ⎝⎠rr ⎝⎠

4 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede Now if ()dr 1, then let us define: 2 2 1 ⎛⎞dd ⎛⎞ 1 ⎛⎞dd ⎛⎞ ε + ≡−⎜⎟ ⎜⎟cosθ and: ε − ≡+⎜⎟ ⎜⎟cosθ 4 ⎝⎠rr ⎝⎠ 4 ⎝⎠rr ⎝⎠

11 11 Then: = and: = r + r 1+ ε + r − r 1+ ε − with: ε + 1 and: ε − 1

Now if ε + 1 and ε − 1, we can use the Binomial Expansion (a specific version of the more generalized Taylor Expansion) of the expression:

−1 12 1 13iii23 135 =+()1εεεε±±±± =− 1 + − +− ...... (Valid on the interval: −≤ 1ε ± ≤+ 1) 1+ ε ± 224246iii

234 Since ε ± is already <<1, then the higher-order terms ()()()εεε±±±, , ,...etc. are incredibly small (<<<<<1), so negligible error is incurred by neglecting these higher-order terms, 1 i.e. keeping only terms linear in ε ± in the binomial expansion of , we have: 1+ ε ±

1111 1111 =− ()1 2 ε + and: =− ()1 2 ε − r + r 1+ ε + r r − r 1+ ε − r

dipole qq⎧⎫11⎧⎫ 111 1 Vr()=−⎨⎬ ⎨()11 −−−22εε+−() ⎬ 44πεoorr +− πε ⎩⎭rr Then: ⎩⎭ q ⎛⎞1 1 11q ⎛⎞1 = ⎜⎟{} 1 −−2 ε + 1 +=22ε −−+⎜⎟{}()()εε − 4πε o ⎝⎠x 4πε o ⎝⎠r

2 2 1 ⎛⎞dd ⎛⎞ 1 ⎛⎞dd ⎛⎞ Now: ε + ≡−⎜⎟ ⎜⎟cosθ and: ε − ≡+⎜⎟ ⎜⎟cosθ 4 ⎝⎠rr ⎝⎠ 4 ⎝⎠rr ⎝⎠

⎧⎫2 2 qd11⎡⎤⎛⎞ 1 dd⎛⎞1 d ⎛⎞⎛⎞⎛⎞⎛⎞⎪⎪⎢⎥⎜⎟⎛⎞⎜⎟ ⎛⎞⎛⎞ ⎛⎞ Vrdipole ()= ⎜⎟⎜⎟⎜⎟⎜⎟⎨⎬+−⎜⎟cosθ ⎜⎟⎜⎟− ⎜⎟cosθ 424πε ⎝⎠⎝⎠⎝⎠⎝⎠rr⎢⎥⎜⎟⎝⎠rr⎜⎟ ⎝⎠⎝⎠4 ⎝⎠r o ⎩⎭⎪⎪⎣⎦⎝⎠⎝⎠ qdd⎛⎞⎛⎞⎛⎞11⎧⎫ ⎛⎞ Then: =+⎜⎟⎜⎟⎜⎟⎨⎬ cosθθ ⎜⎟ cos 42πε o ⎝⎠⎝⎠⎝⎠rr⎩⎭ ⎝⎠ r q ⎛⎞11⎛⎞⎧⎫⎛⎞dqd ⎛⎞⎛⎞1 = ⎜⎟⎜⎟⎨⎬2 ⎜⎟cosθθ= ⎜⎟⎜⎟ cos 4πε o ⎝⎠r ⎝⎠2 ⎩⎭⎝⎠rrr4πε o ⎝⎠⎝⎠

qd⎛⎞⎛⎞11 qdqd ⎛ ⎞ ⎛ ⎞ Thus: Vrdipole () ⎜⎟⎜⎟cosθ == ⎜22 ⎟ cosθθ ⎜ ⎟ cos 444πεooo⎝⎠⎝⎠rr πε ⎝ r ⎠ πε ⎝ r ⎠

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 5 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede The Magnitude of the Electric Dipole : p ≡ qd= p

Thus, we may also express the potential of a pure physical dipole as: qd⎛⎞11 p ⎛⎞ Vrdipole ()==⎜⎟22cosθ ⎜⎟ cosθ (valid for d << r) 44πεoo⎝⎠rr πε ⎝⎠

1 1 Note that: Vr()∼ whereas Vr()∼ (valid for point charge q located at origin) dipole r 2 monopole r

We define the vector as: p ≡ qd where the charge-separation distance vector d points (by convention) from –q to +q:

+q

p ≡ qd SI Units of p = Coulomb-meters d d

−q

In our current situation here we see that ddz= ˆ :

zˆ

P (field point) +q θ r d d p ≡ qd Ο yˆ

−q xˆ

Thus here if: p ==qd qdzˆ but: zrˆ = cosθ ˆ then: p ==qd qdzˆ = qdcosθ rˆˆ = pcosθ r

qd⎛⎞1 qd cosθθ ⎛⎞ 1 p cos ⎛⎞ 1 Then: Vrdipole () ⎜⎟222cosθ == ⎜⎟ ⎜⎟ 444πεooo⎝⎠rrr πε ⎝⎠ πε ⎝⎠

The potential Vrdipole ()associated with an electric dipole moment p ( p ==qd qdzˆ) from a pure, physical electric dipole oriented with ddz= ˆ , for rd is thus given by: pprcosθ 1iˆ 1 ⎛⎞ ⎛⎞ ˆ Vrdipole () ⎜⎟22= ⎜⎟ where: pri = pcosθ = qd cosθ 44πεoo⎝⎠rr πε ⎝⎠

6 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede The electric field Erdipole () associated with a pure, physical electric dipole, with electric dipole moment p ==qd qdzˆ is: dipole dipoleˆ dipole Erdipole()=−∇ VrErrErEr dipole () = r ()ˆ +θϕ( )θ + ( )ϕˆ in spherical-polar coordinates.

The components of Erdipole ()in spherical-polar coordinates are:

dipole ∂Vrdipole ( ) 12p Err ()=− = 3 cosθ ∂rr4πε o dipole 11∂Vrdipole ( ) p Erθ ()=− = 3 sinθ rr∂θπε4 o 1 ∂Vr( ) Erdipole ()=−dipole =0 ϕ r sinθϕ∂

Explicitly, the electric field intensity of a pure, physical electric dipole with electric dipole moment p ==qd qdzˆ (in spherical-polar coordinates) is:

12pp 1 1 p Er=+=cosθ rˆˆ sinθθˆˆ⎡ 2cos θ r + sin θθ ⎤ dipole () 333⎣ ⎦ 444πεooorrr πε πε

1 1 Note that: Er()∼ (c.f. w/ Er()∼ for single point charge q at r = 0 ). dipole r3 monopole r 2 Note also that Vrdipole ()and Erdipole ()have no explicitϕ -dependence, since the charge configuration for an electric dipole is manifestly axially / azimuthally symmetric (i.e. charge configuration for electric dipole is invariant under arbitrary ϕ -rotations). pri ˆ 1 ⎛⎞ ˆ Now: Vrdipole ()= ⎜⎟2 with electric dipole moment p = qdzˆ, and pri == pcosθ qd cosθ , 4πε o ⎝⎠r (since zrˆi ˆ = cosθ ), and rxyz2222=++in Cartesian/rectangular coordinates.

In Cartesian/rectangular coordinates the electric field intensity of a pure, physical electric dipole with electric dipole moment p ==qd qdzˆ (in spherical-polar coordinates) is:

⎛⎞∂∂∂ dipole dipole dipole Er()=−∇ Vrdipole () =−⎜⎟ xˆˆ + y + zVrExEyEzˆˆ dipole () =xyz ˆ + ˆ + dipole ⎝⎠∂∂∂xyz

Transformation from Spherical-Polar → Cartesian Coordinates:

xr==+−sinθ cosϕ xˆˆ sin θ cos ϕ r cos θ cos ϕθˆ sin ϕϕˆ yr==++sinθ sinϕ yˆˆ sin θ sin ϕ r cos θ sin ϕθˆ sin ϕϕˆ zr==−cosθ zˆ cosθθθ rˆ sin ˆ

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 7 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede It is a straight-forward exercise to show that the electric field components associated with a pure physical electric dipole with electric dipole moment p ==qd qdzˆ (in Cartesian coordinates) are:

dipole pxzp⎛⎞33sincos ⎛θθ ⎞ Ex ==⎜⎟53 ⎜ ⎟ 44πεoo⎝⎠rr πε ⎝ ⎠ (since charge configuration pyzp33sincosθθ dipole⎛⎞ ⎛ ⎞ dipole of electric dipole is axially / EEyx==⎜⎟53 ⎜ ⎟ = ⇐ 44πεoo⎝⎠rr πε ⎝ ⎠ azimuthally symmetric) 22 2 dipole pzr⎛⎞⎛33cos1−− p θ ⎞ Ez ==⎜⎟⎜53 ⎟ 44πεoo⎝⎠⎝rr πε ⎠

In coordinate-free form, it is also a straight-forward exercise (try it!!!) to show that the electric field intensity of a pure physical electric dipole with electric dipole moment p ==qd qdzˆ is of the form: 11⎛⎞ Erphysical =−3 prrpiˆˆ dipole () ⎜⎟3 ⎣⎡ () ⎦⎤ 4πε o ⎝⎠r whereas the coordinate-free form of a point electric dipole is of the form:

11⎛⎞ 1 Erpoint =−−3 prrppriˆˆ δ 3 dipole () ⎜⎟3 ⎣⎦⎡⎤() () 43πεoo⎝⎠r ε

E − Field Lines & Equipotentials Associated with a Pure, Physical Electric Dipole:

n.b. Equipotentials are ⊥ to lines of Er() everywhere!

8 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede We explicitly show here that the electric field associated with a pure physical electric dipole with electric dipole moment p ==pzˆˆ qdz can be written in coordinate-free form as: 11⎛⎞ Erphysical =−3 prrpi ˆˆ dipole () ⎜⎟3 ⎣⎡ () ⎦⎤ 4πε o ⎝⎠r We have already shown (above) that: 1 ⎛⎞p Erphysical =+⎡2cosθ rˆ sinθθˆ⎤ dipole () ⎜⎟3 ⎣ ⎦ 4πε o ⎝⎠r Now: p = pzˆ and zrˆ =−cosθ ˆ sinθθˆ (in spherical-polar coordinates) Thus: piiirpprpzrˆˆˆˆ==ˆ zˆ r But: zrˆiiˆˆ=−()cosθ r sinθθˆ r ˆ = cos θ ϕˆ And: rrˆˆi =1, θˆirˆ = 0 Thus: priˆ = pcosθ θ θˆ =( prri ˆˆ) And: pprrp=+()iiˆˆ()θˆˆθθ = pcos rp ˆ − sin θθ ˆ O yˆ So therefore: ϕ ϕˆ ˆ ⎣⎦⎡⎤33coscossin()prrpprprpi ˆˆ−=θ ˆ −θθθ ˆ + =+ 2prp cosθθθˆ sin ˆ xˆ ⎡⎤ˆ ˆ =+pr⎣⎦ 2cosθθθ sin 11⎛⎞ Erphysical 3 prrpiˆˆ Thus: dipole ()=−⎜⎟3 ⎣⎦⎡⎤() Q.E.D. 4πε o ⎝⎠r

The Potential Vrquad () and Electric Field Erquad ( ) Associated with a Pure, Linear Physical Electric

We have seen that a pure, physical electric dipole was constructed by: 1. Starting with a monopole electric moment (i.e. charge +Q) 2. “Copying it” 3. Charge-conjugating (+Q → −Q) the “copied” charge 4. Displacing the conjugated charge –Q from the original charge +Q by a separation distance d

Likewise, we can construct a pure, physical, linear electric quadrupole by: 1. Starting with a pure, physical, linear electric dipole 2. “Copying it” 3. Charge-conjugating the charges associated with the “copied” electric dipole 4. Translating the charge conjugated electric dipole along the symmetry axis of the original electric dipole by amount d, as shown in the figures below:

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 9 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede 1. zˆ 2. zˆ 3. zˆ copy +Q +Q → +Q +Q −Q

d d d

−Q −Q −Q −Q +Q

Original Original Copy Original Charge-Conjugated Copy

4. zˆ

+Q −Q

d

= −2Q −Q +Q −Q d Translation of charge-conjugated copy along axis of original dipole +Q by amount d.

Pure, Physical, Linear Electric Quadrupole:

zˆ P (Field Point) ra Note that this linear electric quadrupole has +Q axial / aximuthal symmetry – i.e. because r all charges (+Q, −2Q, +Q) are co-linear d (all on zˆ axis), problem is invariant under Ο θ (arbitrary) ϕ -rotations. −2Q yˆ ⇒ Vrquad ( ) and Erquad () will have no π −θ explicit ϕ -dependence for the linear d xˆ electric quadrupole. rb

+Q n.b. QTOT = 0for pure electric quadrupole.

Again, we use the principle of (linear) superposition to obtain Vrquad ( ): VrVrVquad()== TOT ()+−+ Q (@@0@ zdV =++=+=− ) 2 Q( z) V Q ( zd) 12⎛⎞QQQ 1⎛⎞ Q⎡⎤ ⎛⎞⎛⎞ r r =−+=⎜⎟⎜⎟⎢⎥ ⎜⎟⎜⎟ −+ 2 44πεoa⎝⎠rrr b πε o⎝⎠ r⎣⎦ ⎝⎠⎝⎠ r a r b

10 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede

222 222 Again, using the Law of Cosines: rrda =+−2cos rdθ and rrdrdb =++cosθ

We obtain: 1 ⎛⎞Qr⎧⎫ r Exact analytic Vrquad ()=−+⎨⎬2 ⇐ ⎜⎟ 22 22 expression 4πε o ⎝⎠r ⎩⎭rd+−2cos rdθ rd ++ 2cos rd θ

Again, for regime where the observation point P is far away from pure, physical, linear electric ⎛⎞r ⎛⎞r quadrupole, i.e. r >> d, we expand ⎜⎟ and ⎜⎟ in a binomial (i.e. Taylor) series ⎝⎠ra ⎝⎠rb (as was done previously for the case of a pure, physical electric dipole).

Neglecting terms in these expansions that are higher order than linear (i.e. > ()dr2 ) we obtain: 2 2 ⎛⎞rd⎛⎞ ⎛⎞ d()3cosθ − 1 ⎜⎟ 1cos−+⎜⎟θ ⎜⎟ ⎝⎠rra ⎝⎠ ⎝⎠ r2 2 2 ⎛⎞rd⎛⎞ ⎛⎞ d()3cosθ − 1 ⎜⎟ 1cos++⎜⎟θ ⎜⎟ ⎝⎠rrb ⎝⎠ ⎝⎠ r2 x=cosθ

Recall that the Ordinary Legendré Polynomials Px () are:

Shorthand notation: Px00()=→1 P( cosθ ) = 1

PP(cosθ )()= θ Px11()=→ x P( cosθ ) = cosθ ()31x22−−() 3cos1θ Px()=→ P()cosθ = 2222 2 2 ⎛⎞rdd⎛⎞ ⎛⎞ ⎛⎞rdd⎛⎞ ⎛⎞ ∴ ⎜⎟ PP01()θ −+⎜⎟()θθ ⎜⎟ P 2() and ⎜⎟ PP01()θ ++⎜⎟()θθ ⎜⎟ P 2() ⎝⎠rrra ⎝⎠ ⎝⎠ ⎝⎠rrrb ⎝⎠ ⎝⎠ 2 2 11⎛⎞Qr⎡⎤⎛⎞ ⎛⎞ r ⎛⎞⎛⎞ Q⎡ d(3cosθ − 1)⎤ ∴ Vrquad ()=−+=⎜⎟⎢⎥⎜⎟ 2 ⎜⎟ ⎜⎟⎜⎟⎢ 2 ⎥ 442πεoa⎝⎠rr⎝⎠ ⎝⎠ r b πε o ⎝⎠⎝⎠ r⎢ r ⎥ ⎣⎦⎣ ⎦ 213cos1Qd 22⎛⎞⎛⎞θ − = ⎜⎟3 ⎜⎟ 42πε o ⎝⎠r ⎝⎠

Then for r >> d: P2 (θ ) 213cos121Qd22⎛⎞⎛⎞θ − Qd 2 ⎛⎞ Vrquad () ⎜⎟33⎜⎟= ⎜⎟ P2 ()θ 424πεoo⎝⎠rr⎝⎠ πε ⎝⎠

1 11 Note that: Vrquad ()∼ (c.f. with Vr()∼∼ and Vr() 2 ) r3 monopoler dipole r

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 11 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede Note also that: Vr∼ Pθ (c.f. with VrPVrP∼∼θ , θ ) quad ( )()2 monopole( ) 01( ) dipole () () 1 2 ==1cosθ 2 ()3cosθ − 1

Note further that: Vrquad ( ) must be proportional to an even power of l, i.e. Pleven= ()θ because a pure, physical, linear electric quadrupole has reflection symmetry about the zˆ -axis (i.e. about θ = π /2) (i.e. a rotation from / by a vector lying in x – y plane e.g. xˆ or yˆ axis).

zˆ zˆ θ →−()πθ

1 2 +Qa +Qb P2 (θθ) = 2 (3cos− 1) is an even under −2Q −2Q θ →−(πθ) reflection:

PP22(π −=+θθ) ( )

+Qb +Qa

We can also see that Vrdipole () must be proportional to an odd power of l, i.e. Pl= odd ()θ because a pure, physical, linear electric dipole has a sign change under reflection symmetry about θ = π /2

zˆ zˆ

θ →−()πθ P1 (θ ) = cosθ +Q −Q is an odd function under θ →−(πθ) reflection: p = Qdzˆ p =−Qdzˆ PP11(π −=−θθ) ( )

=0 cos()π −=θπθθπ cos cos + sin sin −Q +Q = −cosθ

Likewise, Vrmonopole () must be proportional to an even power of l: zˆ θ →−(πθ) zˆ

PP00()θπθ=−= ( ) 1 +Q +Q

12 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede As we have seen for the two previous cases, that of:

1. The electric monopole, with its accompanying electric monopole moment, the electric charge Q (n.b. Q is a scalar quantity) (SI units of Q: Coulombs)

2. The electric dipole with its accompanying electric dipole moment p ≡==Qd, p p Qd (n.b. p is a vector quantity) (SI units of p : Coulomb-meters) 3. The electric quadrupole also has an accompanying electric quadrupole moment QQdd≡ 2 (n.b. Q is a quantity) (SI units of Q : Coulomb-meters2) Tensor QQdd≡ 2 = “double vector” QQddQd≡=222 2-dimensional matrix Formally speaking, Q is a rank-2 tensor (i.e. a 2-dimensional matrix) - the 9 elements of the Q tensor (in general) are: Q Q Q n.b. Q has only six independent components, because Q = Q xx yz zx ij ji Q = Qxy Qyy Qzy i.e. Qxy = Qyx

Qxz Qyz Qzz Qxz = Qzx Qyz = Qzy Also, note that: Qxx + Qyy + Qzz = 0 or: Qzz = −(Qxx+Qyy) {i.e. Q is traceless}

The quadrupole moment tensor can also be written in coordinate-free form, e.g. in Cartesian coordinates as:

n = # discrete charges qi n 1 2 2 Qrrrq≡−2 ∑()31ii i i with rrriii= i i=1 xˆˆx 0 0 Unit Dyadic: 1 ≡ 0 yˆˆy 0 0 0 zzˆˆ

For the case of a pure, linear (i.e. axially/azimuthally symmetric) electric quadrupole with quadrupole moment Q (e.g. oriented along the zˆ -axis):

zˆ Here, Qxx = Qyy, and since: Qxx + Qyy + Qzz = 0 +Q 2 d Then: Qzz = −2Qxx = −2Qyy ≡ 2Qd All other Qij vanish (= 0) for ij≠ −2Q −1 0 0 n.b. conventions / definitions of linear 2 d i.e. QQdquad = 0 −1 0 linear Qquad differ in different textbooks!!! +Q 0 0 +2

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 13 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede For the case of a pure, linear (i.e. axially/azimuthally symmetric) electric quadrupole with quadrupole moment Q (oriented along the zˆ -axis), expressed in Cartesian coordinates:

zˆ P (Field Point) # discrete charges n 1 2 2 i ra Qrrrq≡−2 ∑()31ii i i with rrriii= i=1 +Q r Unit Dyadic: d r xˆˆx 0 0 b −2Q yˆ 1 ≡ 0 yˆˆy 0 d 0 0 zzˆˆ +Q xˆ irdz==+1: 1 ˆ qQ1 = + rrii= irz==2 : 0ˆ qQ= −2 2 2 irdz==−3: 3 ˆ qQ3 = +

12 ⎛⎞=0 =0 1 Thus: QQdzzd=−−313022ˆˆ Qi zz ˆˆ − 01i + Qdzzd313122ˆˆ−= Qdzz 2 ˆˆ − ()⎜⎟()() 22 ⎝⎠2 for charge 1: for charge 3: +Qr @ =+ dzˆ +Qr @ =− dzˆ 1 for charge 2: 3 −=2Qr @ 0 zˆ 2 22⎛⎞31zzˆˆ − ∴ QQdzz=−=() 3ˆˆ 1 2 Qd⎜⎟ ⎝⎠2 223cos2 θ − 1 21Qd⎛⎞() 21 Qd ⎛⎞ 1 2 Then: Vrquad () ⎜⎟33= ⎜⎟ P2 ()cosθ P2 ()cosθθ=−() 3cos 1 424πεoo⎝⎠rr πε ⎝⎠ 2

We can express Vrquad ()in a different (but totally equivalent manner), using the fact(s) that:

rxyzˆˆˆ=++sinθ cosϕθϕθ sin sin cos ˆ zrˆˆiiˆˆ== rz cosθ xxˆˆii=1, xy ˆˆ==0, xz ˆ iˆ 0 33cos()()rzˆˆiiˆˆ zr = 2 θ yxˆˆiii= 0, yy ˆˆ==1, yz ˆˆ 0 rrˆˆii11= zxˆˆˆˆiii = 0, zyˆ ==0, zz 1

14 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede Then for observation/field point P far from quadrupole, i.e. r >> d:

coordinate-free form 2 ⎡⎤ 11⎛⎞ 2Qd ⎛⎞ 1rzzrˆˆii()31ˆˆ − ˆˆii ⎢⎥ Vrquad ()≈=⎜⎟33() rQr ⎜⎟ 442πε⎝⎠rr πε ⎝⎠⎢⎥ oo⎣⎦ 21Qd222⎛⎞⎡⎤31()()rzˆˆˆˆiiˆˆ zr− r ii r 213cos1 Qd ⎛⎞⎡ θ − ⎤ ==⎜⎟33⎢⎥ ⎜⎟ 4242πεrr πε ⎢ ⎥ oo⎝⎠⎢⎥⎣⎦ ⎝⎠⎣ ⎦ ≡P2 ()cosθ 21Qd 2 ⎛⎞ = ⎜⎟3 P2 () cosθ 4πε o ⎝⎠r

axially-symmetric Vrquad () as given above is valid for a pure, linear, physical electric quadrupole oriented along the zˆ -axis, for r (observation / field point) >> d.

The potential Vr()and electric field intensity Er( ) associated with a pure, physical, quad quad linear electric quadrupole with quadrupole moment Q (oriented along the zˆ -axis) are: 213cos1Qd 22⎛⎞⎡⎤θ − Vrquad ()= ⎜⎟3 ⎢⎥ 42πε o ⎝⎠r ⎣⎦

ˆ ErErEEquad()=++ rˆ θϕθϕˆ =−∇ Vr quad ( ) , in spherical-polar coordinates:

∂Vr() 32iiQd22⎛⎞ 1⎡⎤ 3cosθ − 1 32 Qd 2 ⎛⎞ 1 Err ()=− =⎜⎟44⎢⎥ = ⎜⎟ P2 ()cosθ ∂rr424πεoo⎝⎠⎣⎦ πε ⎝⎠ r 1321∂Vr() i Qd 2 ⎛⎞ Erθ ()=− = ⎜⎟4 sinθ cosθ rr∂θπε4 o ⎝⎠ 1 ∂Vr() Er()=− =0 ← No ϕ -dependence because charge configuration is manifestly ϕ r sinθϕ∂ axially / azimuthally symmetric (invariant under arbitrary ϕ -rotations)

Explicitly writing out the form of the electric field intensity Erquad ( ) for a pure, linear, physical electric quadrupole oriented along the zˆ -axis, for r (observation / field point) >> d:

32iiQd22⎛⎞ 1⎡⎤ 3cosθ − 1 32 Qd 2 ⎛⎞ 1 ˆ ˆ Erquad ()=+⎜⎟44⎢⎥ r ⎜⎟sinθ cosθθ 424πεoo⎝⎠rr⎣⎦ πε ⎝⎠

32i Qd 22⎛⎞ 1⎡⎤⎛⎞ 3cosθ − 1 ˆ ˆ =+⎜⎟4 ⎢⎥⎜⎟r sinθθθ cos 42πε o ⎝⎠r ⎣⎦⎝⎠

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 15 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede E -field lines & equipotentials associated with a pure, physical, linear electric quadrupole: n.b. E -field lines ⊥ to equipotentials everywhere in space

Higher-Order Pure, Linear Physical Electric Multipoles

The next higher order pure, linear physical multipole is known as the pure, linear physical electric octupole. We can construct / create it (as before) by:

1. Starting with a pure, linear, physical electric quadrupole 2. “Copying it” 3. Charge-conjugating (Q→ −Q) the charges associated with the “copied” electric quadrupole 4. Translating the charge-conjugated electric quadrupole along the symmetry axis of the original electric quadrupole, this time by an amount 2d:

1. zˆ 2. zˆ zˆ 3. zˆ zˆ copy +Q +Q → +Q +Q −Q d d d −2Q −2Q → −2Q −2Q + +2Q d d d +Q +Q → +Q +Q −Q

Original Original Copy Original Charge-Conjugated Copy

16 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede 4. zˆ zˆ

+Q −Q Original → ← Charge-Conjugated Copy −2Q +2Q

= 0Q +Q −Q −Q

+2Q

−Q

Pure, Linear (Axially/Azimuthally-Symmetric) Physical Electric Octupole:

zˆ P (Observation / Field point) +Q ra

d rb Following the methodology as used in previous cases: 2d −2Q 4 ⎛⎞11 d r VrVrP= ∼ cosθ ∗∗Ο octupole()∑ i () ⎜⎟4 3() i=1 ⎝⎠r 4πε o 1 3 =−2 ()5cosθθ 3cos ⎛⎞Ο 1 ˆ 4d d Ο rc y ErVroctupole()=−∇ octupole ()∼ ⎜⎟5 ∗ ⎝⎠r 4πε o

+2Q rd Ο = Octupole Moment ∼ Qddd (Rank-3 tensor) 3 3 xˆ d Ο ~ Qd (SI units: coulomb-meter )

−Q Note: QTOT = 0

th th In general, for l -order electric multipole, = 0, 1, 2, 3, . . . defining M ≡ -order multipole moment (SI units: coulomb-(meters)b) then the potential associated with a pure, physical, linear multipole moment is of the form:

M ⎛⎞1 Vr()∼ ⎜⎟+1 P()cosθ 4πε o ⎝⎠r

The electric field intensity associated with a pure, physical, linear multipole moment is of the 1 M form: Er()=−∇ Vr ()∼ +2 4πε o r

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 17 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede Multipole Moments, Potential and Electric Field Associated with an Arbitrary Localized Electric Charge Distribution ρ (r′) - Outside of ρ ()r′

Suppose we have an arbitrary, but localized electric charge distribution ρ ()r′ somewhere in space, contained within the volume v′ and bounded by the surface S′ :

r =−rr′ r= r = rr− ′ cosΘ=′′rrˆˆi = cosine of opening angle between vectors r and r ′ . Θ′ = opening angle between vectors rrand ′ - very important!

22 22 Law of Cosines: r 2 =+rr′′′ −2cos rr Θ=+− rr ′ 2 rri ′

If the observation / field point P is far away from electric charge distribution ρ ()r′ such that: rr== aa = maximum distance of ρ (r′) to originϑ then for r >> a (a = max value of r′ ):

2 2 2 ⎡⎤⎛⎞rr′′ ⎛⎞ ⎛⎞rr′′ ⎛⎞ r 2 =+r ⎢⎥12cos⎜⎟ − ⎜⎟ Θ′ or: r = r 12cos+−⎜⎟ ⎜⎟ Θ′ ⎣⎦⎢⎥⎝⎠rr ⎝⎠ ⎝⎠rr ⎝⎠ ≡ε 1 for r a

2 ⎛⎞rr′′ ⎛⎞ Define: ε ≡−⎜⎟2cos ⎜⎟ Θ′ for r >> a (a = max value of r′ ) ⎝⎠rr ⎝⎠

1 ρ ()r′ 11 −12 Now: Vr()= ∫ dτ ′ with: =+()1 ε 4πε o v′ r r r

18 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede

Carry out a (full) binomial expansion of 1/r (for r >> a):

∞ 11−1/2 1⎛⎞−12 n 1⎛⎞ 1 323 5 =+()1εεεεε =∑⎜⎟ =−+−⎜⎟ 1 + ... r rrn=0 ⎝⎠ n r⎝⎠28 16 n 1 ⎛⎞−12 ()−1 Γ−()n 2 where: ⎜⎟= is the binomial coefficient and Γ( x) is the gamma function. ⎝⎠ n nn! Γ() Γ−()n 1 and: 2 =−()11() −+1 ....() −+− 1nn 1 =−() 11() ....() −3 Γ()n 22 2 22 2

2233 11⎡⎤ 1⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞⎛⎞rr′′ 3 r ′ r ′ 5 r ′ r ′ Then: =−⎢⎥1⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟⎜⎟ −Θ+ 2cos′′ −Θ− 2cos −Θ+ 2cos ′ ... r rrr⎣⎦⎢⎥28⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠⎝⎠ rr 16 rr

Collecting together like powers of rr′ :

233 11⎡⎤⎛⎞rr′′′′′′ ⎛⎞⎛⎞⎛ 3cos123Θ− ⎛⎞ r 5cos Θ− 3cos Θ ⎞ =+⎢⎥1⎜⎟ cos Θ+′ ⎜⎟⎜⎟⎜ + ⎜⎟ ⎟ + ... rr r22 r r ⎣⎦⎢⎥⎝⎠ ⎝⎠⎝⎠⎝ ⎝⎠ ⎠

Thus we see that: 23 11⎡⎤⎛⎞rr′′ ⎛⎞ ⎛⎞ r ′ =Θ+Θ+Θ+Θ+⎢⎥PP01()cos′′⎜⎟() cos ⎜⎟ P 2() cos ′ ⎜⎟ P 3() cos ′ ... !!!! r rrr⎣⎦⎢⎥⎝⎠ ⎝⎠ ⎝⎠ r

11∞ r′ ⎛⎞ ′ Hence: =Θ∑⎜⎟P ()cos where Θ′ = opening angle between rrand ′. r rr=0 ⎝⎠

2 1 ⎛⎞rr′′ ⎛⎞ This remarkable result occurs because (where ε ≡ ⎜⎟−Θ2cos ⎜⎟ ′ ) is known as the 1+ ε ⎝⎠rr ⎝⎠ Generating Function for the Legendré Polynomials!!!

11⎛⎞ Then, since Vr()= ∫ ρ () r′′⎜⎟ dτ for r >> a (a = max value of r′ ), the potential outside 4πε o v′ ⎝⎠r the volume v′ containing the charge distribution ρ (r′) is given by: 11⎛⎞∞ ⎛r′ ⎞ Vr=Θρ rPd′ cos ′′τ outside () ∫ ⎜⎟∑ ⎜ ⎟ ()( ) 4πε ⎝⎠rr=0 ⎝ ⎠ o v′ 11∞ ⎛⎞ =ΘrrP′ ρ ′′′ cos dτ ∑⎜⎟+1 ∫ () () ( ) 4πε o =0 ⎝⎠r v′

11⎛⎞ Then defining: Vroutside =Θ rrPd′ ρ ′′′cos τ () ⎜⎟+1 ∫ () () ( ) 4πε o ⎝⎠r v′

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 19 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede

∞∞ 11⎛⎞ We obtain (for r >> a): Vr== Voutside r rrPd′ ρ ′′′cos Θτ outside ()∑∑ () ⎜⎟+1 ∫ () () ( ) ==004πε o ⎝⎠r v′ Linear superposition of Θ′ = opening angle multipole potentials!!! between rrand ′.

This expression is known as the Multipole Expansion of Vr( ) in powers of 1/r. outside It is valid / useful when r >> a (a = max value of r′ ). Note that this is an exact expression.

Having obtained Vroutside (), we can then obtain Eroutside( ) =−∇ Vr outside ( ), and thus we see that: ∞∞ outside outside outside outside Eroutside ()==−∇∑∑ Er () V () r i.e. Er( )()=−∇ Vr ==00 Linear superposition of multipole electric fields!!!

Thus, we see that, for observation / field point distances far away from the (arbitrary) localized electric charge distribution ρ ()r′ (i.e. r >> a (a = max value of r′ )) the electrostatic potential Vroutside () and associated electric field Eroutside( ) =−∇ Vr outside ( ) are linear superpositions of outside outside multipole electrostatic potentials Vr ( ) and multipole electric fields Er () respectively, th each arising from the electric multipole moment M associated with the localized electric charge distribution ρ ()r′ !!!

Order of Electrostatic Potential Electric Field Electric Multipole Electric Multipole outside outside outside Moment M Vr () Er( ) =−∇ Vr( ) = 0 P0 = 1 1 ⎛⎞Q M0 = Q (total/net = ⎜⎟2 Monopole 1 ⎛⎞Q 4πε ⎝⎠r charge, coulombs) = ⎜⎟ o (scalar) 4πε o ⎝⎠r = 1 1 ⎛⎞Qd 1 ⎛⎞Qd M ==Qd p ∼ ∼ 1 Dipole ⎜⎟2 ⎜⎟3 4πε o ⎝⎠r 4πε o ⎝⎠r (coulomb-meters) (vector) = 2 2 2 1 ⎛⎞Qd 1 ⎛⎞Qd M 2 ==2Qdd Q Quadrupole ∼ ⎜⎟3 ∼ ⎜⎟4 2 4πε o ⎝⎠r 4πε o ⎝⎠r (coulomb-meters ) (rank-2 tensor) = 3 3 3 1 ⎛⎞Qd 1 ⎛⎞Qd M3 ∼ Qddd =Ο Octupole ∼ ⎜⎟4 ∼ ⎜⎟5 3 4πε o ⎝⎠r 4πε o ⎝⎠r (coulomb-meters ) (rank-3 tensor) = 4 4 4 1 ⎛⎞Qd 1 ⎛⎞Qd M 4 ∼ Qdddd= S Sextupole ∼ ⎜⎟5 ∼ ⎜⎟6 4 4πε o ⎝⎠r 4πε o ⎝⎠r (coulomb-meters ) (rank-4 tensor) ...... th Order 1 ⎛⎞Qd 1 ⎛⎞Qd M ∼ Qr= M ∼ ∼ () Multipole ⎜⎟+1 ⎜⎟+2 b 4πε o ⎝⎠r 4πε o ⎝⎠r (coulomb-meters ) (rank- tensor)

20 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede Thus we see that: → The higher-order multipole fields fall off 1/r faster than those associated with next lower order multipole. → Must get in closer and closer to charge distribution ρ (r′) in order to sense / observe / detect the higher-order moments!

We can write the electrostatic potential yet another way:

For r >> a (a = max value of r′ )

⎡⎤2 2 ∞ 11 1 1(3()rrˆi ′′− r ) V r== Voutside r⎢⎥ρτ rd′′ + ri r ′ ρ rd ′ τ ′ + ρ rd ′ τ ′ +.... outside()∑ l () ∫∫∫() 23() () l=0 42πε ⎢⎥rr r o ⎣⎦⎢⎥vvv′′′ ⎡ ⎤ ⎢ ⎥ 1 Q priiiˆˆ rQr ˆ ⎢ Net ⎥ Thus, we see that: Vroutside ()=+++23..... 4πε o ⎢ rr r ⎥ ⎢monopole dipole quadrupole ⎥ ⎣ term term term ⎦

2 2 (3()rrˆi ′′− r ) Qrd≡ ρ ′′τ p ≡ rrd′ρ ′′τ Qrd≡ ρ ′′τ with: Net ∫ () , ∫ ( ) and ∫ () ….. v′ v′ v′ 2

Recall / note: rrˆˆii′′== r r r ′cos Θ ′ where Θ′ = opening angle between rrand ′.

The multipole expansion of Vr() which contains the opening angle Θ′ between r (field outside point) and r′ (source point) can be rewritten in terms of (θ and ϕ ) for r and (θ′′ and ϕ ) for r′ using the so-called Addition Theorem for :

zˆ Θ′ = opening angle between rrand ′

S′ (source P (field point) point) Θ′ r θ′ θ r′ Spherical Harmonics Addition Theorem: yˆ 4π + ′ ′ ⎛⎞* ′′ 2π −ϕ ϕ PYY()cosΘ=⎜⎟∑ mm()(θ ,ϕθϕ , ) ⎝⎠21 + m=− ϕ′ n.b. complex conjugate xˆ

11∞∞+()rr′′ ⎛⎞ 4π () ==PYYcos Θ=′ * θ ,ϕθϕ′′ , Then: ∑∑∑++11() ⎜⎟ mm()( ) r rr−+′ lm===−00 r⎝⎠21 r

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 21 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede

∞ 11⎛⎞ Vr=Θ rrPd′′ρτcos ′′ outside () ∑⎜⎟+1 ∫ () () ( ) 4πε o =0 ⎝⎠r v′ ∞+ 11⎛⎞ ⎛ 4π ⎞ Thus: = rrYY′ ρ ′′′′* θϕ , θ , ϕ d τ ∑∑⎜⎟+1 ∫ ⎜ ⎟() (),,mm ( ) ( ) 421πε o ==−0 ⎝⎠rlv′ m ⎝+ ⎠ ∞+ outside = ∑∑Vrm () ==−0 m 14⎛⎞π ()r′ where: Vroutside = ρ rYY′ * θϕ,, θ′′ ϕ d τ ′ mmm() ⎜⎟∫ +1 (),, ( ) ( ) 421πε o ⎝⎠ + v′ r ∞+ 141⎛⎞⎛⎞π Vr= rrY′ ρ ′′′′* θϕ,, Y θ ϕ d τ outside () ∑∑⎜⎟⎜⎟+1 ∫ () (),,mm ( ) ( ) 421πε o ==−0 ⎝⎠⎝⎠lr+ v′ m Thus: ∞+ 141⎛⎞⎛⎞π ⎡ ⎤ = YrrYd* θ ,ϕρθϕτ′ ′′′′ , ∑∑⎜⎟⎜⎟+1 ,,mm( )⎢∫ () () ( ) ⎥ 421πε o ==−0 ⎝⎠⎝⎠lr+ m ⎣v′ ⎦

The Ylm, ()θ,ϕ are the Spherical Harmonics; θ and ϕ are the polar & azimuthal angles for r , the vector from the origin to the field point, P and θ′ and ϕ′ are the polar & azimuthal angles for r′ , the vector from the origin to the source point, S′ .

We can then define qm - the Electric Multipole Moment of order & m: qrrYd≡ ′′ρ θϕ ′′′, τ mm∫ () () ( ) v′

Because of the properties of the Y,m (θ,ϕ ) , namely that:

21()()+−m ! YY()()()θ,1ϕθϕ=− * , YPe()θϕ,cos= () θ imϕ −mm, m 4!π () + m

* We see that: qq−mm=−()1 ,

outside 14⎛⎞π 1* Thus: VYq,,mmm= ⎜⎟+1 ()θϕ, 421πε o ⎝⎠ + r ∞+141 ∞+⎛⎞π Vr== Voutside r Y* θϕ, q Then: outside()∑∑,,, m () ∑∑⎜⎟+1 m() m ==−00mm421πε o ==−⎝⎠ + r

Again, Eroutside()=−∇ Vr outside () which by the principle of linear superposition becomes: ∞+ ∞+ outside outside ==−∇∑∑Er,,mm() ∑∑ Vr () ==−00mm ==−

outside outside i.e. Er,,mm()=−∇ Vr ()

22 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede

outside The main advantage of using these seemingly more complex expressions for Vr,m () * involving the Ym ()θϕ, and Ym ()θϕ′′, spherical harmonics is that they are directly connected to a right-handed xˆˆ−−yzˆ coordinate system. The earlier expression for Vroutside () involving the P ()cos Θ′ Legendré Polynomials, it must be kept in mind at all times that Θ′ = opening angle between field point r and source point r′ .

The explicit derivation of Vroutside () using the Addition Theorem for Spherical Harmonics: ∞+ 141π ⎛⎞⎛⎞* ′ ′′′′ Vroutside()= ⎜⎟⎜⎟+1 Y m(θ,,ϕρ )() rrYd () m ( θϕτ ) 421πε ∑∑ + r ∫ o ==−0 ⎝⎠⎝⎠m v′ ≡qm (electric multipole moment of order &m ) thus makes it explicitly clear that Vrfcnroutside ( ) = ( ,,θ ϕ ) only – all source variable ()r′′′,,θ ϕ dependence has been integrated out, in carrying out the over the volume v′ !!!

Thus Vroutside () is fully capable of correctly/exactly describing many other kinds of multipole moments we have not yet discussed, e.g.:

A. Pure Physical Electric Dipole(s) Lying in the x-y Plane: a. zˆ b. zˆ c. zˆ

d/2 d/2 −Q −Q d/2 d/2 +Q −Q d/2 d/2 yˆ yˆ yˆ

+Q ϕ xˆ xˆ xˆ (x-axis) (y-axis) (x-y plane)

B. Pure, Physical Electric Dipole Randomly Oriented in Space:

zˆ

−Q d/2 (3-D dipole) θ yˆ ϕ xˆ d/2 +Q

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 23 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede C. Pure Physical, but Non-Colinear Electric : a. zˆ b. zˆ c. zˆ

+Q d/2 −Q +Q d/2 +Q −Q d/2 −Q d/2 d/2 −Q d/2 yˆ d/2 yˆ yˆ d/2 d/2 −Q d/2 +Q d/2 d/2 −Q +Q +Q xˆ xˆ xˆ

(x-y plane) (y-z plane) (x-z plane)

D. Pure Physical, but Non-Colinear Electric Octupoles:

Cube Centered on (x,y,z) = (0,0,0)

The Choice of Origin of Coordinates Does Matter!!! Note that the choice of origin of coordinates in the electric multipole expansion of Vroutside ( ) does matter – can affect e.g. determination of electric dipole moment, p if QNET ≠ 0 !!

A point charge Q located at the origin of coordinates Ο (x,y,z) = (0,0,0) is a pure electric monopole. However, a point charge Q located some distance d along dˆ from the origin is no longer a pure electric monopole! The monopole moment Q = QTOT does not change, but Vr0 ( ) 1 ⎛⎞Q (where = 0) does change, because Vr()= ⎜⎟ is not quite correct – the exact potential 4πε o ⎝⎠r 1 ⎛⎞Q is Vr()= ⎜⎟ and r ≠ r; however r r when r >> r′ . 4πε o ⎝⎠r

24 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede

- For higher electric moments, if (and only if) QTOT = QNET = 0, then (pure) electric

moment M (where > 0) is independent of choice of origin of coordinate system. - If net / total charge QNET = QTOT ≠ 0, then the higher-order electric moment(s) M (where > 0) can be made to vanish if one chooses the origin or coordinates to be located at the charge-weighted center of charge, then r′ = 0 .

qrrYd===′′ρθϕτ,0 ′ if r ′0 mm∫ () () ( ) v′

−2Q +Q −Q +Q −Q Ο (origin) d = d zˆ + d zˆ

p1 = Qdzˆ p2 = −Qdzˆ

Note here that: pp= 12+= p 0!!!

If the origin is displaced from the center of charge for electric dipole by an amount a :

e.g. rra*′ =+′ where a = vector displacement of origin of coordinate system,

prrdp**=→=+∫∫′ρτ()′′ ′ * ( rard ′ )() ρ ′ τ ′ vv′′ then: =+∫∫rrd′′′ρτρτ() ard () ′′ vv′′ =+paρτ rd′′ =+ pQa=+pp ∫ () Net origin v′ ==QQNet() Tot

* - If QNET ≠ 0, then p =+pporigin ≠ p because the origin-dependent electric dipole moment, pQaorigin≡≠ Net 0 !!!

If QNET ≠ 0, then the choice of origin does matter; because the electric dipole moment p depends on the choice of origin !!!

If QNET ≠ 0, then higher-order electric multipole moments must be accompanied by explicitly specifying the choice of origin of coordinates!!!

* - Iff QNET = 0, then p = p , i.e. p is independent of choice or origin of coordinate system. +Q d −5Q Origin,Ο

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 25 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede The Potential for a Pure Physical Electric Quadrupole (in Cartesian Coordinates) Not Necessarily With Colinear Charges

The potential for a pure, physical electric quadrupole (not necessarily with collinear charges) can be written in Cartesian coordinates as: 1133 ⎛⎞xx Vr=−ij 3 xxr′2δ ρτ rd′′ quad() ∑∑ ⎜⎟5 ∫ () i j ij () 42πε o ij==11⎝⎠r v′ 1133 ⎛⎞xx Vrij Q or as: quad()= ∑∑ ⎜⎟5 ij 42πε o ij==11⎝⎠r with elements of the quadrupole moment tensor Qxxrrd≡−3 ′ ′ ′′′2δ ρτ ij∫ ( i j ij ) () v′ 2222222 with rxyzxxx′′′′′′′=++=++123 1, 2,3 and where the summations i = 1, 2, 3 and j = 1, 2, 3 represent sums over the{ x,,yz} components respectively; i.e. i, j = 1: x1 ≡ x i, j = 2: x2 ≡ y and i, j = 3: x3 ≡ z =≠ 0 if i j and where δδij=−Kroenecker function { == 1 if i j }

⎛⎞QQQ11 12 13 ⎜⎟ QQQQ= The 9 elements of the quadrupole moment tensor Q are the Qij’s: ⎜⎟21 22 23 ⎜⎟ ⎝⎠QQQ31 32 33 sum of diagonal elements =0 3 Where: ∑QieQQQii =++=0 . . 11 22 33 0 (i.e. Q is a traceless rank-2 tensor /33× matrix) i=1 and also: QQij=≠ ji for ij, i.e. QQ12= 21, QQ13= 31 and QQ23= 32.

In general, if rxiyjzk=++ˆˆˆ and rxiyjzk′′=++ˆˆ ′ ′ˆ then:

11⎛⎞⎧ V r=++333 xy xy′′′′ρτ r d zx xz ′′′′ ρτ r d yz yz ′′′′ ρτ r d quad () ⎜⎟5 ⎨ ∫∫∫() () () 4πε o ⎝⎠r ⎩ vvv′′′

11122 22 22 ⎫ +−() 3x 1∫∫∫xrd′′′ρ ()τρτρτ +−() 3 y 1 yrd ′′′() +−() 3 z 1 zrd ′′′() ⎬ 222vvv′′′⎭

26 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede The 9 elements of the quadrupole moment tensor Q (in Cartesian coordinates) are thus:

Mean square of xixj (multiplied by q). 2 2 Qxrdqxqx===′′′2 ρτ ′ ′ xx ∫ () v′ 2 2 Qyrdqyqy===′′′2 ρτ ′ ′ yy ∫ () v′ 2 2 Qzrdqzqz===′′′2 ρτ() ′ ′ zz ∫ n.b. The Quadrupole Moment Tensor v′ ⇐ QxyrdqxyqxyQ====′′ρτ ′ ′ ′′ ′′ Q has only 6 independent components xyyx∫ () v′ QyzrdqyzqyzQ====′′ρτ ′ ′ ′′ ′′ yz∫ () zy v′ QzxrdqzxqzxQ====′′ρτ ′ ′ ′′ ′′ zx∫ () xz v′

Then:

11⎛⎞⎡ 1 1 1 V r=+++−+−+−333 xyQ yzQ xzQ 313131 x222 Q y Q z Q quad() ⎜⎟5 ⎢ xy yz xz() xx() yy() zz 4222πε o ⎝⎠⎣r

A relationship exists between multipole moments expressed using spherical-polar coordinates

qm and those expressed using Cartesian coordinates Qij . The first few of these are given below:

115 qq== q Q 004π 2024π 33

3115 m qp==−−=− q() QiQq with () 1 q* 10438ππz 21 13 23 −mm 3115 q11=−() pxy − ip q 22 = () Q 11 − 2 iQ 12 − Q 22 8122ππ

©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 27 2005 - 2008. All rights reserved. UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede The Energy / Work Associated With a Charge Distribution ρ (r′) Located at (or Near) the Origin of the Coordinate System in an External Electric Field Erext ()

For r >> a (a = max value of r′ ), the energy / work associated with a charge distribution in an external field Erext () is given by:

1 33 ∂E ext WQVr==−=−0 pEri 0 Q j − .... ext() ext ()∑∑ ij 6 ij==11 ∂x r =0 i x =0 =− Edli i ∫ref . ext pt . Erext()==−∇= 0 Vr ext () 0

1, 2,3 Where the summations i = 1, 2, 3 and j = 1, 2, 3 represent sums over the { x,,yz} components respectively; i.e. i, j = 1: x1 ≡ x i, j = 2: x2 ≡ y i, j = 3: x3 ≡ z

And: Qxxrrd≡−3 ′′ ′2δ ρτ ′ ′ with rxyzxxx′2222222=++=++′′′′′′ ij∫() i j ij () 123 v′ 3 And with: QQij= ji , and ∑QQQii=++=++=11 22 Q 33 QQ xx yy Q zz 0 i=1

14∞+⎛⎞⎛⎞π 1 Note: The multipole expansion method for Vr= Y* θϕ, q outside() ∑∑⎜⎟⎜⎟+1 m() m 421πε o ==−0 ⎝⎠⎝⎠lr+ m with qrYrd = ()′′′′′ (θϕ, )() ρ τ is analogous to the taking of an inner product!!! mlm∫v′

It can then be seen that the electric multipole moments qm are the strengths (i.e. coefficients) th associated with the (),m -order multipoles of the electric charge distribution ρ ()r′ !!!

Electrostatic Forces and Torques Acting on Multipole Moments of the Charge Distribution The net force and torque acting on the charge distribution as an expansion in multipole moments are given below: 33 ext ⎡⎤1 ∂=Erj ()0 F() r==+∇ qE ( r0 ) pi E () r +∇⎢⎥ Qij +.... ()r =0 ∑∑ ⎣⎦⎢⎥6 ij==11 ∂xi xi =0

⎧ 33 1 ⎪⎡⎤∂∂⎛⎞⎛⎞ext ext τ ()rpEr=× () +⎨⎢⎥⎜⎟⎜⎟ QEr23jj() =00 − QEr jj() = ()r =0 ∑∑ 3 ⎢⎥∂∂xx32jj==11 ⎩⎪⎣⎦⎝⎠⎝⎠r =0

33 ⎡⎤∂∂⎛⎞⎛⎞ext ext +=−=⎢⎥⎜⎟⎜⎟∑∑QE31jj() r 0 QE jj() r 0 ∂∂xx13jj==11 ⎢⎥⎣⎦⎝⎠⎝⎠r =0

33⎫ ⎡⎤∂∂⎛⎞⎛⎞ext ext ⎪ +=−=+⎢⎥⎜⎟⎜⎟∑∑QE12jj() r 0 QE jj() r 0⎬ .... ⎢⎥∂∂xx21jj==11 ⎣⎦⎝⎠⎝⎠r =0 ⎭⎪

28 ©Professor Steven Errede, Department of Physics, University of Illinois at Urbana-Champaign, Illinois 2005 - 2008. All rights reserved.