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Lecture Notes 08: Electric Multipole Expansion

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Lecture Notes 08: Electric Multipole Expansion UIUC Physics 435 EM Fields & Sources I Fall Semester, 2007 Lecture Notes 8 Prof. Steven Errede LECTURE NOTES 8 POTENTIAL APPROXIMATION TECHNIQUES: THE ELECTRIC MULTIPOLE EXPANSION 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 field 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 dipole 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 Electric Field Er( ) of a Pure Physical Electric Dipole “Pure” → Qnet = 0 “Physical” → Spatially extended electric eipole d ≠ 0 , d > 0 {n.b. ∃ “point” electric dipoles with d = 0, e.g. neutral atoms & molecules…} First, let us be very careful / wise as to our choice of coordinate system. A wrong choice of coordinate system will unnecessarily complicate the mathematics 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. Angle the dipole axis makes with respect to zˆ & xˆ axes must be described by two angles - θ and ϕ . Smart / wise choice of coordinate system: Exploit intrinsic symmetry 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 =+−⎜⎟ rdrcosθ =++⎜⎟ rdrcosθ ⎝⎠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 =+rr1⎜⎟ −⎜⎟ cosθ =+rr1⎜⎟ +⎜⎟ 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 Series 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 Moment: 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 electric dipole moment 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.
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