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ROCHESTER PHY 217 - Lecture 21B Notes - Electric Dipoles and their Interactions

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23 October 2002 Physics 217, Fall 2002 1Today in Physics 217: electric dipoles and their interactions Origin of coordinates for multipole moments Force, torque, and potential energy for dipoles in uniform electric fields Force on dipoles in nonuniform electric fields Example: dipole vs. dipole Permanent and induced dipoles in atoms and moleculesEprotonFelectronFInduced dipole momentEprotonFelectronFInduced dipole moment23 October 2002 Physics 217, Fall 2002 2Choice of coordinate origin matters a lot in multipole expansionChanging the origin doesn’t change the physics, but it can radically change the terms in the series. Consider a point charge at the origin:and one not at the origin, say at How did a point charge get all these nonzero multipolemoments? It didn’t, really; the situation still amounts to one point charge, but the accounting is more complicated, when Monopole only, of courseqVr=()()0,, , ,0:raθφ θ=() ()23020 301 cos cos cosqaa aVPPrr r rθθ θ =+ + + +  …11.r≠r23 October 2002 Physics 217, Fall 2002 3Choice of coordinate origin matters a lot inmultipole expansion (continued) So when someone gives you a charge distribution and asks what all the moments are, they also have to tell you what coordinate system to use. One useful exception: if the total charge is zero, then the monopole moment is zero, and the dipole moment is independent of the choice of coordinate origin.Consider such a situation, and suppose the dipole moment were p originally and pain a coordinate system with its origin displaced by some vector a:() ( )()() ().addddqρτ ρτρτ ρτ′′′ ′ ′′==−′′′ ′′=−=−=∫∫∫∫aprr rarrr a r pap023 October 2002 Physics 217, Fall 2002 4Force and torque on dipole in uniform EfieldIf the dipole moment is constant, the net force is zero, becausethe charges get pulled equally and oppositely.There is a torque, though, that tends to align the dipole moment vector with the applied field:qE-qE+q-qdExz()22ˆ , in this case.qqqpE++−−=×+×=× +− ×−=×=×=−Nr F r FddEEdE pEy23 October 2002 Physics 217, Fall 2002 5Potential energy of dipole in uniform Efield (Griffiths problem 4.7)Consider a dipole initially perpendicular to the field (0). The field tends to pull it into alignment (toward 1); we have to push to make it move toward 2. Thus the work we do is(Work wedo = potential energy.)E012xzθ22sincos.WdpE dpEUθπθπθθθθ′=×′′==− =− ⋅=∫∫pEpE23 October 2002 Physics 217, Fall 2002 6Force on dipole in nonuniform EfieldIf Echanges with position, the forces on each charge in a simple dipole will no longer in general be equal in magnitude, leading to a net force F on the dipole. Suppose the dipole is very short; infinitesimal, in fact:xzqE-qE+q-qdE()()()()Definition of gradient:coszzzzxxxyyqqEEEEdEEEEqα+−=−=∆∆=⇒∆=⋅∆=⋅∆=⋅∴= ⋅ = ⋅FEE EdddFdEpE——————(if we had chosen togive E a y or an xcomponent)α23 October 2002 Physics 217, Fall 2002 7Torque on dipole in nonuniformEfieldThe torque on a smalldipole, about its own center, is stillsince the same arguments apply as before. But since there’s a net force F on the dipole, there’s an extra torque of r×F about any other point:ExzƒExzθƒ=×NpEFNFNr=×+×NpErF23 October 2002 Physics 217, Fall 2002 8Dipole vs.dipole: force and torqueGriffiths problems 4.5 and 4.29: Two perfect (infinitesimal) dipoles p1and p2are perpendicular and lie a distance rapart. What is the torque on p1(about its center) due to p2? What is the torque on p2(about its center) due to p1? What are the forces on each, due to each? Why are the torques not equal and opposite? zxyp1p2r23 October 2002 Physics 217, Fall 2002 9Dipole vs.dipole: force and torque (continued)The field of each, at each other’s position:The torque on each:()()11 11113333322 2222333332cos sinˆˆˆˆˆ22cos sin 2 2 2ˆˆˆˆˆ1ppppprrryapp ppprrryaθθθθ=+==−=−′′′′′=+=−==′′′Erθθ zzErθ ryy()()212112 133112221 23322ˆˆ ˆ1ˆˆ ˆ2ppppaapppNpaa=× = × =−=× = ×− =−NpE z y xpE y z x23 October 2002 Physics 217, Fall 2002 10Dipole vs.dipole: force and torque (continued)Force on 2 due to field of 1:By Newton’s third law, the force on 2 by 1 is()() ()22 1 21112234223ˆˆ.yayapyppppyya==∂=⋅ =∂∂=− =∂Fp E Ezz—()()1211 243ˆ1.ppa=⋅ =−Fp E z—23 October 2002 Physics 217, Fall 2002 11Dipole vs.dipole: force and torque (continued)The forces the dipoles exert on each other are equal and opposite. Why aren’t the torques? Because we calculated the torques about different centers. If we refer both torques to thecoordinate origin (i.e. the position of dipole 1), thenas you always thought it should be.() ()() ()()12112312 12221 23412 12 12133 32ˆ0 1 (still)3ˆˆ ˆ0232ˆˆ ˆ0,ppapp ppNaaapp pp ppaa a=× =−=× +×=− +×=− + = =−NpE xpE rF xy zxx xN23 October 2002 Physics 217, Fall 2002 12--Permanent and induced dipoles in atoms and moleculesSome naturally-occurring charge distributions, such as many simple molecules, have permanent, built-in dipole moments:and the electrical properties of materialsmade with these components are decisively influenced by the behaviour of these dipoles in each others’ fields. Some materials are of course made up of neutral, non-polar components, but even these can have dipole moments inducedby external fields. Consider even the lowly hydrogen atom:OHH++--NHH++H+pp23 October 2002 Physics 217, Fall 2002 13Permanent and induced dipoles in atoms and molecules (continued)EProtonElectronprotonFelectronFInduced dipole momentax23 October 2002 Physics 217, Fall 2002 14Permanent and induced dipoles in atoms and molecules (continued)The electron and proton move in opposite directions until the force on the proton by the displaced electron balances the force on the proton by the external field. Suppose (crudely) that the electron has uniform charge density and stays spherical through this process. If the equilibrium position of the proton is a distance dfrom the center of the electron, thenα, called the polarizability, is thus proportional to atomic volume. ()()3inducedexternal2333330induced external413ˆˆˆ, or43,4 in MKS.qdqd pqqdaaaaaππααπε−=− =− =− =−===Ex xx EpE23 October 2002 Physics 217, Fall 2002 15Atomic polarizabilitiesElementαH 0.66Li 12.0Na


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ROCHESTER PHY 217 - Lecture 21B Notes - Electric Dipoles and their Interactions

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