Density of Energy Stored in the Electric Field
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1 Density of Energy Stored in the Electric Field Deprtment of Physics, Cornell University c Tomás A. Aris October 14, 01 Figure 1: Digrm of Crtesin vortices from René Descrtes Principi philosophie, published in Contents 1 Introduction: Review of previous concepts 1.1 Electric field nd ction t distnce Electric field nd energy Electric potentil energy in terms of the electric field 4.1 Bsic strtegy Integrtion by prts Fundmentl result Appliction nd double-check 7 1
2 1 Introduction: Review of previous concepts 1.1 Electric field nd ction t distnce As we hve discussed since the very beginning of the course, mny experiments hve confirmed Coulomb s lw for the force cting on chrge q due to the presence of chrge q 1 t distnce r, q 1 q F k C ˆr. 1 r This fct is tht this lw works very well when the chrges re not moving. Nonetheless, lw like this presents number of deep, fundmentl questions for both Physics nd Philosophy. Prime mong these questions is the mystery of ction t distnce, the possibility of mtter cting on other mtter cross reches of empty spce. Erly scientist-philosophers, including René Descrtes nd Gottfried Wilhelm Leibniz who invented differentil clculus independently of Sir Isc Newton, when fced with the ction-t--distnce concept in the context of the grvittionl forces keeping the erth nd plnets in their orbits, rejected it outright s metphysicl impossibility. Indeed, modern physics lso rejects this concept, but for different reson nmely tht it would imply the possibility of trnsmitting informtion t speeds fster thn light, thus violting the Theory of Specil Reltivity. Wht this mens is tht Eq. 1 is relly only n pproximtion tht will hve to be corrected when we study wht hppens s the chrges begin to move. To see how ction t distnce would llow for fster-thn-light communiction, imgine tht someone, Person, hs chrged object q ttched to force detector nd tht person Person 1 hs chrge q 1 under his control. Then, s soon s Person 1 moves q 1, the ction t distnce inherent in Eq. 1 would imply tht Person could detect the slight chnge in the direction of the electric force instntneously. Nothing sys tht this would not work even if Person were one light-yer wy so fr wy tht it would tke light one yer to trvel between Persons 1 nd. If Eq. 1 relly worked, the two people would be ble to communicte fster thn light, nd the Theory of Specil Reltivity would be proven flse! See Figure. q1 "I d better tell my friend bout the problem set prty..." 1 light yer q "Hey I felt tht! Wow, tht WAS fst!" Figure : Possibility of fster-thn-light communiction enbled by ction t distnce implied by Coulomb s lw. In erlier lectures, we resolved this difficulty in two steps tht correspond exctly to how Descrtes resolved the problem philosophiclly. First, we decided tht the force on chrge q does not depend on some distnt thing hppening t the loction of q 1. Rther, we decided tht chrge q feels the effects of some locl property of spce t the loction r where the chrge q sits. In Descrtes s lnguge, this corresponds to the vortex ner the loction of the Erth pushing on the Erth, rther thn the Sun cting directly on the erth, s in Figure 1. In modern physics lnguge, rther thn clling this locl property of spce vortex, we cll it the Electric Field E r, nd we write F q E r,
3 so tht the force on q relly only depends on wht is hppening loclly t the loction r. Compring this eqution to Coulomb s lw, Eq. 1, this pproch works so long s we define E r k Cq 1 ˆr. r There, however, is still problem, but now with Eq.. The problem is tht we still hve the question of how it is tht chrge q 1 cn hve n influence like this t points r which my be fr wy from q 1. The nswer to this ws our second key result, which corresponds to how the Crtesin vortices in Figure 1 ffect only their neighbors, not distnt points in spce. In modern physics lnguge, this mens we wnt n eqution for E r tht only connects points tht re right next to, infinitesimlly close to, ech other so tht they re seprted by differentil distnces, dy, nd dz. Mthemticlly, this mens we need some differentil eqution for E r. In fct, we found just such reltionship in the differentil form of Guss s lw, E r ρ r/ɛ 0. 3 Indeed, it turns out tht this eqution, is n exct eqution of physics tht works in ll circumstnces, even if the chrges re moving. The gret dvntge of voiding ction t distnce nd lwys finding wys to express our physicl lws in terms of differentil equtions is tht we re much more likely to find equtions nd physicl lws tht lwys work! 1. Electric field nd energy The key concept we lerned bout connecting energy nd electric fields is the electrosttic, or electric, potentil V r, which is defined to be the potentil energy per unit chrge of smll positive test chrge when plced t loction r. By definition, this hs the vlue V r 1 Q U Q r 4 1 r F Q r d r Q r r F Q r Q d r E r d r, where in the lst step we use the fct tht the force per unit chrge is precisely the definition of the electric field. Note lso tht we use d r to be more mthemticlly precise nd void hving our integrtion vrible be the sme s our limit of integrtion. This eqution for the electric potentil hs similr ction-t-distnce problem. The potentil t loction r depends on wht is hppening t who series of other points r, some of which re very fr wy becuse of the lower limit on the integrl. This gin could led to fstthn-light communictions: hve Person mesure the potentil V r while Person 1 does things to chnge the electric field t the points ner him, nd Person gin could know instntneously tht something is hppening. A differentil eqution tht we found for V r solves this problem lso, giving description where V r depends on wht is hppening t nerby points. Specilly, the differentil eqution we found for V r ws E r V r. 5 The finl energy-relted quntity we hd ws the totl electric potentil energy. To compute this, we divided the totl chrge distribution of the system into tiny little chunks of chrge dq. We then used the electric potentil to compute the potentil energy of ech of these chrges by multiplying the potentil energy 3
4 per chnge V r by ech chrge dq to find the energy ssocited with ech chunk chrge, du V r dq. Finlly, we dded up ll of these contributions, tking into ccount the fmous double-counting correction fctor of 1/, to find U tot 1 V r dq, 6 where the formul for dq depends upon the type of distribution we re looking t, λ dl liner distribution dq σ da surfce distribution. ρ dv volume distribution The min problem with Eq. 6 s fundmentl result is tht it involves V r, nd so it still my hve vestiges of ction t distnce. Also, we would like to think of the electric field s some sort of rel, physicl disturbnce t ech point in spce like one of Descrtes vortices, nd not just convenient mthemticl fiction. If the electric field were true disturbnce in spce, we would expect there to be n energy ssocited with it nd, thus, tht the electric energy should be relted somehow directly to the electric field itself. As we shll see in the next section, the nswer to both of these concerns is the sme: it is possible to eliminte V r from Eq. 6 nd in the process write the stored energy U tot entirely in terms of contributions from the electric field t ech point in spce r. The result is n eqution which is not only correct in the Theory of Specil Reltivity, but which is lso used without ny further corrections in the modern Theory of Quntum Mechnics! Electric potentil energy in terms of the electric field.1 Bsic strtegy To re-express Eq. 6 in terms of the electric field lone, we begin by considering the one form for dq which cn describe ny chrge distribution. Specificlly, this is the form for volume chrge distribution, dq ρ r dv. This form ensures tht we integrte over ll of spce nd so include ll chrges in ny system. Also, even if some of the chrges re rrnged into point, line or surfces chrges, these chrges cn be viewed s loctions with very high volume densities of chrges ρ r concentrted in tiny regions ner the corresponding points, lines or surfces. Our strting point is thus U tot 1 V r ρ r dv, 7 where the triple integrl reminds us tht we re doing three-dimensionl integrl over ll of spce. Now, keeping in mind our gol of expressing this in terms of the electric field, we note right wy tht, rerrnging Eq. 3, we hve ρ r ɛ 0 E r, so tht Eq. 7 becomes U tot 1 V r ɛ 0 E r dv ɛ 0 V r E r dv, 8 where we hve fctored out the constnt fctor ɛ 0. The result Eq. 8 hs very interesting mthemticl structure. We would rther not hve the derivtive cting on E we would rther just hve the electric field E to think bout. Also, we relly would like the derivtive to be cting on V r, especilly in the form V r, which we cn see from Eq. 5, is E r. Thus, if we could just move the derivtive from the fctor E r over to the fctor V r, we could hve the totl potentil energy U tot entirely nd directly in terms of the electric field, which is our gol! This kind of mneuver of moving derivtive from one fctor to nother inside of n integrl is done ll the time for one-dimensionl integrls in second semester clculus clsses. This procedure is known s integrtion by prts. We now only hve to lern how to generlize the procedure to multidimensionl integrls, nd we will hve the result we wnt. 4
5 . Integrtion by prts The whole point of this subsection is to derive the formul for integrtion by prts in multiple dimensions, Eq. 1. If you re hppy enough to ccept tht result, you cn skip now directly to Section.3. It turns out tht integrtion by prts works in the sme bsic wy for multidimensionl integrls s it does for one-dimensionl integrls. Recll tht one-dimensionl integrtion by prts begins with the product rule for derivtives, d df dg fx gx gx + fx, nd then integrting both sides, d fx gx fx gx b df gx + df gx + fx dg fx dg, 9 where, in the second step, we use the fct tht the integrl of the derivtive of ny function returns bck exctly tht function. Finlly, we rerrnge the terms in Eq. 9 to get the finl fmous formul for integrtion by prts, fx dg fx gx b df gx. 10 To generlize the bove to the cse we hve in mind Eq. 8, note tht the derivtion begins by looking t the derivtive of the product of the two functions fx nd gx. Compring Eqs. 10 nd 8, we see tht V r plys the role of fx nd E r plys the role of gx. Thus, we should begin by considering the derivtive of the product V r E r. Becuse this product is vector quntity, the nturl type of derivtive to consider is the divergence simply becuse you cnnot tke the grdient of vector. Fortuntely, it turns out tht there is simple product rule for the divergence of the product of sclr quntity with vector quntity, V E V E + V E. 11 You cn prove this by using the definition of the divergence, pplying the ordinry product rule, nd rerrnging terms, 1. You cn remember this result by noting it works just like the ordinry product rule, s long s you keep trck of wht kinds of derivtives nd products you cn tke for different combintions of sclrs nd vectors. As with the one-dimensionl cse, the next step fter identifying nd tking the proper derivtive is to integrte both sides of the resulting eqution. Integrting Eq. 11 over ll of spce gives V E dv V E dv + V E dv, 1 Using the definitions of divergence nd of the product of sclr nd vector, we hve V E V Ex + x y V Ey + V Ez z V Ex Ex + V x x + sme terms for y nd z V Ex + sme terms for y nd z x V E + V E, + V Ex + sme terms for y nd z x where in the lst step, we mke two key identifictions. For the first term, we note tht we hve the sum of the product of ech component of V with the corresponding component of E, thus giving the dot product of these two quntities. For the second term, we fctor out the common fctor of V, nd wht is left is the definition of E. 5
6 which rerrnges to wht we wnt, V E dv V E dv V E dv, 1 which is the generliztion of integrtion by prts to multidimensionl integrls..3 Fundmentl result We re now in position to quickly derive our fundmentl result. There re three terms in Eq. 1. The left-hnd side of the eqution, prt from simple constnt fctor of ɛ 0 /, is exctly the totl energy tht we wnt from Eq. 8. The lst term on the right side of the eqution will ultimtely give us something very simple in terms of the electric field becuse V E. This leves only one term to consider in detil, the first term on the right side of the eqution. The first term on the right of Eq. 1 is triple integrl of divergence nd, thus, cn be simplified by the divergence theorem, V E dv V E da, 13 V V where V represents the volume of integrtion nd V represents the surfce bounding this volume. In the present cse, this volume V represents ll of spce becuse we re trying to clculte the totl potentil energy of ll of the chrges in our system. The boundry of this volume V, then represents ny very lrge surfce tht contins the whole universe! Mthemticlly, we cn represent this s the surfce of sphere of rdius R where we tke the limit R. The point of choosing such surfce is tht we know tht, very fr wy from ny chrge distribution of totl chrge Q, the electric field nd thus lso potentil from tht chrge distribution will look just like tht coming from point chrge of chrge Q. Thus, for the surfce integrl in Eq. 13 we cn tke E k C Q/R ˆr nd V k C Q/R. Moreover, becuse the surfce is sphericl, we hve da ˆr da nd our fr wy forms for E nd V re constnt over the surfce of this sphere. Thus, when we include the surfce re of the sphere 4πR we find tht this entire second term in Eq. 1 becomes for us V V E da k C Q/R k C Q/R 4πR 4πkCQ /R 0, where in the lst step we use the fct tht R 0 becuse our surfce is supposed to contin ll of spce. The conclusion of ll of this nlysis, then, is the simple fct tht the first term on the right of Eq. 1 cn be completely ignored! Following this result, we drop the first term on the right side of Eq. 1. Then, multiplying the resulting eqution by ɛ 0 / nd using the fct tht V r E, we hve our finl result U tot ɛ 0 U tot ɛ 0 ɛ 0 ɛ 0 V r E r dv V r E r dv E r E r dv E r dv ɛ0 E r dv. 14 The wy we interpret this finl, beutifully simple result, is tht potentil energy ssocited with ech volume of spce dv tht we re dding up to form our totl energy U tot must be ɛ0 E r dv, so tht the energy 6
7 density, the energy per unit volume of spce ssocited with the electric field t ech point in spce r is u r ɛ 0 E r, 15 nd we cn write the totl potentil energy s U tot u r dv Appliction nd double-check We cn double-check our finl result 15 by pplying it to prllel plte cpcitor, where we hd the key results tht U tot 1/CV nd C ɛ 0 A/d, where V is the voltge between the pltes, A is the re of the pltes nd d the seprtion between them. To clculte the energy density, the energy per unit volume, we tke the totl energy U tot nd divide by the volume of the cpcitor, which is Ad. This gives u U tot Ad 1 CV 1 Ad ɛ 0A d V Ad ɛ 0AV Ad ɛ 0 V. d But, becuse we know tht the mgnitude of the electric field between the pltes is constnt nd hs vlue E V/d, we see tht the cpcitor indeed exctly obeys Eq. 15! In most courses t this level, s you cn see from the textbook, one simply does this clcultion for the energy density in the cpcitor nd then guesses tht the energy density formul Eq. 15 works for ll cses. But, becuse we hve lerned the differentil form for Guss s lw, Eq. 3, nd the vlue voiding equtions which include ction t distnce, we re ctully ble to prove tht Eq. 15 works for ll cses!!! 7
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