Gee Fuctios Jauay, 6 The aplacia of a the Diac elta fuctio Cosie the potetial of a isolate poit chage q at x Φ = q 4πɛ x x Fo coveiece, choose cooiates so that x is at the oigi. The i spheical cooiates, the potetial is popotioal to. As a fuctio, f = is efie o the ope iteval,, but ot at the oigi. Its aplacia is also efie o this iteval, a is quickly see to vaish eveywhee, = = = This leas to a ifficulty whe we cosie the ivegece theoe, fo which the volue itegal iclues the oigi 3 x = x ce the ight ha sie is well-efie but the left is ot. Iee, fo a sphee S ε, of aius ε, the itegal o the ight becoes S S ε x = π π = 4π ε ε θθϕ Howeve, the itegal o the left is uefie. The igoous way to hale this is to exte the fuctio f = to a istibutio. A istibutio is efie as the liit of a sequece of fuctios, givig a object which is oly eaigful whe itegate. Thus, if we efie a istibutio f to be the liit f x li a f a x whee f a x is a collectio of fuctios epeig o a paaete a. A istibutio is ofte calle a fuctioal, a we use the two tes itechageably. The itegal of the istibutio is efie as the liit of the well-behave itegals of the seies of fuctios
f x x li a f a x x a this ay be pefectly fiite eve if f x is ot a tue fuctio. With this i i, let f a x =. This is efie fo the close iteval [, ], a so is its +a aplacia = + a + a = + a 3/ = 3 + a 3/ = 3 + a 3 4 3/ + a 5/ 3 = + a + 3 3/ + a 5/ 3 = + a 3 + a 5/ + a 5/ 3a = + a 5/ We ay theefoe efie a istibutio to exte δ x = by δ x = li f a x a 3a = li a + a 5/ The itegal is ow well-efie: 3 x li a = π li a a = 4π li 3 x + a a = li a 4πε 3 ε + a 3/ = 4π + a 5/ 3 + a fo ay fiite ε. As a pleasat bous, the ivegece theoe is ow satisfie as log as we uesta to be a istibutio. 3/
Notice that the fuctioal δ iveges at = a vaishes fo all >, while its itegal is fiite. Futheoe, if f is ay sooth, boue fuctio of whih vaishes outsie soe copact set, the f 3 x = li f 3 x a + a whee the eaie R satisfies which vaishes as ε, leavig This eas that = 4πf li a = 4πf 4πR = R = li x a < εf li 3 4π li + a 3/ a f +... + a 3/ a f 3 3 +... + a 3/ 3 x = 4πf f +... + a 3/ is popotioal to a Diac elta fuctioal at the oigi, = 4πδ 3 x x Retuig to a abitay oigi, we ay wite this as x x = 4πδ 3 x x Notice that what is uivesally efee to as the Diac elta fuctio is, popely speakig, a fuctioal. Geeal solutio fo the potetial of a poit chage with bouay coitios I tes of the potetial, Gauss s law i fee space is Φ = ɛ ρ x The chage esity fo a isolate chage q at positio x is ρ x = qδ 3 x x We wish to solve fo the potetial Φ fo this poit souce, a bouay coitios give o soe suface, S. The bouay ay be copise of ultiple pieces. Fo the peceig sectio, we see that the solutio fo the potetial is Φ x = q 4πɛ x x + Φ 3 3
whee Φ is ay solutio to the aplace equatio, Φ = Now, we kow that the solutio to the aplace equatio is uique oce we specify bouay coitios, a a foal poof of this will be give below. Suppose we have bouay coitios Φ x S = Φ x S fo ay poit x S o S. The if we equie Φ x S = Φ x S q 4πɛ x S x thee is a uique solutio fo Φ, a theefoe a uique solutio fo Φ satisfyig the give bouay coitios. Alteatively, we ay fi the solutio iectly by solvig Φ = q δ 3 x x 4πɛ with bouay coitios Φ x S. This is oe staightfowa tha it appeas, because the Diac elta fuctio vaishes alost eveywhee. Theefoe, uless x = x, we ae solvig the aplace equatio. As a esult, we ay costuct ou solutio fo Φ fo solutios to the coespoig aplace equatio.. Exaple: Isolate poit chage The siplest exaple is fo a isolate poit chage, with the potetial vaishig at ifiity. aleay show that x x = 4πδ3 x x so we ieiately have Φ x, x = 4πɛ x x + F x We have whee F satisfies the aplace equatio, F =. By uiqueess, the fuctio F ust be eteie by the bouay coitios. I the peset case, we ask fo Φ x, x to vaish at x. Sice the fist te i Φ x, x aleay satisfies this, we equie the sae coitio fo F : F = F = The aguet of the peceeig sectio shows that F x = is the uique solutio to this, so the Gee fuctio fo a isolate poit chage is G x, x = x x. If we have a localize istibutio of chage, ρ x, i epty space, the potetial vaishes at ifiity a we ca use this Gee fuctio to fi the potetial eveywhee by itegatig Φ x = G x, x ρ x 3 x 4πɛ = ρ x ɛ x x 3 x. Exaple: Bouay coitios o a squae.. The aplace equatio Cosie a -iesioal exaple, with bouay coitios give o a squae of sie with oe coe at the oigi. et the bouay at x = have potetial, with the eaiig bouay segets havig 4
Φ =. The with the gle chage q at x = x, y, the Poisso equatio becoes x + y Φ = q δ x x δ y y 4πɛ We begi by solvig the hoogeeous aplace equatio, x + y Φ = by sepaatig i Catesia cooiates. Witig Φ = X x Y y, the iviig by Φ, the aplace equatio is XY x + y XY = X X x + Y Y y = Sice the fist te epes oly o x a the seco oly o y, each ust be costat, so with the ieiate solutios X X x = Y Y y = α X α x = A α h αx + C α cosh αx Y α y = B α αy + D α cos αy These fuctios will satisfy the bouay coitios fo y a at x = if we set α = π a C α = D α =, leavig X α x = A α h πx Y α y = B α πy Cobiig coefficiets, the geeal solutio is the a su Φ x, y = A h πx The eaiig bouay coitio at x = is fou by settig = Φ, y = πy A h π πy This is just a Fouie seies fo a costat. Multiplyig by πy so that a fially y πy π cos π = = Φ, y = A δ h π π = A h π A = Φ x, y = o A h π π h π a itegatig, y πy 4 πx h π h π πy πy 5
.. Fouie epesetatio of the Diac elta fuctio Notig that the y-epeece is escibe by a e seies, we ake use of the fact that the Diac elta fuctioal ay also be witte as a Fouie seies, To fi the A, ultiply by πy δ y y = A πy a itegate to fi the coefficiets, yδ y y πy πy = = y A δ A πy πy y πy = A δ y πy + πy cos = A δ = A δ y cos πy = A so that A = πy Itegatig twice shows that a we have y δ y y = πy πy πy π πy = δ y y..3 A asatz fo a paticula solutio While thee ae systeatic appoaches to solvig the poit paticle Poisso equatio i vaious cooiate systes, we take a siple appoach hee. Suppose we guess that we ca fi a solutio of the fo Φ p x, y = π a substitute eqs. a ito the Poisso equatio x + y f x πy π x + f x πy f x πy πy Φ = q 4πɛ δ x x δ y y 6 = q 4πɛ δ x x πy πy
a theefoe, equatig like tes, Now expa f a the seco elta fuctio, a substitute, Theefoe, π Fially, check that x + y B π Φ p x, y = π f x + f = q 4πɛ δ x x f x = δ x x = πx q π 3 ɛ B πx πx + B πx +, Φ p x, y = q π 3 ɛ..4 Bouay coitios, = q π 3 ɛ π πx + πx = q 4πɛ B = q πx 4πɛ πx B = q πx 4πɛ + πx πy π + + π πx πx = q 4πɛ δ x x δ y y πx πy πy πx πy πx πy πy The potetial Φ p x, y is a paticula solutio to ou Poissoi equatio, but it oes ot satisfy the bouay coitio at x =, istea vaishig at all fou bouay lies. To get the coplete solutio, we ee to a the hoogeeous solutio that satisfies the bouay coitios. The full solutio is theefoe Φ x, y = q π 3 ɛ, 3 Supepositio πx + πx πy πy + o The geeal poble of electostatics is to solve the Poissio equatio, Φ = ɛ ρ x 4 πx h π h π πy 7
with give chage esity ρ x a bouay coitios Φ x S. Kowig the solutio fo a poit chage at x allows us to o this ieiately by takig the supepositio of ifiitesial chages ρ x 3 x a suig itegatig ove ou etie volue: Φ x = 4πɛ ρ x 3 x x x + Φ x whee Φ satisfies the aplace equatio. We choose Φ so that Φ satisfies the bouay coitio. We will exaie etails of this solutio i the ext Note. 8