L8b - Laplacias i a cicle Rev //04 `Give you evidece,' the Kig epeated agily, `o I'll have you executed, whethe you'e evous o ot.' `I'm a poo ma, you Majesty,' the Hatte bega, i a temblig voice, `--ad I had't begu my tea--ot above a week o so--ad what with the bead-ad-butte gettig so thi--ad the twiklig of the tea--' `The twiklig of the what?' said the Kig. `It bega with the tea,' the Hatte eplied. `Of couse twiklig begis with a T!' said the Kig shaply. `Do you take me fo a duce? Disks: Covetig the Laplace equatio to pola coodiates If the chage distibutio is a two-dimesio cicula shape (o i 3 dimesios, a ifiitely log cylide), we use the Laplacia i pola coodiates: We ae give u xx + u yy = 0 ad x = cos, y = si The Chai Rule gives the fist patials: u u u u u u ad x x x y y y. u With symmety about the oigi, thee is o vaiatio i agle so that 0 : u u u u ad x x y y. Now take the secod patials usig the poduct ule:
u u u xx x x xx u u u yy y y yy Let s simplify by gettig id of the patials of, statig with implicit patial deivatives of x + y =. Veify each of the followig: x y x x y xx x y y x yy y 3 3 Fiish the tasfomatio, obtaiig u u u u xx yy Example: cylidical capacito Let a solid cylidical coducto of adius a be coaxial with a thi cylide of ie adius b; let the ie coducto have chage Q ad the oute have chage Q. Sice we oly have vaiatio i, why ot ty DSolve o V ''( ) V '( ) 0? The costats will be obtaied by lettig V(a) = V 0 ad V(b) = -V 0. As befoe, we expect V = 0 iside this cylide, somewhee betwee a ad b but ot at the midpoit. Develop a expessio fo V(), i tems of a, b ad V 0 by elimiatig both Cs. Use this to solve fo the positio of the V = 0 equipotetial i tems of a ad b. Sample plot of V() fo values of a, b ad V 0 show.
u Vaiatio with agle: If we have 0, we eed the full covesio of Laplace s equatio to pola. To begi, we ca epeat some of the steps take above: Give u xx + u yy = 0 ad x = cos, y = si We use the Chai Rule to fid the fist patials: u u u u u u ad x x x y y y, The secod patials come fom the poduct ule o these fist deivatives: uxx uxx uxx ux x u xx u u u u u yy y y yy y y yy We ve aleady established each of the followig elatioships: x y x x y xx x y y x yy y 3 3 x y Sice cos ad si, ad x y y x ad the secod patials ae xx y x ad yy y x x, with y y x ad. x x x Fiish the tasfomatio, obtaiig uxx uyy u u u 3
Hee s what Mathematica thiks about all this: 0 It woks! 0 Now, we ae eady to use sepaatio of vaiables o the pola Laplace poblem: Let u F() G() so that F'' G F' G FG'' 0, with F sigifyig deivatives wt to ad G sigifyig deivatives wt. Veify that this sepaates ito G'' G 0 ad F'' F' F 0. The G fuctio has a slight vaiatio o the usual: G( ) Acos Bsi, whee have = fo ay ozeo itege, because G must be peiodic (ie, G( ) G( ) ). The F equatio (which is kow as a Eule diffeetial equatio) solves via a cleve substitutio: F() =. D Show that this yields the solutio u (, ) ( C )( Acos Bsi ), as. What about = 0? The F '' F ' 0, which you should ecogize, as it is solved above by F () C Dl. 4
Solutios usig a vaiety of BCs Fist case: Iside a disk, < a. If we equie that the solutio fuctios be fiite at = 0, we ae foced to set D = 0. We ca costuct a geeal solutio by summig what emais: u (, ) A0 ( Acos Bsi ). We ca see that A 0 is just twice the value of u at = 0; we ae t give ay ifomatio about this value of u. Ad what ae the A, B? Hag o a bit loge Bouday coditios o the iside of the disk: Diichlet poblems agai! Now suppose we kow that o the bouday of the disk, = a, we have u(a, ) = h() fo some fuctio h. Pluggig this ifomatio ito the solutio, h( ) A0 a ( Acos Bsi ), which is the Fouie Seies fo h. Ad we kow how to do that! A0 h( ) d, whee is just a dummy vaiable of itegatio. 0 A h ( )cos d ad B h ( )si d 0 a a 0 Example a; at a, h( ) Cos, This is a easy eough fuctio to show i a PolaPlot: 5
We calculate Fouie Coefficiets as show above, such as It speeds up the plot if we calculate some coefficiets befoehad: We ca the wite the sum to obtai u(, ) fo ay umbe of coefficiets: It is ice to see that we obtai the same plot at = a = : Keep i mid that this plot is the value of u as a fuctio of o the bouday of the disk of adius. Poblems Geeate plots of u fo vaious values of iside the disk of adius. This makes a vey cool aimatio! Also ty some multiple-lobed fuctios, such as h() = + si 3, etc. Fid the potetials iside the disk of adius if 00 if 0 u(, ) 0 if Note that while these ae valid potetial poblems, they ae ot electostatic poblems if the disk is a coducto. The exteal suface of a coducto is a equipotetial, whe the chage distibutio is at equilibium, esultig i u(, q) = 0 fo < a. Ad a equipotetial suface has a field of 0. 6
Icedibly, the expessios fo h(), A ad B ca be combied ito a sigle expessio, kow as Poisso s Itegal Fomula: < a: h( ) u (, ) ( a ) d 0 a acos( ) I this expessio, is the dummy vaiable of itegatio. Named fo Simeo Deis Poisso, Fech mathematicia ad physicist who was egaded by Laplace as almost a so. Poisso took Fouie s spot at the Ecole Polytechique i Pais whe Napolea set Fouie to Geoble. Use the Fouie Seies techique to plot a aimatio of the solutio u(,) fo < a with h() = / at a =. You will eed at least 9 tems i the Fouie Seies to get a decet appoximatio to h at = a. The use the Poisso Fomula. Math factoid: This paticula bouday coditio is kow as a Achimedes spial. We ae foced to use NItegate to obtai values fo Poisso s Fomula ad eve that beaks dow as gets close to ad gets close to. 7
Secod case -- outside the disk: Suppose we have a bouday fuctio at = a ad equie that u 0 as goes to ifiity D The geeal solutio agai: u (, ) ( C )( Acos Bsi ) We ow set C = 0 ad thus we must keep D ozeo (ad that meas we ca t have both u 0 at = 0 ad as ifiity). Now u (, ) A0 ( Acos Bsi ) Agai, the Fouie Seies: h( ) A0 a ( Acos Bsi ) with a a coefficiets A h( )cos d ad B ( )si 0 h d 0 Fially, the Poisso Fomula outside the disk: > a: h( ) 0 u (, ) ( a) d a acos( ) Example Let h() = + cos ad a =. Outside the disk ( > ), u(, ) 0 as iceases. The effects of the bouday value fuctio ae see whe we ae close to the disk; at lage distaces, the solutio loses detail. These PolaPlots wee geeated usig the NItegate vesio of the Poisso Fomula. Red cuve is h(), ie u(,); blue cuve is u(.5,); gee cuve is u(,). 8
Do t be fooled by the appeaace of deceasig adius, makig it look like we e iside the disk. These ae values of u that seem to be deceasig with iceasig (fom ed to blue to gee), as advetised. What is with the two poits whee all cuves itesect? What would be the value of the patial deivatives of u(, ) at those poits? Plot moe of these cuves. Build the same set of plots (ad use lage values of ) fo the sowflake h() = + si 3. What about showig the equipotetials? Ou old fied, the MeshFuctios to the escue. Note that the plot fuctio of choice is PaameticPlot, which equies a tiple of fuctios {x,y,z}, each fuctios of the paametes ad. The plot s axes ae the coodiates x ad y, with u the vetical, but sice x ad y descibe a cicle, the plot looks like it is limited to the disk of adius. Figue out a cleve way of showig the egative of the gadiet of the potetial fuctio (epesetig field vectos), which ae eveywhee pepedicula to the equipotetials. Compae the distibutio of equipotetials fo seveal diffeet h() bouday fuctios. 9
Recallig that the electic field is the egative gadiet of potetial, the spacig of the equipotetials is a measue of the field stegth. Whee is the electic field stogest: ea smoothly cuved segmets of the bouday fuctio o ea moe poity segmets? Does the field behavio help explai why you should t stad ude a tee duig a thudestom? Math At! 0