3.5 Method of Images. ds R Q (

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1 3.5 Metho of Images Metho of images epaces the oigina bouna b appopiate image chages in ieu of a foma soution of Poissonʼs o Lapaceʼs equation so that the oigina pobem is geat simpifie. The basic pincipe of the metho of images is the uniqueness theoem. As ong as the soution satisfies Poissonʼs o Lapaceʼs equation an the soution satisfies the given bouna conition, the simpest soution shou be taken. Point chage ove goune pane conucto B iect soution B metho of images,, 4$% Q + " + whee R is the istance fom s to the point une consieation an S is the suface of the entie conucting pane. + 4$% S s s R,, Q 4" R R + On vai in the egion of >.

2 3.5 Metho of Images Eampe A positive point chage Q is ocate at istances an, espective, fom two goune pepenicua conucting haf-panes, as shown in the figue. Detemine the foce on Q cause b the chages inuce on the panes.

3 3.5 Metho of Images Line Chage An Paae Conucting Cine Letʼs take a tia soution an inteigent guess that i " We have foun that the E fie geneate b a ine chage is E a / m The eectic potentia at a istance fom a ine chage of ensit integating the eectic fie intensit E $ % E $ " % " n " can be obtaine b Note that the efeence point fo eo potentia,, cannot be at infinit. Let us eave unspecifie fo the time being.

4 3.5 Metho of Images The potentia at a point on o outsie the cinica suface is obtaine b aing the contibutions of an i. In paticua, at a point M on the cinica suface, we have M n $ n " " " i Equipotentia sufaces ae specifie b i n We have simpifie the soution b consieing the is so age that the istance of the efeence point to an is negigibe. i Theefoe, tianges OMP i an OPM simia. We have o i i a a Constant i i a / Constant PM PM OPi OM i OM OP " Fom the above equation we see that if the image ine chage, togethe with, wi make the ashe cinica suface in the figue equipotentia. As the point M changes its ocation on the ashe cice, both i an wi change; but thei atio emains a constant that equas a/.

5 Two Paae Conucting Cines of The Same Raius 3.5 Metho of Images Since M " n i an i a i a We have n " a an $ n " a The capacitance pe unit ength is whee Consequent, C " $ n / a a D i D fom which we obtain / D + D 4a " " C F / m n[ D / a + D / a ] cosh D / a

6 Two Paae Conucting Cines of The Same Raius cont. 3.5 Metho of Images

7 We now eveop a metho fo soving 3-D pobems whee the bounaies, ove which the potentia o its noma eivative is specifie, coincie with the cooinate sufaces of an othogona, cuviinea cooinate sstem. In such cases, the soution can be epesse as a pouct of thee one-imensiona functions, each epening sepaate on one cooinate vaiabe on, The poceue is cae the metho of sepaation of vaiabes. Cassifications of bouna-vaue pobems: 3.6 Metho of Sepaation of aiabes Diichet pobems, in which the vaue of the potentia is specifie evewhee on the bounaies; Neumann pobems, in which the noma eivative of the potentia is specifie evewhee on the bounaies; Mie bouna-vaue pobems, in which the potentia is specifie ove some bounaies an the noma eivative of the potentia is specifie ove the emaining ones.

8 Letʼs investigate Lapaceʼs equation fo scaa eectic potentia in Catesian cooinates sstem fist:,, To app the metho of sepaation of vaiabes, we assume I oe fo the above equation to be satisfie fo a vaues of,,, we must have: Theefoe, 3.6 Metho of Sepaation of aiabes That is + +, k, k k, + k, + k + k Consequent, an, + + k k k

9 3.6 Metho of Sepaation of aiabes Constant k an the function foms ae etemine b the given bouna conitions. Fo eampe, if the potentia appoaches to when appoaches to infinite, the possibe soution fom is De k with a positive ea k.

10 3.6 Metho of Sepaation of aiabes EAMPLE Two goune,semi-infinite,paae-pane eectoes ae sepaate b a istance b.a thi eectoe pepenicua to an insuate fom both is maintaine at a constant potentia. Detemine the potentia istibution in the egion encose b the eectoes. Soution: With inepenent of, we have,,,, In the -iection:,,, In the -iection:,,, b, B D e Since, we choose, whee k is a positive ea numbe an k k " k k k jk, k k A sin k The bouna conitions in -iection suggest that An appopiate soution of the Lapaceʼs equation satisfing patia bouna conitions name, hamonic functions is n k, BD A e sin k 3 C n

11 3.6 Metho of Sepaation of aiabes n k, BD A e sin k 3 C The constant k is etemine b the BC at b, that is k, b C e sin kb Theefoe, the hamonic function becomes n n n sin kb k n / b, n,,3,... " n / b n n, Cne sin b In oe to satisf the BC at, we use the pincipe of inea supeposition of n to fin the specific soution of the given BC: " " n$ / b n$ n, n, Cne sin n ", Cn sin, < < b n n b n b m In oe to evauate the coefficients C, we mutip both sies of the equation b sin n b An integate the poucts fom to b: b n$ m$ m$ sin n b b b b " Cn sin sin

12 b " Cn sin sin 3.6 Metho of Sepaation of aiabes b n$ m$ m$ sin n b b b

13 3.6 Metho of Sepaation of aiabes Metho of sepaation of vaiabes in Cinica cooinates

14 3.6 Metho of Sepaation of aiabes

15 3.6 Metho of Sepaation of aiabes

16 3.6 Metho of Sepaation of aiabes

17 3.6 Metho of Sepaation of aiabes

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