Question

Consider a potential problem in the half-space defined by 2 20, with Dirichlet boundary conditions on the plane z = 0 and at infinity). (a) Write down the appropriate Green function G(x, x'). (b) If the potential on the plane z = 0 is specified to be 0 Vinside a circle of radius a centered at the origin, and Ø = 0 outside that circle, find an integral expression for the potential at the point P specified in terms of cylindrical coordinates (p, 0, z). (e) Show that, along the axis of the circle (p = 0), the potential is given by 1 = v1- va oz)

Answer 1

Answer - Page No -o (2.7) Given that, Consider a potential problem in the half- space defined by Z 20 Dirichlet boundary conditions on the plane 2-0 B (and at infinity). 1. (a) Green function , G(x,x): It satisfies the conditions that, G(2,x') = Text +F(x,x') - Go (x,x) = al la pell where P: (6.,z) → (M, ,-2) is the Minor reflection operator. H-x) = f(x-1) + (y-y") * (z-z)? 1 x-px'l, Vix=x++ (4 7' )*+/2-(-2) >> Go (X,X') (-) + (- )+ - ) /(x-x)* 4-4) (+ which conresponds to unit charge q= 4TTE (6) obtain an integral expression for the potential at the point p specified in terms of Cylindrical Coordinates CP,Q,2)

d = v inside a circle 1. Page No - of radius a centered at 0= o outside a cicle. solution of Dirichlet problems. 06) og her pa's dogfx, x'da! – 03. dGo (lix') = -d/dGo (x,x') 2-0 dn' = z'z P7 p'2pp! Cos (0-6') #2-2!*j Z-6-2) prt pi? app'Cos (8-6²)+(2-6-3'))²)/ Jz oo z'tz_ 근구근 - z -z P7 p'2.pp'Cos($-$]+(2z)-J% pip'=2 pp'casco-6') +8+2)744 zo O+Z o-z pt Pl_app'cos (8-0)+(2-0) 3/2 (PFP _2 pp'Cos (0-0)+(2+273/ - z -Z I pt pt = 2pp'Cos (0-ø+{2))%Ptp = zeppicosco-}+* -z-z p zp'teppicos (0-) +(234)372

- (2) on' substitute pt plz app'Cos (0-0) +23") / (2) in o) r - 22 ofessor de overy enco-6 secarte de ple-417)- En Sol bid. on ou de cocos varande 1 (s'dodd') - sve op for bep caco-d'tes] since o dependence can be eliminated by letting 0'>0'+, Uz z 0 (0,0,2)= Il do pap' 21 to DI DS-2.pp'casco-6)+(2) % (c) show that p="[1 - 0 1 2] along the anis of the circle (p=0) - from part (b),

- Page No-9 0 (0o,0z) - men som forever store the For p=0 2TT A II Ø (0,2) = 2 T o dos Pap' To'+ =260) p'casco-6)+(21)3% e t de poder entretien - Stop (1xc293)** ve (682+2) You - Veľ costzy * (1972)3] --V={1037629* (2) *] ve (clasiceyn ] - constante ] -[-comhr ..$= v[ hare] Hence showed.