Greens.function.cylindrical.coords.pdf - Section 4.9...

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Section 4.9Laplace’s equation in cylindrical coordinatesand on the inner surface of the outer sphereQb=b24πZZ ·Φ(r, θ, φ)∂r¸r=bd=2πσ0bbaZba(r0a)dr0=πσ0b(ba)where we have used the orthogonality of the Legendre polynomialsZ11Pc(x)dx= 2δc0.(4.51)As a variation on this problem we relax the condition that the two spheres are at the same potential. Instead we requirethat the inner sphere be neutral and determine the potential of the inner sphere minus the potential of the outer sphere. Theresult can be readily obtained using the superposition principle. The inner sphere is made neutral by distributing a chargeπσ0a(ba)uniformly over its surface. The inner surface of the outer sphere will then have a total charge ofπ¡b2a2¢σ0.Applying the superposition principle yields thatΦ2(a)Φ2(b) =πσ0(ba)2b=Q0(ba)b(b+a)Example 2: Inner sphere potential isΦ0P1(cosθ),outer sphere is groundedTo demonstrate another application of the Green’s function we consider a boundary value problem. Again the systemconsists of concentric spheres, radiia < b.The boundary conditions areΦ(a, θ, φ) =Φ0cosθ,Φ(b, θ, φ) = 0.The green’s function solution to this problem is given byΦ(r, θ, φ) =a24πZZΦ0cosθ0·∂r0G(r;r0)¸r0=ad0Using the orthogonality of the Legendre polynomials this reduces toΦ(r, θ, φ)=a23Φ0cosθ∂r0ãb3r3¤ £r03a3¤(b3a3) (rr0)2!r0=a=a2Φ0cosθb3r3r2(b3a3)This result illustrates the general result that the boundary value problems. The boundary conditions are expanded inspherical harmonics and the boundary value problem is solved for each coefficient in the expansion. The solution of theproblem is then given by the superposition principle4.9Laplace’s equation in cylindrical coordinatesProblems in which the system is confined to cylindrical volumes are best analyzed in cylindrical coordinates. Incylindrical coordinates Laplace’s equation is1ρ∂ρ·ρ∂ρΦ(ρ, ϕ, z)¸+1ρ22∂ϕ2Φ(ρ, ϕ, z) +2∂z2Φ(ρ, ϕ, z) = 0.(4.52)86
Section 4.9Laplace’s equation in cylindrical coordinatesAs in the case of spherical coordinates, this equation is solved by a series expansion in terms of products of functions of theindividual cylindrical coordinates. That is, we use separation of variables.Φ(ρ, ϕ, z) =Xl,m,nαlmnUl(ρ)Vm(ϕ)Wk(z).(4.53)SubstitutingΦlmk(ρ, ϕ, z) =Ul(ρ)Vm(ϕ)Wk(z).into Eq 4.52 and dividing byΦlmk(ρ, ϕ, z), we can sequentiallyseparate all the variable .dependence:1Ulρd·ρdUl¸+1Vmρ2d22Vm=1Wnd2dz2Wk=k2(4.54)ρUld·ρdUl¸ρ2k2=1Vmd22Vm=m2(4.55)These functions satisfy the Eqs. 4.54, 4.55 and 4.56:d2Wk(z)dz2=λzWk(z);λz=k2(4.54a)Wk(z) =Akekz+Bkekz(4.54b)Note thatkcan be complex, pure imaginary or real.

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Term
Spring
Professor
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Tags
Boundary value problem, Dirac delta function, Bessel function

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