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2 Pages

### Laplace02

Course: PH 316, Fall 2009
School: Rose-Hulman
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Word Count: 669

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for Laplace02.doc 12/9/01 PH 317 rev 5 Axisymmetric problem satisfying laplace's equation : ( no - dependence in the equation : 2V/r2 + 1/r V/r + 2V/z2 = 0 ) All derivatives evaluated at the center of the cell(r,z). r = a and z = b . A taylor series expansion gives V(r+a,z+b) = V(r,z) + a V/r + b V/z + ab/2 2V/rz + a2/2 2V/r2 + b2/2 2V/z2 . + higher terms in a and b. In slightly different notation we have...

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for Laplace02.doc 12/9/01 PH 317 rev 5 Axisymmetric problem satisfying laplace's equation : ( no - dependence in the equation : 2V/r2 + 1/r V/r + 2V/z2 = 0 ) All derivatives evaluated at the center of the cell(r,z). r = a and z = b . A taylor series expansion gives V(r+a,z+b) = V(r,z) + a V/r + b V/z + ab/2 2V/rz + a2/2 2V/r2 + b2/2 2V/z2 . + higher terms in a and b. In slightly different notation we have V(r,z+b) = V(r,z) + b V/z + b2 2V/z2 = Vo + b Vz + 1/2 b2 Vzz . V(r,z-b) = V(r,z) - b V/z + b2 2V/z2 = Vo - b Vz + 1/2 b2 Vzz . Then Vzz = ( V(r,z+b) + V(r,z-b) - 2Vo)/b2 . V(r+a,z) = V(r,z) + a V/r + a2 2V/r2 + ... = Vo + aVr + 1/2 a2 Vrr + .. V(r-a,z) = V(r,z) - a V/r + a2 2V/r2 + ... = Vo - aVr + 1/2 a2 Vrr + .. . Then we find Vr = (V(r+a,z)-V(r-a,z))/(2a), and Vrr = (V(r+a,z)+V(r-a,z) - 2 Vo)/a2 . Since Laplace'e equation with no phi dependence is Vrr + 1/r Vr + Vzz = 0, we have 0 = (V(r+a,z)+V(r-a,z) -2 Vo)/a2 +1/r (V(r+a,z)-V(r-a,z))/(2a) +(V(r,z+b) + V(r,z-b) - 2 Vo)/b2 . This can be solved for Vo: Vo ( 2/a2 + 2/b2)= (V(r+a,z)+V(r-a,z))/a2 +1/r (V(r+a,z)-V(r-a,z))/(2a) +( V(r,z+b) +V(r,z-b))/b2 . When we are away from r = 0 and z = 0 we have Vo = Crr (V(r+a,z)+V(r-a,z)) +Cr/r (V(r+a,z)-V(r-a,z)) + Czz (V(r,z+b)+V(r,z-b)) , where the constants Crr, Cr, and Czz are Crr = 1/2 b2/(a2 + b2), Cr = 1/4 ab2/( a2 + b2), and Czz = 1/2 a2/( a2 + b2) . At r=0, catastrophe? If Vr(0) 0, then at r=0 Vr/r definitely blows up. For Laplace's equation to hold, Vr(0) = 0. Now let s equal the radial distance from the z-axis and taylor-expand Vr: (s<<1) 12/9/01 rev 5 = Vr(s) Vr(0) + s Vrr(0) + higher powers of s, and since Vr(0) = 0, we have Vr(s) = s Vrr(0) + ... . Laplace's equation at s is Vrr(s) + 1/s( Vr(0) + s Vrr(0)) + Vzz(s) . Now using Vr(0) = 0 we see that for tiny s near r=0 we are ok because the limit of s0 gives Vrr(0) + Vrr(0) + Vzz(0) = 0 , or 2 Vrr + Vzz = 0 Another way to deal with r=0 is to sit on the z-axis and write Vxx + Vyy + Vzz = 0. Since we are at r=0, an x-derivative is identical to a y-derivative and also identical to an rderivative. This means that Vxx = Vrr = Vyy. Then Laplace's equation becomes 2 Vrr + Vzz = 0 at r=0. At r=0 we have V(0+a,z) = Vo + 0 + 1/2 a2 Vrr, so Vrr = (2/a2) (V(0+a,z)- Vo) . At r=0 for 2Vrr + Vzz = 0, we find 0 = 2 (2/a2) (V(0+a,z)- Vo) + ( V(0,z+b) + V(0,z-b) - 2 Vo)/b2 . Then Vo is found to be Vo = [(4/a2)V(0+a,z) + (V(0,z+b) + V(0,z-b))/b2]/[4/a2 + 2/b2] . Or at r=0 Vo = Cro V(0+a,z) + Czo (V(0,z+b) + V(0,z-b)), where Cro = 2b2/(a2+2b2) and Czo = 1/2 a2/(...

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