ring source

Ring source in Toroidal Coordinates PhL 12.31.13 1. Ring source in cylindrical coordinates In a potential theory problem (Laplace equation) and in electrostatics language, a ring source of total charge q in cylindrical coordinates with radius ρc and at z = 0 would be written as source = q δ(z) δ(ρ-ρc)/[2πρ] . The reason for the 2πρ is that we need the 3D integral of the source to be q: ∫dφ ∫ρdρ ∫dz q δ(z) δ(ρ-ρc)/[2πρ] = 2π ∫ρdρ ∫dz q δ(z) δ(ρ-ρc)/[2πρ] = q ∫dρ ∫dz δ(z) δ(ρ-ρc) = q The problem then is to express this ring source in toroidal coordinates. 2. Digression on the equations for the toroidal coordinate level curves (I have derived these equations elsewhere and they probably appear somewhere in M&F or elsewhere. ) horizontal circles (tori cross sections, let ρ = horizontal coordinate, z = vertical coordinate) (ρ - acothξ)2 + z2 = a2/sh2ξ ρc = acothξ R = a/|shξ| ( Historically (200BC), the horizontal circles are called the "circles of Appollonius" and they have some very simple properties, notably, | r - a | / | r + a | = e-ξ. ) vertical circles (when truncated at z=0, they give the bowls) ρ2 + (z - acotu)2 = a2/sin2u zc = a cot(u) R = a/|sinu| The picture going with these equations showing the various circles is this Remember that you rotate this bipolar 2D system around the vertical z axis by φ to get toroidals. 3. Geometry associated with the ring source Now consider the following picture, where on the right is one of our "horizontal circles" being a cross section of a gray toroid, We know at once from the previous section that the center of the circle is at ρc = a coth(ξ0). We also know that R = a/|shξ0| . Then ρc2 - R2 = a2 coth2(ξ0) - a2/(shξ0)2 = a2[coth2(ξ0) - csch2(ξ0)] = a2 ρc/R = a coth(ξ0)* shξ0/a = ch(ξ0) // 0 < ξ0 < ∞ in normal usage We have now derived this set of equations which appeared in my earlier email and which appear in the bowl paper, ρc = a cothξ0 R = a /shξ0 => = a , ρc/R = chξ0 (10.9) Also, we can get an expression for cylindrical ρ expressed in toroidals as follows (these x,y,z equations are those which in fact define toroidal coordinates (ξ,u,φ) ) x = a cosφ shξ/(chξ - cosu) = cosφ hφ y = a sinφ shξ/(chξ - cosu) = sinφ hφ z = a sinu/(chξ - cosu) = sinu hξ we find that ρ2 = x2+y2 = a2sh2ξ / (chξ - cosu)2 => ρ = a shξ/(chξ - cosu) . The inverse equations are given by ξ = tanh-1[2aρ/(a2+ ρ2+ z2)] tan(u) = [ -2az/(a2-ρ2-z2)] // note that z = 0 => u = 0 as one possibility tanφ = y/x However, one really needs to execute the following algorithm to find compute ξ,u,φ from x,y,z : 1. Compute φ from the last equation above, in range (0,2π) say. Then sinφ and cosφ are known. 2. Compute ξ from the unambiguous formula shown above involving tanh-1. 3. Compute cosu from the " x = " formula above. 3. Compute sinu from the "z = " formula above. 4. Then compute u manually from these values. The scale factors are given by hξ = hu = a/[ chξ - cosu] hφ = a shξ/[ chξ - cosu] = shξ hμ hξ hu hφ = shξ hξ3 = a3 shξ /[ chξ - cosu]3 which will be used below. Note: M&F deal with this stuff on page 1210. Instead of using our angle u, they use an angle θ. Here is the connection between their θ and our u: The relationship between u and θ is this: u = (θ+π) mod (2π) or u = θ+π for θ in (0,π) u = θ-π for θ in (π,2π) or u = θ + π sign(π-θ) For any θ in (0,2π) we know the following to be true: sinθ = - sinu cosθ = - cosu tanθ = + tanu . 4. Jacobians and final expression for the ring source We know in spherical coordinates that the Jacobian tells how volume elements transform, dxdydz = J(r,θ,φ)drdθdφ = r2sinθ drdθdφ J(r,θ,φ) = r2sinθ . Here J is the Jacobian for these curvilinear coordinates and for orthogonal coordinates ξ1,ξ2,ξ3 we have J(ξ1,ξ2,ξ3) = h1h2h3 . // product of the scale factors In cylindrical coordinates we have dxdydz = J(ρ,z,φ)dρdzdφ = ρdρdzdφ J(ρ,z,φ) = ρ In toroidal coordinates we have instead dxdydz = J(ξ,u,φ)dξdudφ J(ξ,u,φ) = h1 h2 h3 = shξ h13 = shξ a3 / [chξ - cosu]3 Thus, dxdydz = a3 shξ (chξ - cosu)-3 dξdudφ . We can then get this connection between cylindrical and toroidal coordinates using dxdydz = dxdydz" ρdρdzdφ = a3 shξ (chξ - cosu)-3 dξdudφ . // note that dimensionally correct But above we had ρ = a shξ (chξ - cosu)-1 so then a shξ (chξ - cosu)-1dρdzdφ = a3 shξ (chξ - cosu)-3 dξdudφ or dρdzdφ = a2 (chξ - cosu)-2 dξdudφ or dρdz = a2 (chξ - cosu)-2 dξdu . Since delta functions work as the inverse of volume elements, we then know that δ(ρ-ρ1)δ(z-z1) = a-2(chξ - cosu)2 δ(ξ-ξ1)δ(u-u1) where ξ1 and u1 are the toroidal coordinates matching ρ1 and z1 in cylindrical coordinates. We are more interested in [ from above we know that 1/ρ = (chξ - cosu)/(ashξ) ] δ(z-z1) δ(ρ-ρ1)/ [2πρ] = a-2(chξ - cosu)2 δ(ξ-ξ1)δ(u-u1) * (chξ - cosu)/(2πashξ) = (1/2π) a-3(chξ - cosu)3 δ(ξ-ξ1)δ(u-u1) /shξ For the desired ring source at the cross hairs of the above drawing, we have ρ1 = ρc // ρc = a cothξ0 z1 = 0 u1 = 0 // the ring source lies on the "iris" limit of the bowl ξ1 = 2ξ0 . To verify this last equation from our inversion formula above, ξ1 = tanh-1[2aρ/(a2+ ρ12+ z12)] = tanh-1[2aρc/(a2+ ρc2)] = tanh-1[2a a cothξ0/(a2+ a2 coth2ξ0)] = tanh-1[2 cothξ0/(1 + coth2ξ0)] = tanh-1[2 /(tanhξ0 + cothξ0)] = tanh-1[2 tanhξ0 /(tanh2ξ0 +1)] = tanh-1[ tanh(2ξ0)] = 2ξ0 . Is this really true? To check, compute ρ1 using formula above: ρ1 = a shξ1/(chξ1 - cosu1) = a sh(2ξ0)/( ch(2ξ0) - 1) = a 2 sh(ξ0)ch(ξ0) / [2sh2(ξ0)] = a ch(ξ0)/ sh(ξ0) = a coth(ξ0) which is correct. Then we have shown that δ(z-z1) δ(ρ-ρ1)/ [2πρ] = (1/2π) a-3(chξ - cosu)3 δ(ξ-ξ1)δ(u-u1) /shξ = (1/2π) a-3(chξ1 - cosu1)3 δ(ξ-ξ1)δ(u-u1) /shξ1 = (1/2π) a-3(chξ1 - 1)3 δ(ξ-ξ1)δ(u) /shξ1 // u1 = 0 = (1/2π) a-3(2sh2(ξ1/2))3 δ(ξ-ξ1)δ(u) /[2sh(ξ1/2)ch(ξ1/2)] = (1/2π) a-3(8sh6(ξ1/2)) δ(ξ-ξ1)δ(u) /[2sh(ξ1/2)ch(ξ1/2)] = (1/2π) a-3(4sh5(ξ1/2)) δ(ξ-ξ1)δ(u) /[ch(ξ1/2)] = (1/2π) a-3(4sh5(ξ0)) δ(ξ-2ξ0)δ(u) /[ch(ξ0)] = (1/2π) a-3 4sh4(ξ0) th(ξ0) δ(ξ-2ξ0)δ(u) = (2/π) a-3sh4(ξ0) th(ξ0) δ(ξ-2ξ0)δ(u) . Then the ring source of total charge q expressed in toroidal coordinates is given by source = q δ(z) δ(ρ-ρc)/[2πρ] or source = q (2/π) a-3sh4(ξ0) th(ξ0) δ(ξ-2ξ0) δ(u) I guess the main fact is that source = constant δ(ξ-2ξ0) δ(u) where the details are only needed for normalization with respect to q. I am a little surprised at the 2ξ0 , maybe it is wrong. I originally thought it would just be δ(ξ-ξ0). The work above is just a "quick shot" and is not fully checked!!