A2 3 STARTING VOLTAGE FOR A CAPILLARY coordinates called m=const "Prolate Spheroidal and is an angle about the line Ff Fig. 4a Here and so, lines of n= const are confocal hyperboloids( foci at F, F)while 5= const. lines are confocal ellipsoids with the same foci. The surface n=o is the symmetry plane, S and one of the n-surfaces, n=no, can be chosen to represent(at least near its tip) the protruding liquid surface from a capillary as in Fig 4a If the potential o is assumed to be constant(V)on n=no, and zero on the plane s, then the entire solution for o will depend on n alone. The n part of Laplaces equation in these coordinates is which, with the stated boundary conditions, integrates easily to th-ln (Al2) Let R=x2+y(cylindrical radius). From n=2, the(z, R)relationship for an n= const. hyp 16.522 spel m artipezssanch Lecture 23-25A2.3 STARTING VOLTAGE FOR A CAPILLARY Fig. 4b shows an orthogonal system of coordinates called “Prolate Spheroidal Coordinates”, in which η = r 1 − r2 a ; ξ = r1 + r2 a and ϕ is an angle about the line FF’. Here r1 = x 2 + y 2 + z + a 2 ⎛ ⎝ ⎞ ⎠ 2 r2 = x 2 + y 2 + z − a 2 ⎛ ⎝ ⎞ ⎠ 2 and so, lines of η = const.are confocal hyperboloids (foci at F, F’) while ξ = const. lines are confocal ellipsoids with the same foci. The surface η = o is the symmetry plane, S, and one of the η-surfaces, η=ηo, can be chosen to represent (at least near its tip) the protruding liquid surface from a capillary as in Fig. 4a. If the potential φ is assumed to be constant (V) on η=ηo , and zero on the plane S, then the entire solution for φ will depend on η alone. The η part of Laplace’s equation in these coordinates is ∂ ∂η 1 −η 2 ( )∂φ ∂η ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = o (A11) which, with the stated boundary conditions, integrates easily to φ = V th−1 η th−1 ηo (A12) Let R (cylindrical radius). From 2 = x 2 + y 2 η = r 1 − r2 a , the (z,R) relationship for an η = const. hyperboloid is 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 8 of 36