the projection on it of that of the circular section through the same point R Meusnier's theorem E 1_(1 R UR cosa cosa cot a (A20) This means that Fig. 7 EE y cot a E A21 The question then is to find an external electrostatic field such that the cone is an equipotential (say, o=o), with a normal field varying as in(A21),ie, proportional to 1/Nr. Notice that the spheroids of Sec. A2.3 do generate cones in the limit when r>>a (with no= cos a), but this type of electrostatic field has En 1/r, and cannot be the desired equilibrium solution If we adopt a spherical system of coordinates( Fig. 8), it is known that Laplace's equation admit axi-symmetric"product"solutions of the type P=AP(cos 9)r B=A@(cos 9) where P, Q, are Legendre functions of the 1 and nd, respectively. Of the two, P, has Fig. 8 ingularity when 9=180, and Q The latter is acceptable, because 9=0 is inside the liquid cone, and we only need the solution outside. The normal field, from(A22b)is 16.522, Space Pl Lecture 23-25 Prof. Manuel marti Page 12 of 36the projection on it of that of the circular section through the same point (Meusnier’s theorem): 1 Rc = 1 R ⎛ ⎝ ⎞ ⎠ cosα = cosα rsinα = cotα r (A20) This means that 1 2 ε oEn 2 = γ cotα r En = 2γ cotα ε or (A21) The question then is to find an external electrostatic field such that the cone is an equipotential (say, φ = o ), with a normal field varying as in (A21), i.e., proportional to 1/ r . Notice that the spheroids of Sec. A2.3 do generate cones in the limit when r>>a (with ηo = cosα) , but this type of electrostatic field has En ≈ 1/r , and cannot be the desired equilibrium solution. If we adopt a spherical system of coordinates (Fig. 8), it is known that Laplace’s equation admit axi-symmetric “product” solutions of the type φ = APν (cosϑ)r ν (A22a) or φ = A Qν(cosϑ)r ν (A22b) where Pν,Qν are Legendre functions of the 1st and 2nd kind, respectively. Of the two, Pν has a singularity when ϑ = 180o , and Qν has one at ϑ = o. The latter is acceptable, because ϑ = o is inside the liquid cone, and we only need the solution outside. The normal field, from (A22b) is then 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 12 of 36