En=Eg= sin 9- d(cos 9) and. in order to haves Thus P=Aron(cos 9 (A23) The function @,(cos 9)is shown in Fig.(9). The essential point is that this function has a single zero, at 9=a=49290 24) which can therefore be taken as the equipotential liquid surface. Notice that this angle is universal (independent of fluid properties, applied voltage, etc). Taylor(and others)have verified experimentally this value, as long as no strong space charge effects are present and as long as the electrode geometry is"reasonably similar to what is implied by eq (A27). This latter point is clarified by Fig. 10, where one generic equipotential of (A23) is shown together with the Taylor cone; notice that all other equipotentials have shapes which can be simply scaled from the one shown, according to r /r=o,1.)for a given angle 9 The experimental fact that stable taylor cones do form even when the electrodes applying the voltage are substantially different from the shape in Fig. 10 apparently indicates that the external potential distribution near the cone is dictated by the 16.522 spel e artiles snch Lecture 23-25En = Eϑ = − 1 r ∂φ ∂ϑ = +A dQν d( ) cosϑ sinϑ 1 r 1−ν and, in order to have En ≈ 1 r 1/ 2 , we need ν = 1 2 . Thus, φ = A r 1 / 2 Q1 / 2 (cosϑ) (A23) The function Q1 / 2 (cosϑ) is shown in Fig. (9)• . The essential point is that this function has a single zero, at ϑ = α = 49.290o (A24) which can therefore be taken as the equipotential liquid surface. Notice that this angle is universal (independent of fluid properties, applied voltage, etc). Taylor (and others) have verified experimentally this value, as long as no strong space charge effects are present, and as long as the electrode geometry is “reasonably similar” to what is implied by Eq. (A27). This latter point is clarified by Fig. 10, where one generic equipotential of (A23) is shown together with the Taylor cone; notice that all other equipotentials have shapes which can be simply scaled from the one shown, according to r2 /r1 = φ 2 / φ1 ( ) 2 for a given angle ϑ . The experimental fact that stable Taylor cones do form even when the electrodes applying the voltage are substantially different from the shape in Fig. 10 apparently indicates that the external potential distribution near the cone is dictated by the 16.522, Space Propulsion Lecture 23-25 Prof. Manuel Martinez-Sanchez Page 15 of 36