A New Modified Helmholtz Equation for the Expression of the Gravity Gradient and the Intensity of an Electrostatic Field in Spherical Harmonics

Gerassimos Manoussakis ()
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Gerassimos Manoussakis: Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, Iroon Polytechneiou 9, 157 80 Zografos, Greece

Mathematics, 2023, vol. 11, issue 20, 1-12

Abstract: In this work, it is shown that the geometry of a gravity field generated by a spheroid with low eccentricity can be described with the help of a newly modified Helmholtz equation. To distinguish this equation from the modified Helmholtz equation, we refer to it as the G-modified Helmholtz equation. The use of this new equation to study the spheroid’s gravity field is advantageous in expressing the gravity vector as a vector series of spherical harmonics. The solution of the G-modified Helmholtz equation involves both the gravity intensity g (or simply gravity g ) and the intensity E of an electrostatic field as shown in sequel. An electrostatic field generated by an oblate spheroid charged with l electrons (uniform ellipsoidal charge distribution) is demonstrated to be a special case. Both gravity intensity g and intensity E are governed by the same law and can be expressed as a series of spherical harmonics, and thus the G-modified Helmholtz equation is useful for describing the gravity and electrostatic fields.

Keywords: electrostatic field; intensity; gravity; equipotential spheroid; spherical harmonics (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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