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The force described by the [[Yukawa potential]]
:<math> U(r) = -\frac{G M m}{r} e^{-\lambda r}</math>
has the property that the force outside of a spherical shell is also a Yukawa potential with the same range <math>1/\lambda</math> and centered at the shell's center, but for <math>\lambda > 0</math> the equivalent point mass is not the same as the mass of the shell.<ref>{{cite web |last=Kuhn |first=Paulo |title=Debye-Hückel interaction, or Yukawa potential, in different geometries |url=https://wp.ufpel.edu.br/pskuhn/files/2024/09/debye-huckel-interaction-or-yukawa-in-different-geometries.pdf |access-date=14 February 2025}}</ref><ref>{{cite web |last=McDonald | first=Kirk |orig-date=April 17, 1984 |date=December 20, 2021 |title=A Naïve Estimate of the Coupling Constant in Yukawa Theory |url=http://kirkmcd.princeton.edu/examples/yukawa.pdf |access-date=14 February 2025}}</ref><ref>{{cite web |title=Shell theorem for a general potential |url=https://math.stackexchange.com/questions/296180/shell-theorem-for-a-general-potential |website=Mathematics Stack Exchange |access-date=14 February 2025 |language=en}}</ref> For a shell of radius <math>R</math> and mass <math>M</math>, the equivalent point mass is:
:<math>M_\text{eff} = M \frac{\sinh \lambda r}{\lambda r}</math>.
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