Editing Openai/67b71e2c-cc10-8008-9939-dfe494d1384d (section)

=== Assistant:  ===
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The capacitance of a sphere is given by:

C=4πϵ0rC = 4 \pi \epsilon_0 rC=4πϵ0r
where:
* ϵ0=8.854×10−12\epsilon_0 = 8.854 \times 10^{-12}ϵ0=8.854×10−12 F/m (vacuum permittivity),
* r=80r = 80r=80 microns = 80×10−680 \times 10^{-6}80×10−6 m.

C=4π(8.854×10−12)(80×10−6)C = 4 \pi (8.854 \times 10^{-12}) (80 \times 10^{-6})C=4π(8.854×10−12)(80×10−6)
C≈8.91×10−15 FC \approx 8.91 \times 10^{-15} \text{ F}C≈8.91×10−15 F
====== Charge is calculated as: ======

Q=CVQ = C VQ=CV
where V=−100VV = -100VV=−100V:

Q=(8.91×10−15)×(−100)Q = (8.91 \times 10^{-15}) \times (-100)Q=(8.91×10−15)×(−100)
Q≈−8.91×10−13 CQ \approx -8.91 \times 10^{-13} \text{ C}Q≈−8.91×10−13 C
====== The electrostatic force between two particles separated by distance ddd is given by Coulomb’s law: ======

F=14πϵ0Q2d2F = \frac{1}{4\pi\epsilon_0} \frac{Q^2}{d^2}F=4πϵ01d2Q2
Substituting:
* Q=−8.91×10−13Q = -8.91 \times 10^{-13}Q=−8.91×10−13 C,
* d=100d = 100d=100 microns = 100×10−6100 \times 10^{-6}100×10−6 m,

F=(9×109)(100×10−6)2×(8.91×10−13)2F = \frac{(9 \times 10^9)}{(100 \times 10^{-6})^2} \times (8.91 \times 10^{-13})^2F=(100×10−6)2(9×109)×(8.91×10−13)2
F=(9×109)×(7.94×10−2510−8)F = (9 \times 10^9) \times \left(\frac{7.94 \times 10^{-25}}{10^{-8}}\right)F=(9×109)×(10−87.94×10−25)
F≈7.12×10−7 NF \approx 7.12 \times 10^{-7} \text{ N}F≈7.12×10−7 N
====== The gravitational force acting on the particle is: ======

Fg=mg=43πr3ρgF_g = mg = \frac{4}{3} \pi r^3 \rho gFg=mg=34πr3ρg
where:
* ρ=3000\rho = 3000ρ=3000 kg/m³ (density of lunar regolith),
* g=1.62g = 1.62g=1.62 m/s² (lunar gravity).

Fg=43π(80×10−6)3(3000)(1.62)F_g = \frac{4}{3} \pi (80 \times 10^{-6})^3 (3000) (1.62)Fg=34π(80×10−6)3(3000)(1.62)
Fg≈1.04×10−8 NF_g \approx 1.04 \times 10^{-8} \text{ N}Fg≈1.04×10−8 N

===== - Electrostatic Repulsion Force: Felectric≈7.12×10−7 NF_{\text{electric}} \approx 7.12 \times 10^{-7} \text{ N}Felectric≈7.12×10−7 N =====
* Gravitational Force: Fgravity≈1.04×10−8 NF_{\text{gravity}} \approx 1.04 \times 10^{-8} \text{ N}Fgravity≈1.04×10−8 N

===== - The electrostatic force is about 68 times stronger than lunar gravity at this spacing of 100 microns. =====
* This means that electrostatic repulsion can still strongly influence dust motion, even for larger grains.
* Though less extreme than smaller grains at closer distances, dust could still be lofted or transported due to charging effects.

Would you like to extend this to different charge conditions, such as -500V in Earth's magnetotail?