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

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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=40r = 40r=40 microns = 40×10−640 \times 10^{-6}40×10−6 m.

C=4π(8.854×10−12)(40×10−6)C = 4 \pi (8.854 \times 10^{-12}) (40 \times 10^{-6})C=4π(8.854×10−12)(40×10−6)
C≈4.45×10−15 FC \approx 4.45 \times 10^{-15} \text{ F}C≈4.45×10−15 F
====== Charge is calculated as: ======

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

Q=(4.45×10−15)×(−100)Q = (4.45 \times 10^{-15}) \times (-100)Q=(4.45×10−15)×(−100)
Q≈−4.45×10−13 CQ \approx -4.45 \times 10^{-13} \text{ C}Q≈−4.45×10−13 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=−4.45×10−13Q = -4.45 \times 10^{-13}Q=−4.45×10−13 C,
* d=55d = 55d=55 microns = 55×10−655 \times 10^{-6}55×10−6 m,

F=(9×109)(55×10−6)2×(4.45×10−13)2F = \frac{(9 \times 10^9)}{(55 \times 10^{-6})^2} \times (4.45 \times 10^{-13})^2F=(55×10−6)2(9×109)×(4.45×10−13)2
F=(9×109)×(1.98×10−253.025×10−9)F = (9 \times 10^9) \times \left(\frac{1.98 \times 10^{-25}}{3.025 \times 10^{-9}}\right)F=(9×109)×(3.025×10−91.98×10−25)
F≈5.88×10−7 NF \approx 5.88 \times 10^{-7} \text{ N}F≈5.88×10−7 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(40×10−6)3(3000)(1.62)F_g = \frac{4}{3} \pi (40 \times 10^{-6})^3 (3000) (1.62)Fg=34π(40×10−6)3(3000)(1.62)
Fg≈1.30×10−9 NF_g \approx 1.30 \times 10^{-9} \text{ N}Fg≈1.30×10−9 N