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

=== Assistant: To estimate the self-repulsion force among lunar regolith particles due to electrostatic charging at -100 V during the … ===
To estimate the self-repulsion force among lunar regolith particles due to electrostatic charging at -100 V during the lunar night, we need to consider the Coulomb force acting between similarly charged dust particles.

===== The charge (QQQ) on a dust particle can be estimated using capacitance theory, assuming the regolith grains act like small conducting spheres: =====

Q=CVQ = C VQ=CV
where:
* C=4πϵ0rC = 4 \pi \epsilon_0 rC=4πϵ0r is the capacitance of a spherical dust grain,
* V=−100VV = -100VV=−100V is the surface potential,
* ϵ0=8.854×10−12\epsilon_0 = 8.854 \times 10^{-12}ϵ0=8.854×10−12 F/m is the vacuum permittivity,
* rrr is the radius of the dust grain.

For typical lunar dust grains, let’s consider a radius of r=1r = 1r=1 μm (10−610^{-6}10−6 m):

C=4π(8.854×10−12)(10−6)C = 4 \pi (8.854 \times 10^{-12}) (10^{-6})C=4π(8.854×10−12)(10−6)
C≈1.11×10−16 FC \approx 1.11 \times 10^{-16} \text{ F}C≈1.11×10−16 F
Thus, the charge on a 1-μm grain is:

Q=(1.11×10−16 F)(−100 V)Q = (1.11 \times 10^{-16} \text{ F}) (-100 \text{ V})Q=(1.11×10−16 F)(−100 V)
Q≈−1.11×10−14 CQ \approx -1.11 \times 10^{-14} \text{ C}Q≈−1.11×10−14 C

===== The Coulomb force (FFF) between two like-charged particles separated by a distance ddd is given by: =====

F=14πϵ0Q2d2F = \frac{1}{4\pi\epsilon_0} \frac{Q^2}{d^2}F=4πϵ01d2Q2
Assuming the two particles are 10 μm apart (d=10−5d = 10^{-5}d=10−5 m):

F=(9×109)(10−10)×(1.11×10−14)2F = \frac{(9 \times 10^9)}{(10^{-10})} \times (1.11 \times 10^{-14})^2F=(10−10)(9×109)×(1.11×10−14)2
F=(9×109)×(1.23×10−28)F = (9 \times 10^9) \times (1.23 \times 10^{-28})F=(9×109)×(1.23×10−28)
F≈1.1×10−18 NF \approx 1.1 \times 10^{-18} \text{ N}F≈1.1×10−18 N

===== - Compared to Lunar Gravity: The gravitational force acting on a single 1-μm particle of density ρ≈3000\rho \approx 3000ρ≈3000 kg/m³ is: Fg=mg=43πr3ρgF_g = mg = \frac{4}{3} \pi r^3 \rho gFg=mg=34πr3ρg =43π(10−6)3(3000)(1.62)= \frac{4}{3} \pi (10^{-6})^3 (3000) (1.62)=34π(10−6)3(3000)(1.62) ≈2.0×10−14 N\approx 2.0 \times 10^{-14} \text{ N}≈2.0×10−14 N The electrostatic repulsion force (~10−18^{-18}−18 N) is much weaker than gravity for 1-μm grains, meaning it won’t easily lift regolith off the surface. =====
* For Smaller Grains (0.1 μm): Since charge scales with radius, but weight scales with r3r^3r3, smaller grains experience relatively stronger repulsion. For 0.1 μm grains, the electrostatic force can approach or exceed their weight, possibly leading to dust lofting.

===== - The electrostatic self-repulsion force between 1-μm lunar dust particles at -100V is about 10−1810^{-18}10−18 N. =====
* This force is much weaker than lunar gravity for 1-μm grains, so large grains won’t levitate.
* However, submicron grains (0.1 μm and smaller) can experience forces comparable to or greater than their weight, supporting the idea of electrostatic dust lofting on the Moon.

Would you like calculations for different dust sizes or conditions (e.g., -500V charging in Earth's magnetotail)?