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

===== Step-by-Step Calculations for 10-Micron and 80-Micron Particles with 100-Micron Spacing =====

====== Step 1: Capacitance of Both Particles ======

The capacitance of a spherical particle 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).

For the 10-micron particle (r=10×10−6r = 10 \times 10^{-6}r=10×10−6 m):

C10=4π(8.854×10−12)(10×10−6)C_{10} = 4 \pi (8.854 \times 10^{-12}) (10 \times 10^{-6})C10=4π(8.854×10−12)(10×10−6)
C10≈1.11×10−15 FC_{10} \approx 1.11 \times 10^{-15} \text{ F}C10≈1.11×10−15 F
For the 80-micron particle (r=80×10−6r = 80 \times 10^{-6}r=80×10−6 m):

C80=4π(8.854×10−12)(80×10−6)C_{80} = 4 \pi (8.854 \times 10^{-12}) (80 \times 10^{-6})C80=4π(8.854×10−12)(80×10−6)
C80≈8.91×10−15 FC_{80} \approx 8.91 \times 10^{-15} \text{ F}C80≈8.91×10−15 F

====== Step 2: Charge on the Particles ======

Charge is calculated as:

Q=CVQ = C VQ=CV
For the 10-micron particle (V=−100VV = -100VV=−100V):

Q10=(1.11×10−15)×(−100)Q_{10} = (1.11 \times 10^{-15}) \times (-100)Q10=(1.11×10−15)×(−100)
Q10≈−1.11×10−13 CQ_{10} \approx -1.11 \times 10^{-13} \text{ C}Q10≈−1.11×10−13 C
For the 80-micron particle:

Q80=(8.91×10−15)×(−100)Q_{80} = (8.91 \times 10^{-15}) \times (-100)Q80=(8.91×10−15)×(−100)
Q80≈−8.91×10−13 CQ_{80} \approx -8.91 \times 10^{-13} \text{ C}Q80≈−8.91×10−13 C

====== Step 3: Electrostatic Repulsion Force ======

The electrostatic force between two differently sized charged particles is given by Coulomb’s law:

F=14πϵ0Q10Q80d2F = \frac{1}{4\pi\epsilon_0} \frac{Q_{10} Q_{80}}{d^2}F=4πϵ01d2Q10Q80
Substituting:
* Q10=−1.11×10−13Q_{10} = -1.11 \times 10^{-13}Q10=−1.11×10−13 C,
* Q80=−8.91×10−13Q_{80} = -8.91 \times 10^{-13}Q80=−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×(−1.11×10−13)×(−8.91×10−13)F = \frac{(9 \times 10^9)}{(100 \times 10^{-6})^2} \times (-1.11 \times 10^{-13}) \times (-8.91 \times 10^{-13})F=(100×10−6)2(9×109)×(−1.11×10−13)×(−8.91×10−13)
F=(9×109)×(9.89×10−2610−8)F = (9 \times 10^9) \times \left(\frac{9.89 \times 10^{-26}}{10^{-8}}\right)F=(9×109)×(10−89.89×10−26)
F≈8.90×10−8 NF \approx 8.90 \times 10^{-8} \text{ N}F≈8.90×10−8 N

====== Step 4: Gravitational Force on Each Particle ======

The gravitational force acting on each 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).

For the 10-micron particle:

Fg=43π(10×10−6)3(3000)(1.62)F_g = \frac{4}{3} \pi (10 \times 10^{-6})^3 (3000) (1.62)Fg=34π(10×10−6)3(3000)(1.62)
Fg≈2.04×10−11 NF_g \approx 2.04 \times 10^{-11} \text{ N}Fg≈2.04×10−11 N
For the 80-micron particle:

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