# A force comparison for a free grain

An electric field can oppose gravity for one charge polarity and reinforce it for the other. This analytical study compares those forces on hypothetical free spherical grains. It does not calculate detachment, dust removal or hardware performance.

Upward is positive. The chosen uniform upward field is E=10,000 V/m, density is 2,500 kg/m³, and g=9.80665 m/s². Charges are prescribed at -10, -1, 0, +1 and +10 femtocoulombs (1 fC=10^-15 C), independently of radius. Radii range from 1 to 50 micrometres in 0.5 micrometre steps, giving 99 sizes and 495 rows. These are illustrative inputs, not a measured particle population or charging law. Charge is treated as a continuum approximation.

For radius r in metres, m=rho*(4*pi/3)*r³. Downward weight is mg; signed upward electric force is qE; net upward acceleration is qE/m-g. A signed electric-force/weight ratio of +1 balances the two forces in this ideal model. Ratios greater than +1 give upward initial acceleration; negative ratios mean electric force points downward. The positive balancing charge for the stated positive field is mg/E.

At radius 10 micrometres and charge +1 fC: mass=1.04719755e-11 kg, weight=1.02694999e-10 N, upward electric force=1e-11 N, and net upward acceleration=-8.85172034 m/s². The particle still initially accelerates downward. At this size, balancing weight requires +10.26949987 fC with the chosen field. Doubling radius multiplies mass and balancing charge by eight at constant density and field.

The signed plot uses a symmetric logarithmic axis to show positive, zero and negative forces. Its central region is linear; its outer regions are logarithmic. The separate positive balancing-charge graph is logarithmic on both axes. Neither graph includes uncertainty bars because inputs are deterministic teaching choices.

## Boundary of the lesson

The grain has no contact with a substrate. Adhesion, drag, buoyancy, polarization, image forces, charge exchange, collisions and field gradients are absent. Initial force balance is not stable levitation or a trajectory. A neutral grain has zero qE here; neutral polarizable grains can experience forces in nonuniform fields. The calculation does not describe an electrodynamic dust shield, whose travelling nonuniform field and substrate interaction require different modelling. No voltage, electrode spacing or construction steps are supplied.

## Reproduction and independent checks

Run `python reproduce.py` with matplotlib. Outputs: force-weight.csv, metadata.json, checks.json and free-grain-forces.png/svg/pdf. All 495 rows are checked using a diameter-based independent volume calculation and direct qE/m-g acceleration. Further checks verify charge/field polarity, zero-charge gravity, the balancing charge and cubic radius scaling. These are implementation checks, not experimental validation.

## Primary references

- [OpenStax electric-field definition](https://openstax.org/books/university-physics-volume-2/pages/5-4-electric-field): F=qE and polarity. No source figures or prose are reproduced.
- [NASA EDS development paper](https://ntrs.nasa.gov/citations/20150016082): independent comparator using embedded electrodes and a travelling nonuniform field with electrostatic and dielectrophoretic effects. It does not supply our hypothetical parameters or validate this free-grain calculation.

References checked 27 September 2026. Figure is an original analytical plot.