Coulomb wrote his force law a century after Newton wrote his, and the two are the same sentence with different nouns: a product of two properties, divided by the square of the distance. Swap mass for charge and G for k and you have moved from one to the other. The only real difference is the size of the constant — and it is so lopsided that two protons repel each other about 1036 times harder than they attract. Everything below is computed from that one formula, with nothing tuned.
F = k q₁q₂ / r² vs F = G m₁m₂ / r²
Set two charges and pull them apart. The force follows the inverse square exactly: double the distance and it drops to a quarter, which on the log–log plot means a perfectly straight line of slope −2. Gravity between two 1 kg masses is drawn on the same axes for comparison — same slope, same shape, sitting eight decades lower. Like charges push apart, opposite charges pull together; gravity only ever pulls, because there is no negative mass.
Both forces fall off as 1/r², so their ratio does not depend on distance at all. Slide the separation from nuclear scale to interplanetary scale below: both markers sweep across seventy decades of force together, and the bracket between them never changes width. For two protons that bracket is a factor of about 1.24 × 10³⁶ — a number with no useful comparison, since the whole observable universe is only about 10²⁷ metres across. Yet look at the last row of the table: give two 1 kg balls a single extra electron each and gravity wins easily. Charge is overwhelmingly stronger per particle, and almost perfectly cancelled per kilogram.
| pair | electric | gravitational | electric ÷ gravitational |
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A charge does not reach across space to another charge — it fills the space around itself with a field, and the other charge responds to the field where it sits. Drag any charge and the whole picture rebuilds. Field lines leave positive charges and end on negative ones, never cross, and are drawn in proportion to charge, so a +12 nC charge sprouts three times the lines of a +4 nC one. The closed loops are equipotentials — surfaces of equal voltage — and they always cross the field lines at right angles, because the field points straight down the steepest slope of the potential. Move the pointer anywhere on the canvas to read the field and voltage at that point.
Nothing in the previous picture needed a new rule. Each charge produces its own field as if the others were not there, and the total is the vector sum — that is the whole of superposition, and it is why the tangled pictures above come out of a two-line formula. Drag the white probe (or the charges) and watch each contribution separately: every arrow is k·q/r² pointing away from a positive charge or toward a negative one, and the white arrow is what you get by laying them nose to tail. The table adds up the components; the sum row is not computed independently, it is literally the column totals.
| source | q | distance r | k·|q|/r² | Eₓ | E_y |
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If electricity is that much stronger than gravity, why is gravity the force you notice? Because matter comes with both signs and pays them off almost exactly. A kilogram of iron contains about 2.8 × 10²⁶ electrons and precisely as many protons, and the smallest deviation from that balance is felt immediately. Slide the imbalance below: strip away just two parts in a billion billion of the electrons and the electric repulsion already matches the gravity between the objects. That break-even fraction does not depend on how heavy they are or how far apart they sit — both forces scale identically — which is exactly why matter had no choice but to end up neutral.
Every number on this page comes from F = kq₁q₂/r² with k = 8.9875518 × 10⁹ N·m²/C² and the CODATA values of e, G and the particle masses, evaluated in double precision — nothing is fitted. The checks: two 1 C charges 1 m apart come out at 8.988 × 10⁹ N, the proton–proton force ratio at 1.2356 × 10³⁶, the electron–electron ratio at 4.166 × 10⁴², the electric force in a hydrogen atom at the Bohr radius at 8.24 × 10⁻⁸ N, and the log–log slope in section 1 at −2.000000. All of those match the published figures.
The pictures simplify in three honest ways. First, the charges here are modelled as small uniformly charged spheres rather than true points, so the field inside one grows linearly from zero instead of blowing up — that is the correct result for a charged ball, and it keeps the drawing finite, but a genuine point charge has no such interior. Second, this is a flat slice through a three-dimensional field: line density only measures field strength properly in 3-D, so a crowded-looking region on screen is suggestive rather than quantitative. Third, the drifting markers move along the field direction at a speed that saturates with field strength — they trace the geometry, and are not a dynamics simulation. A real charge released in this field would accelerate, gain momentum and overshoot, orbiting or spiralling rather than following a line.
Two pieces of physics are left out entirely. Everything here is electrostatics: charges are held where you put them, so there is no radiation, no magnetism and no finite speed of light — in reality a field change propagates outward at c, and an accelerating charge radiates energy away. And nothing here can actually be built at these charges: air breaks down at about 3 × 10⁶ V/m, so a sphere 1 cm across sparks once it holds more than roughly 33 nC. The section 5 slider will happily show you a 1% imbalance, which is a force of order 10²⁵ N; no such object has ever existed, and that is the point.