Electric Force Compared with Gravity

Coulomb's law, F = k q₁ q₂ / r², and Newton's law of gravitation, F = G m₁ m₂ / r², are built the same way. Each has a constant out front, two source quantities multiplied together, and a 1/r² fall-off. Charge plays the role mass plays; k plays the role G plays. That shared structure is why you can compare them directly: take the ratio F_electric / F_gravity for the same two particles and the r² cancels completely, leaving a plain number that does not care how far apart the particles are.

That number turns out gigantic for things like protons or electrons. k is about 8.99x10⁹ N m²/C², while G is about 6.674x10⁻¹¹ N m²/kg² — k beats G by twenty orders of magnitude before you even touch the charges and masses. On top of that, the elementary charge is a large quantity relative to the tiny mass of a proton or electron, so q² in the numerator stays meaningful while m² in the denominator is minuscule. Multiply those effects together and electric repulsion between two protons outweighs their mutual gravity by roughly 10^36. Gravity never vanishes; it just never has a chance to compete at particle scales.

The one real structural difference is sign. Mass is only positive, so gravity only pulls together. Charge comes in two flavors, so the electric force can pull or push depending on whether the charges match or differ. That single difference is what Gauss's law will exploit later: flux through a closed surface can be positive, negative, or zero, something that has no gravitational parallel in this course.