Coulomb's Law

Multiply the two charges in coulombs, divide by the square of their separation in metres and multiply by the Coulomb constant k_e = 8.99×10⁹ N·m²/C². For the magnitude you insert the magnitudes of the charges: |F| = k_e·|q₁·q₂|/r². Two protons at a separation of 10⁻¹⁵ m thus experience about 230 N, which shows how strong the electrostatic repulsion inside an atomic nucleus is. As with the law of gravitation r² sits in the denominator, so the force falls with the square of the distance. Watch the tiny charge magnitudes: an elementary charge is only 1.6×10⁻¹⁹ C.

What matters are the signs of the charges, not the magnitude of the force. Like signs, that is two positive or two negative charges, repel. Unlike signs, that is plus and minus, attract. If you insert the charges with their signs into the formula, a positive product gives a repulsive force and a negative product an attractive one. In practice the magnitude is usually computed from the absolute values of the charges and the direction is determined separately from the signs. A common mistake is to infer the direction from the formula alone without applying the sign logic.

Both laws share the same mathematical form: a force proportional to the product of two source quantities and inversely proportional to the square of the distance. There are two important differences, however. First, gravity is always attractive, because there is no negative mass, whereas the Coulomb force can be attractive or repulsive depending on the charge signs. Second, the electrostatic force is enormously stronger: between two protons the Coulomb repulsion is about 10³⁶ times larger than their gravitational attraction. That is why the electric interaction dominates at the atomic scale, while gravity only becomes noticeable for large masses such as planets.

Consider the factors separately. Doubling one charge puts a factor of 2 in the numerator, so the force would double. Doubling the distance at the same time puts (2)² = 4 in the denominator, so the force would fall to one quarter. Together this gives a factor of 2/4 = 1/2, so the force is halved. Such ratio problems are solved fastest without inserting concrete numbers: you multiply the linear factors from the numerator by the squared factors from the denominator. This avoids arithmetic errors with the very small charge values and the large constant k_e.

The Coulomb law with k_e strictly holds only in vacuum. In a medium such as water or air the molecules there weaken the electric field, because they polarize themselves and oppose the charge. This weakening is described by the relative permittivity ε_r of the substance. In a medium you divide the force by ε_r, or replace k_e by k_e/ε_r. Water has an ε_r of about 80, so the Coulomb force there is roughly 80 times weaker than in vacuum. This explains why ionic salts separate and dissolve easily in water, while they stay tightly bound as a solid.