Coulomb’s Law
The force between two electric charges: like charges push each other away, opposite charges pull together, and the force falls off with the square of the distance. Solve for the force, either charge or the distance.
What Coulomb’s Law Says
Two charges push or pull on each other with a force proportional to both charges and to 1/r²: twice as far apart means a quarter of the force. Each charge feels the same size of force, in opposite directions (Newton’s third law).
The electric force is enormous compared with gravity: between an electron and a proton it is about 1039 times stronger. We hardly notice because ordinary matter has almost exactly as many positive as negative charges. A material between the charges weakens the force by its relative permittivity εr; water, at about 80, is why salt dissolves.
Coulomb’s Law, Step by Step
FKey Relations
Charges in coulombs, distance in metres, force in newtons. Rearranged: r = √(kq1q2/F) and q2 = Fr²/(kq1).
Charge q1 fills the space around it with a field; a second charge placed in it feels the force q2E.
The work needed to bring the charges together from far apart. Negative for opposite charges: you get energy out.
Sources & Verification
- D. Halliday, R. Resnick & J. Walker, Fundamentals of Physics, 10th ed., Wiley, 2014 — Ch. 21 (Coulomb’s Law) and Ch. 25 (dielectrics).
- CODATA 2022 recommended values (NIST) — ε0 = 8.854 187 8188 × 10−12 F/m; exact e = 1.602 176 634 × 10−19 C.
- CRC Handbook of Chemistry and Physics — relative permittivity of water (80.1 at 20 °C) and air (1.00059).
Frequently Asked Questions
How do you calculate the force between two charges?
With Coulomb's law: F = k × q₁ × q₂ ÷ r², with the charges in coulombs, the distance in metres and k = 8.988 × 10⁹ N·m²/C². Two charges of 1 µC at 10 cm push each other away with 8.988 × 10⁹ × 10⁻⁶ × 10⁻⁶ ÷ 0.01 = 0.899 N, about the weight of 92 grams.
What is Coulomb's constant?
k = 1/(4πε₀) = 8.988 × 10⁹ N·m²/C², where ε₀ = 8.854 × 10⁻¹² F/m is the permittivity of vacuum. In a material the force is weaker by its relative permittivity εr: about 1.0006 for air and 80 for water.
How does the distance change the electric force?
The force goes with 1/r², the inverse square of the distance. Twice as far apart gives a quarter of the force, three times as far a ninth; halving the distance makes it four times as strong.
How do I find the distance between two charges from the force?
Rearrange Coulomb's law: r = √(k × q₁ × q₂ ÷ F). Two 1 µC charges push with 1 N when they are √(8.988 × 10⁹ × 10⁻¹² ÷ 1) = 0.0948 m, about 9.5 cm, apart. Click Solve next to the distance.
How much stronger is the electric force than gravity?
Between an electron and a proton the electric attraction is about 2.3 × 10³⁹ times stronger than their gravitational pull, at any distance, since both fall off as 1/r². In the hydrogen atom (r = 52.9 pm) the electric force is 82 nN. Gravity only wins for large bodies, because they carry almost no net charge.
Why does salt dissolve in water?
Water has a relative permittivity of about 80: its molecules turn towards the ions and weaken the attraction between Na⁺ and Cl⁻ about 80 times. At 0.5 nm the pull drops from 0.92 nN in vacuum to 0.012 nN in water, weak enough for the thermal motion of the water to pull the ions apart.