Coulomb’s law and electric field
F = k · |q₁ q₂| / r²
Coulomb’s law in its two readings: the force between two charges, F = k·q₁q₂/r², and the field created by a single one, E = k·q/r², followed by the force a test charge would feel there. The constant k is derived from the CODATA vacuum permittivity rather than copied rounded; the sign of the charges decides the nature of the force, named in words instead of an ambiguous negative number; and every result recalls the parallel with Newton, same 1/r² form but thirty-nine orders of magnitude apart between a proton and an electron.
0.898755 N
- Calculation
- F = k·|q₁q₂|/r² = 8987550000 × |1E-6 × 1E-6| / 0.1² = 0.898755 N
- Nature
- repulsive, the charges have the same sign
The force is 0.898755 N; it is repulsive, the charges have the same sign.
The sign of the charges decides the nature of the force, never its magnitude: like signs repel, opposite signs attract. The tool separates the two pieces of information instead of returning an ambiguous negative number.
Same form as Newton’s gravitation, in 1/r², but two differences that make the world: gravity only attracts, and electricity is immensely stronger. Between a proton and an electron, the electric force exceeds the gravitational one by a factor of 10³⁹. If matter neither collapses nor explodes, it is because charges cancel almost exactly.
The law holds for point charges or spherically symmetric ones, in vacuum; inside a material, the relative permittivity divides the force. At the atomic scale it stays exact on average but quantum mechanics governs bound states: it gives the order of magnitude of the bond, not its energy levels.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasF = k · |q₁ q₂| / r²
F = k · |q₁ q₂| / r²
E = k · |q| / r²
F = q E
k = 1/(4πε₀) = 8.988 × 10⁹ N·m²/C²
The force falls as the square of the distance, exactly like gravitation: the same geometry, that of an influence diluted over a sphere. The constant k is not an arbitrary number, it derives from the vacuum permittivity ε₀, and the tool computes it from that definition rather than copying a rounded value.
- q
- · electric charge, in coulombs. The sign is part of the charge.
- Elementary charge
- · e = 1.602 176 634 × 10⁻¹⁹ C, exact since the 2019 SI: the charge of the proton, the opposite of the electron’s.
- E
- · electric field, in volts per metre, which is the same unit as newtons per coulomb.
- ε₀
- · vacuum permittivity, 8.854 × 10⁻¹² F/m (CODATA 2018).
Validity domainThe law holds for point charges, or spherically symmetric ones seen from outside, at rest and in vacuum.
The law holds for point charges, or spherically symmetric ones seen from outside, at rest and in vacuum. Inside a material the relative permittivity divides the force; for fast-moving charges, magnetism adds on. At the atomic scale it stays exact on average but quantum mechanics governs bound states: it gives the order of magnitude of the hydrogen bond, not its energy levels. Zero distance is refused, the law diverging as 1/r².
Reading the resultMagnitude and nature are two separate pieces of information: the sign of the charges decides whether they attract or repel, never the intensity.
Magnitude and nature are two separate pieces of information: the sign of the charges decides whether they attract or repel, never the intensity. The field, for its part, exists with no one to feel it: it describes what a charge would experience at that point, and the test charge merely reveals it. The parallel with gravitation is the most fruitful reading: the same 1/r² law, but electricity dominates by a factor of 10³⁹ between proton and electron, and if matter holds without collapsing or exploding, it is because charges cancel almost exactly.