PHYSICS / FIELDS · 03

Charge can pull. Charge can push.

Gravity depends on mass. Electric force depends on charge. Begin with one charge, then bring in a second: the same inverse-square geometry now allows two different directions of force.

01The same spherical geometry. A different source.

Electric charge is a property of matter, measured in coulombs (C). A proton carries +e and an electron carries −e; e ≈ 1.602 × 10⁻¹⁹ C. Positive and negative name two signs, not two different amounts. They allow electric forces to repel as well as attract.

One isolated point charge · vacuumEqual distance, equal field magnitude

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Radius / reference1.50×

Spherical area2.25×

Field magnitude44.4%

The blue sphere is an imaginary measuring surface. Arrows show the direction a tiny positive test charge would be pushed, not particles flowing out of the source. For a negative source they point inward. Moving the surface does not send out a wave.

02Give the second charge a sign.

Choose + and − and release them. Reset, choose + and +, and repeat. Then try − and −. The signs choose attraction or repulsion; the amounts of charge and the centre distance determine the strength.

Two isolated point-charge carriersGold + · blue − · no other forces

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Force on charge 1

F_{2\to1,x}=0.0046\,\mathrm N

Force on charge 2

F_{1\to2,x}=-0.0046\,\mathrm N

Here +x points from charge 1 to charge 2. Equal opposite forces act on different bodies. The carriers each have mass 0.10 kg. Ball sizes mark positions; these are point charges, not conducting spheres whose charge redistributes. Motion stops before contact or leaving the viewing region.

F_r=k\frac{q_1q_2}{r^2}=-0.0046\,\mathrm N,\qquad |F|=0.0046\,\mathrm N

A magnitude is never negative. Fᵣ is a signed radial component: positive means charge 2 is pushed away from charge 1; negative means it is pulled towards charge 1. A negative force is not automatically attraction on every coordinate axis. The direction convention is essential.

03What do k and ε₀ mean?

Experiments give a force proportional to both charges and to 1/r². A proportionality constant converts this pattern into a force in newtons. In vacuum, that constant is k. It is conventionally written using the vacuum permittivity ε₀ (“epsilon zero”). This is a useful way of writing the same measured relation, not an extra force.

k=\frac{1}{4\pi\epsilon_0}\approx8.988\times10^9\ \mathrm{N\,m^2\,C^{-2}}

\epsilon_0\approx8.854\times10^{-12}\ \mathrm{C^2\,N^{-1}\,m^{-2}},\qquad |F|=\frac{|q_1q_2|}{4\pi\epsilon_0r^2}

q₁, q₂: Signed electric charges, in C. A minus sign identifies negative charge.
r: Distance between the two point charges, in m; always positive.
|F|: Force magnitude on either charge, in N. The magnitudes are equal.
ε₀: Permittivity of vacuum. Other materials require a model of their electrical response.

Similar to gravity

Both laws contain a product of source properties and an inverse square of separation. Each interaction gives two equal, opposite forces on different bodies.

|F_g|=G\frac{m_1m_2}{r^2}

Different from gravity

Newtonian gravity between ordinary positive masses attracts. Charge has two signs: unlike charges attract, like charges repel. Electric force depends on charge, not on mass; the resulting acceleration still depends on mass.

a=|F|/m

Scope: electrostatics in vacuum, point charges and slow motion. The release animation uses the quasistatic Coulomb approximation and neglects radiation. Protons and electrons introduce charge signs; the metal-coloured markers are not images of subatomic particles.

Next: how force changes motion

OpenStax · Coulomb’s law