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Comparing electric and gravitational fields

Set the two field theories side by side and the equations pair off row for row, one mathematical shape wearing two costumes. Then put two protons a femtometre apart, compute both forces, and discover which force actually runs the universe at each scale.

Year 13AQA 3.7.1, 3.7.3.1

Builds on Electric potential and Newton's law of gravitation.

IN THIS TOPIC

  • Pair the gravitational and electric equations and name what transfers between them.
  • State the structural difference: attraction only, against attraction or repulsion.
  • Compare the magnitudes of the two forces between subatomic particles, and interpret the answer.

WHAT YOU PROBABLY THINK

Gravity is the strongest force.

One shape, two theories

The two field theories side by side: the same mathematical shape four rows deepgravityelectricF = Gm₁m₂/r²F = Q₁Q₂/4πε₀r²g = F/mE = F/QV = −GM/rV = Q/4πε₀rg = −ΔV/ΔrE = ΔV/Δrthe same mathematics, four rows deepgravity keeps its minus signs; AQA quotes E as a magnitude
FIG. 1The formula ladder: force, field strength, potential and gradient pair off between the two theories, the same shape four rows deep.

Every tool this unit built for gravity reappears for charge with the cast changed: G becomes 1/4πε0, mass becomes charge, and each equation keeps its shape. Both are inverse-square laws, and the shared mathematics is why both come with the same apparatus of field lines, potential, and equipotential surfaces. The minus signs mark the one asymmetry: gravity's potential is always negative because gravity only ever attracts, while the electric potential takes the sign of its charge; and AQA prints g = −ΔV/Δr with its sign but quotes the electric E = ΔV/Δr as a magnitude.

The structural difference

Masses always attract. Charges may attract or repel, because charge comes in two signs and mass comes in one. That single fact decides which force governs which world. Bulk matter is almost perfectly neutral, its positive and negative charges cancelling to fantastic precision, so the electric force, for all its strength, largely switches itself off over large scales. Mass has no cancelling partner: every kilogram adds, nothing subtracts, and an unshieldable attraction that only accumulates ends up sculpting planets, stars and galaxies by default.

Two protons settle it

Two protons, both forces at once: the electric repulsion beats the gravitational attraction by thirty-six orders of magnitudepp1.0 fm apartelectric push: 230 Ngravitational pull: 1.9 × 10⁻³⁴ Nelectricity wins by a factor of 10³⁶
FIG. 2Both forces between two protons a femtometre apart: 230 newtons of electric repulsion against 10⁻³⁴ newtons of gravity.

The spec asks for the comparison between subatomic particles, and two protons inside a nucleus make the cleanest case. At a separation of 1.0 fm the electric repulsion is about 230 N, a startlingly human-sized force on a particle of 10−27 kg, while the gravitational attraction between the same pair is about 1.9 × 10−34 N. The ratio is near 1036, and it is independent of separation, since both forces share the same 1/r2. This is why particle physics ignores gravity entirely, and why something stronger still, the strong nuclear force from the particles unit, must exist to hold nuclei together against that 230 N of repulsion.

THE EXAM BIT

  • The comparison question has a fixed skeleton: similarities are the inverse-square law and the shared machinery of field lines, potential and equipotentials; the difference is attract-only against attract-or-repel. Give both halves.
  • For the proton calculation, take both masses and charges from the data booklet, keep r identical for the two forces, and present the answer as a ratio near 1036.
  • The ratio needs no r: both laws carry 1/r2, so it cancels. Saying so is usually a mark.
  • “Why does gravity dominate astronomy?” wants the cancellation argument: matter is neutral, so electric forces cancel, while masses only add and gravity cannot be screened.
  • Cross-topic link worth one line: the 230 N repulsion inside a nucleus is exactly what the strong force must beat, tying this unit to particle physics.

CHECK YOURSELF

Two protons sit 1.0 fm apart. Using data-booklet values, find the electric and gravitational forces between them and the ratio of the two. Why, despite this ratio, does gravity and not electricity shape galaxies?

Show a hint

Same r in both laws, so the ratio outlives it.

Show the answer

Electric: F = Q1Q2/4πε0r2 = (8.99 × 109 × (1.60 × 10−19)2) / (1.0 × 10−15)2 ≈ 230 N, repulsive.

Gravitational: F = Gm2/r2 = (6.67 × 10−11 × (1.67 × 10−27)2) / (1.0 × 10−15)2 ≈ 1.9 × 10−34 N.

Ratio ≈ 1.2 × 1036, independent of r. Galaxies still belong to gravity because bulk matter is neutral: charges cancel and electric forces screen themselves out, while masses only ever add.

One inverse-square shape, two casts of characters.

Electricity wins by 10³⁶; neutrality hands the universe to gravity.

No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.