Coulomb's Law

Coulomb's law is the electrostatic equivalent of Newton's law of gravitation. It gives the force between two point charges — a simple 1/r² inverse-square law that underpins all of electrostatics. Charles-Augustin de Coulomb measured the force with a delicate torsion balance in the 1780s and discovered that, to the precision of his experiments, the force between two charges was exactly proportional to the inverse square of their separation. More than two centuries later, the law has been verified to one part in 10¹⁶ — it is one of the best-tested laws in physics.

This lesson continues OCR H556 Module 6.2.

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The Law

For two point charges Q₁ and Q₂ separated by a distance r, the magnitude of the force each exerts on the other is

F = Q₁Q₂ / (4πε₀r²)

where ε₀ = 8.85 × 10⁻¹² F m⁻¹ is the permittivity of free space, and 1/(4πε₀) ≈ 8.99 × 10⁹ N m² C⁻².

The force is:

• Attractive if the charges have opposite signs.
• Repulsive if the charges have the same sign.
• Along the line joining the two charges (Newton's third law — the two charges exert equal and opposite forces on each other).

Units Check

[F] = [C²] / [F m⁻¹ × m²]
    = [C²] / [(C² N⁻¹ m⁻²) × m²]
    = [C² × N m² C⁻² × m⁻²]
    = [N]

(Using 1 F = 1 C² N⁻¹ m⁻², which follows from the capacitor energy formula.)

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Comparison with Newton's Law of Gravitation

Compare

F_grav = Gm₁m₂ / r²       (always attractive)
F_elec = Q₁Q₂ / (4πε₀r²)   (attractive or repulsive)

The two laws have exactly the same mathematical form: a product of source strengths divided by 4π r² (with a dimensional constant). But they differ in crucial physical ways:

Feature	Gravitation	Electrostatics
Source	Mass m	Charge Q
Constant	G = 6.67 × 10⁻¹¹	1/(4πε₀) = 8.99 × 10⁹
Sign	Always attractive	Attractive or repulsive
Strength (per coulomb vs per kg)	Very weak	Vastly stronger
Distance dependence	1/r²	1/r²
Shielding	None	Possible (conductors)

The dramatic difference in strength is worth dwelling on. For two electrons, the ratio of the electrostatic repulsion to the gravitational attraction is about 10⁴². Electric forces utterly dominate on the atomic scale; gravity only wins on astronomical scales because matter is overwhelmingly electrically neutral — positive and negative charges cancel — whereas mass is always positive and cannot cancel itself.

Exam Tip: OCR often asks you to "compare gravitational and electric fields". The standard marks are: same 1/r² dependence, gravity always attractive while electric can be either, electric much stronger per unit source, electric can be shielded, gravitational cannot.

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Worked Example — Two Protons

Calculate (a) the electrostatic force, and (b) the gravitational force between two protons separated by 1.0 × 10⁻¹⁰ m (about an Ångström, the scale of an atom). Find the ratio of the two.

Data: e = 1.6 × 10⁻¹⁹ C, m_p = 1.67 × 10⁻²⁷ kg, G = 6.67 × 10⁻¹¹ N m² kg⁻²

(a) Electric force

F_e = (1.6 × 10⁻¹⁹)² × (8.99 × 10⁹) / (1.0 × 10⁻¹⁰)²
    = (2.56 × 10⁻³⁸) × (8.99 × 10⁹) / (1.0 × 10⁻²⁰)
    = 2.30 × 10⁻⁸ N

(b) Gravitational force

F_g = (6.67 × 10⁻¹¹) × (1.67 × 10⁻²⁷)² / (1.0 × 10⁻¹⁰)²
    = (6.67 × 10⁻¹¹) × (2.79 × 10⁻⁵⁴) / (1.0 × 10⁻²⁰)
    = 1.86 × 10⁻⁴⁴ N

(c) Ratio