18.3 Electric Force Between Point Charges

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Understand that, for a point outside a spherical conductor, the charge on the sphere may be treated as a point charge at its centre.
  2. Recall and use Coulomb's law: F = Q₁Q₂ / (4πε₀r²) for the force between two point charges in free space.

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1. The Point Charge Approximation for Spherical Conductors

Imagine you have a metal sphere with electric charge spread evenly across its surface. If you are standing outside that sphere and want to calculate the electric force it exerts, you do not need to worry about the exact shape or size of the sphere. You can treat all of the sphere's charge as if it were concentrated at a single point right at the centre of the sphere.

This is called the point charge approximation.

Why does this work? Because the charge on a uniform spherical conductor is spread out symmetrically (evenly in all directions). The electric field lines it produces outside the sphere are radial — they point straight outward (or straight inward) from the centre, exactly as they would if all the charge sat at one point in the middle.

  • If the sphere is positively charged, the field lines point away from the centre.
  • If the sphere is negatively charged, the field lines point towards the centre.

This means that, as far as any external point is concerned, the sphere behaves identically to a point charge of the same total charge placed at the sphere's centre.

Important: This approximation only works for points outside the sphere. Inside a hollow conductor, the electric field is actually zero — but that is beyond what you need for this subtopic.

2. Coulomb's Law

Coulomb's Law describes the electric force between two point charges — that is, two charged objects that are either very small, or far enough apart that their size doesn't matter.

The law states:

The electric force between two point charges is directly proportional to the product of the charges, and inversely proportional to the square of the distance between them.

The formula is:

F=Q1Q24πε0r2F = \frac{Q_1 Q_2}{4\pi\varepsilon_0 r^2}

Breaking down every symbol:

SymbolWhat it meansUnit
FElectric force between the two chargesNewtons (N)
Q₁Magnitude of the first chargeCoulombs (C)
Q₂Magnitude of the second chargeCoulombs (C)
rDistance between the centres of the two chargesMetres (m)
ε₀Permittivity of free space — a fixed constant of natureC² N⁻¹ m⁻²

The value of ε₀:

ε0=8.85×10−12 C2 N−1 m−2\varepsilon_0 = 8.85 \times 10^{-12} \text{ C}^2 \text{ N}^{-1} \text{ m}^{-2}

You do not need to memorise this — it is given on your data sheet in the exam.

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