18.4 Electric Field of a Point Charge


2026 Syllabus Objectives

By the end of these notes, you should be able to:

  1. Recall and use the equation E=Q4πε0r2E = \dfrac{Q}{4\pi\varepsilon_0 r^2} to calculate the electric field strength due to a point charge in free space.

What is a Point Charge?

A point charge is an idealised (simplified) model of a charged object where all the electric charge is imagined to be concentrated at a single point. In reality, objects have size, but when we are far enough away from them, we can treat them as if all their charge sits at one point. This makes calculations much simpler.


Recap: Electric Field Strength

Before using the formula, let's quickly recall what electric field strength means.

Electric field strength (E) is defined as the force experienced per unit of positive charge at a given point in an electric field. In simpler terms: if you place a tiny positive charge at a point in a field, EE tells you how much force each coulomb of that charge would feel.

E=FqE = \frac{F}{q}

  • EE is measured in newtons per coulomb (N C⁻¹), which is the same as volts per metre (V m⁻¹).
  • It is a vector quantity — meaning it has both a size and a direction.

The Formula: Electric Field Strength of a Point Charge

When we want to find the electric field strength at a certain distance from a point charge, we use the following equation:

E=Q4πε0r2\boxed{E = \frac{Q}{4\pi\varepsilon_0 r^2}}

Let's look at each part carefully:

SymbolWhat it meansUnit
EEElectric field strength at the pointN C⁻¹ or V m⁻¹
QQThe charge producing the electric field (the source charge)Coulombs (C)
ε0\varepsilon_0Permittivity of free space — a fixed constant of natureF m⁻¹
rrThe distance from the centre of the point charge to the point where you want to find EEmetres (m)
4πε04\pi\varepsilon_0A combination of constants that appears in all electric field equations

The value of the permittivity of free space is:

ε0=8.85×1012 F m1\varepsilon_0 = 8.85 \times 10^{-12} \text{ F m}^{-1}

This constant tells us how well a vacuum (empty space) allows an electric field to pass through it.

You will sometimes see the formula written using the shorthand constant kk, where:

k=14πε08.99×109 N m2 C2k = \frac{1}{4\pi\varepsilon_0} \approx 8.99 \times 10^9 \text{ N m}^2 \text{ C}^{-2}

So the formula can also be written as:

E=kQr2E = \frac{kQ}{r^2}

Both versions mean exactly the same thing. In A-Level Physics, you are expected to use the full version with 4πε04\pi\varepsilon_0.

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