In 1785 Charles-Augustin de Coulomb measured exactly how two charges push or pull on each other and found a law that looks remarkably like Newton's gravity: force grows with each charge and dies away with the square of the distance. It is the master equation of electrostatics, every charge-on-charge problem in this course starts here.
Two charged spheres repel each other with a force $F$. You then move them twice as far apart.
Before reading: do you expect the force to become one-half, one-quarter, or something else? Write your prediction and a one-sentence reason.
Three things set the force: two charges and a distance
The force between two charges grows when either charge grows, and falls away sharply as they move apart.
Coulomb's law gives the magnitude of the force between two point charges $q_1$ and $q_2$ separated by a distance $r$:
$$F = \frac{kq_1 q_2}{r^2} \qquad k = 8.99 \times 10^{9}\ \text{N m}^2\,\text{C}^{-2}$$
The factors that affect the force are:
Coulomb's law is $F = \dfrac{kq_1 q_2}{r^2}$. The force is proportional to each charge and inversely proportional to the square of the separation; like charges repel, unlike charges attract, and the pair always feels equal and opposite forces.
Pause, copy the highlighted law and factors into your book before moving on.
Did you get this? True or false: when a large charge and a small charge interact, the larger charge experiences the larger force.
Substituting both charges and the separation
We just saw what each factor does to the force. That raises a question: how do we calculate the actual force for a real pair of charges? This card answers it → substitute both charges and the distance into $F = \dfrac{kq_1 q_2}{r^2}$.
A Coulomb-force calculation has the same shape as the field calculation, just with two charges multiplied together on top.
Use charges in coulombs and distance in metres; the force comes out in newtons. Use the magnitudes of the charges for the size, then decide attract/repel from their signs.
Two point charges, $+4.0 \times 10^{-6}\ \text{C}$ and $-3.0 \times 10^{-6}\ \text{C}$, are $0.50\ \text{m}$ apart in air. Find the magnitude and nature of the force between them.
To find the Coulomb force, substitute the charge magnitudes and separation into $F = \dfrac{kq_1 q_2}{r^2}$ (answer in newtons), then state attractive (unlike charges) or repulsive (like charges) from the signs.
Add the highlighted worked method to your notes before the check below.
Quick check: Two charges experience a force $F$. If the distance between them is tripled with the charges unchanged, the new force is:
Pattern B, Reason about proportionality
Two charges sit a distance $r$ apart and feel a force $F$. In your book, predict the new force (as a multiple of $F$) for each change, and justify each in one line: (1) one charge is doubled; (2) both charges are doubled; (3) the separation is halved; (4) both charges are doubled and the separation is doubled.
Odd one out: three of these changes INCREASE the Coulomb force between two charges. Which one does not?
Dot point: Analyse the factors that affect the force between 2 charged particles using the electrostatic force law and F = kq₁q₂/r² (Coulomb's law)
Why electrostatics and gravitation share the same shape
We just learned to calculate and scale the Coulomb force. That raises a question: why does this law look so much like gravity? This card answers it → both are inverse-square laws between two source quantities.
Coulomb's law and Newton's law of gravitation are built the same way: a constant, the product of two "sources", and an inverse square of distance.
Compare them side by side. Newton's law of gravitation is $F = \dfrac{GMm}{r^2}$ and Coulomb's law is $F = \dfrac{kq_1 q_2}{r^2}$. Both have:
The key difference: gravity is always attractive, while the electrostatic force can attract or repel because charge comes in two signs. Recognising this shared structure makes the Year 12 gravitation topic feel familiar from day one.
Coulomb's law $F = \dfrac{kq_1 q_2}{r^2}$ has the same inverse-square, product-of-sources form as Newton's gravitation $F = \dfrac{GMm}{r^2}$. The difference is that gravity is only attractive, while the electrostatic force can attract or repel.
Pause, copy the highlighted comparison into your book before the check below.
Last check: Decide if the statement is true or false.
Both Coulomb's law and Newton's law of gravitation are inverse-square laws, but only the electrostatic force can be repulsive.
Match the pairs: Match each change to the factor by which the Coulomb force changes.
Dot point: Analyse the factors that affect the force between 2 charged particles using the electrostatic force law and F = kq₁q₂/r² (Coulomb's law)
A fresh set drawn from this lesson's question bank, feedback shown immediately. +5 XP per correct · +25 XP all correct
Pick your answer, then rate your confidence, that tells the system what to drill next.
ApplyBand 5(4 marks) 1. Two point charges of $+2.0 \times 10^{-6}\ \text{C}$ and $+5.0 \times 10^{-6}\ \text{C}$ are placed $0.20\ \text{m}$ apart in air. Calculate the magnitude of the electrostatic force between them and state whether it is attractive or repulsive, with a reason.
1 mark: uses $F = \dfrac{kq_1 q_2}{r^2}$ · 1 mark: correct substitution · 1 mark: $F \approx 2.2\ \text{N}$ · 1 mark: repulsive because both charges are positive (like charges)
AnalyseBand 5(3 marks) 2. Two point charges separated by a distance $r$ repel each other with a force of $0.36\ \text{N}$. One of the charges is then doubled and the separation is increased to $3r$. Determine the new force, showing your reasoning from the structure of Coulomb's law rather than by substituting values of $k$.
1 mark: doubling one charge multiplies the force by 2 · 1 mark: tripling the separation divides the force by 9 · 1 mark: net factor $2/9$, so $F = 0.080\ \text{N}$
Dot point: Analyse the factors that affect the force between 2 charged particles using the electrostatic force law and F = kq₁q₂/r² (Coulomb's law)
AnalyseBand 5(3 marks) 3. Coulomb's law and Newton's law of universal gravitation have the same mathematical form. State two features they share and one important way in which they differ.
1 mark: both are inverse-square in separation · 1 mark: both are proportional to the product of the two source quantities (charge, mass) · 1 mark: gravitation is always attractive whereas the electrostatic force can attract or repel
Dot point: Analyse the factors that affect the force between 2 charged particles using the electrostatic force law and F = kq₁q₂/r² (Coulomb's law)
MC answers and full explanations are shown inline as you complete each question. Use the retry button to attempt a fresh set from the lesson bank.
Q1 (4 marks): Using Coulomb's law, $F = \dfrac{kq_1 q_2}{r^2} = \dfrac{(8.99 \times 10^{9})(2.0 \times 10^{-6})(5.0 \times 10^{-6})}{(0.20)^2} = \dfrac{0.0899}{0.040} \approx 2.2\ \text{N}$. The force is repulsive because both charges are positive, like charges repel, so each charge is pushed away from the other along the line joining them.
Q2 (3 marks): Coulomb's law gives $F = kq_1q_2/r^2$, so the force is directly proportional to each charge and inversely proportional to the square of the separation. Doubling one charge multiplies the force by $2$. Tripling the separation multiplies $r^2$ by $9$, which divides the force by $9$. Combining the two independent effects, the force changes by a factor of $2/9$. The new force is therefore $0.36 \times 2/9 = 0.080\ \text{N}$, still repulsive because neither sign has changed. Reasoning proportionally like this avoids ever needing the value of $k$, since it cancels in the ratio.
Q3 (3 marks): Both laws are inverse-square: the force falls as $1/r^2$ as the separation grows, so doubling the distance quarters the force in each case. Both are also directly proportional to the product of the two source quantities, the two charges in Coulomb's law and the two masses in Newton's law, and both carry a constant of proportionality ($k$ and $G$). The crucial difference is sign. Mass comes in only one kind, so gravitation is always attractive. Charge comes in two kinds, so the electrostatic force is attractive between unlike charges and repulsive between like charges. The electrostatic force is also enormously stronger for everyday particles.
You can now calculate and scale the force between two point charges with Coulomb's law and recognise its inverse-square structure. Next lesson leaves point charges behind to look at the uniform field between two parallel plates.
↩ Back to focus-area hubAt the start you predicted what doubling the separation does to the force. The answer is the force drops to one-quarter, because $F \propto 1/r^2$ and $2^2 = 4$. If you said one-half, revisit Card 1, the squared distance is what makes the Coulomb force fall off so steeply.
A short spaced-retrieval set, mixing this lesson with earlier ones. It gets harder to guess and easier to remember.
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