IB Physics SLTopic 4 — Electric & Magnetic FieldsPaper 1 & 2F = kq₁q₂/r²~8 min read
Coulomb’s Law
Last page gave the rule: opposites attract, likes repel. Coulomb’s law now tells you exactly how hard. It’s the electric twin of Newton’s law of gravitation — same inverse-square shape — but with one dramatic twist: this force can push as well as pull.
📘 What you need to know
Coulomb’s law: the force between two point charges is F = kq1q2 ÷ r², with k = 8.99 × 10⁹ N m² C⁻² and r the centre-to-centre distance
The force is proportional to the product of the charges and to 1 ÷ r² — an inverse-square law, just like gravity
Like charges give a repulsive force; opposite charges give an attractive one. The two charges always feel equal and opposite forces (Newton’s third law)
k = 1 ÷ (4πε0), where ε0 = 8.85 × 10⁻¹² C² N⁻¹ m⁻² is the permittivity of free space (air ≈ ε0)
A medium’s relative permittivity (dielectric constant) εr = ε ÷ ε0 has no units; a bigger εweakens the force, so the force in a medium is 1 ÷ εr of the vacuum value
Coulomb and Newton are twins — both ∝ 1 ÷ r² — but gravity only attracts, while the electric force can attract or repel
Only valid for point charges (or uniform spheres treated as a point at the centre) whose size is much smaller than their separation
The Law Itself
Every charge sits in its own electric field, and any other charge nearby feels a force from it. Coulomb measured exactly how that force depends on the charges and their separation:
The electric force between two point charges is proportional to the product of the charges and inversely proportional to the square of their separation.
Coulomb’s lawF = kq1q2 ÷ r²
Where:
F = electric force between the charges (N)
q1, q2 = the two charges (C)
r = distance between their centres (m)
k = the Coulomb constant, 8.99 × 10⁹ N m² C⁻² (in the data booklet)
Attractive or Repulsive?
The size of the force comes from the formula; the direction comes from the signs. Two charges of the same type push apart; two opposite charges pull together. Whatever the charges, the pair always feel equal and opposite forces — Newton’s third law never takes a day off.
The magnitude comes from F = kq1q2 ÷ r²; the direction comes from the signs. Opposite charges attract, like charges repel — and each charge feels the same size force. Remember to measure r centre to centre.
It’s an Inverse-Square Law
That r² on the bottom does the heavy lifting. Because the force depends on 1 ÷ r², moving the charges apart makes it fall away fast: double the separation and the force drops to a quarter; treble it and the force is just a ninth.
The inverse-square shape: the force plunges as the charges separate. Go from r to 2r and the force falls to a quarter — exactly the same curve as gravity, just far stronger.
Newton’s gravity F = GMm ÷ r²
both ∝ 1 ÷ r² (same shape)
Coulomb’s law F = kq1q2 ÷ r²
The two laws are near-identical twins. The big difference: mass only ever attracts, but charge comes in two signs, so the electric force can pull or push — and it’s enormously stronger than gravity for everyday amounts of charge.
Where k Comes From: Permittivity
The constant k isn’t random — it’s built from a property of space itself, the permittivity of free spaceε0:
The Coulomb constantk = 1 ÷ (4πε0)
Here ε0 = 8.85 × 10⁻¹² C² N⁻¹ m⁻². Permittivity measures how easily an electric field forms in a material. Air is so close to a vacuum that we take its permittivity as ε0. Every other material has a higher permittivity, and we compare them using the relative permittivity (or dielectric constant):
It’s a ratio of two permittivities, so it has no units. Put a material between the charges and the Coulomb constant becomes k = 1 ÷ (4πε): a bigger permittivity means a smaller force. In practice the force in a medium is just the vacuum force divided by εr — water (εr ≈ 80) weakens it dramatically.
The Point-Charge Rule
One catch worth memorising: Coulomb’s law only works for point charges, or for uniformly charged spheres that you can treat as a point charge sitting at the centre — and only when the objects are much smaller than their separation. Always measure r from the centres. You can’t use it for oddly shaped charged objects, where the charge is smeared out unevenly.
🧭 Using Coulomb’s law
List the charges in coulombs and the separation r in metres, measured centre to centre
Put the magnitudes into F = kq1q2 ÷ r² to get the size of the force
Read the direction off the signs — like charges repel, opposite charges attract
In a medium, swap ε0 for ε = εrε0 — or simply divide the vacuum force by εr
Sanity-check with the inverse square — ×2 distance means ×¼ force
Quick recap:F = kq1q2 ÷ r² with k = 8.99 × 10⁹ = 1 ÷ (4πε0). Like charges repel, opposite attract; it’s inverse-square like gravity but two-signed; and it only applies to point charges with r taken centre to centre.
WE 1
Two small charged spheres carry charges of +4.0 μC and +6.0 μC and are held 0.30 m apart in air. (a) Calculate the electric force between them. (b) State whether it is attractive or repulsive. (c) The separation is now tripled. State the new force.
Part (a) — Coulomb’s law (convert: μC → C, cm → m)F = kq₁q₂ ÷ r² = (8.99 × 10⁹ × 4.0 × 10⁻⁶ × 6.0 × 10⁻⁶) ÷ (0.30)²F = (8.99 × 10⁹ × 2.4 × 10⁻¹¹) ÷ 0.090F ≈ 2.4 NPart (b) — direction from the signs
Both charges are positive (like charges), so the force is
repulsivePart (c) — inverse square: ×3 distance → ×1/9 forceF’ = 2.4 ÷ 9F’ ≈ 0.27 N
WE 2
(a) A dielectric material has a relative permittivity of 5.0. Calculate its permittivity ε and state the unit. (b) Two point charges are held a fixed distance apart. By what factor does the electric force between them change when they are moved from a vacuum into this material?
Part (a) — rearrange εr = ε ÷ ε₀ε = εr × ε₀ = 5.0 × (8.85 × 10⁻¹²)ε = 4.4 × 10⁻¹¹ F m⁻¹Part (b) — a medium changes k = 1 ÷ (4πε)
The permittivity is εr = 5.0 times bigger, so k (and the force) is 5.0 times smallerforce becomes 1/5 = 0.20 × the vacuum valueBigger permittivity → weaker force. That’s why εr is called the material’s “resistance” to forming a field.
💡 Top tips
k and ε0 are in the data booklet — you don’t memorise the numbers, but do know k = 1 ÷ (4πε0)
Convert first: μC and nC → C, mm and cm → m. Most Coulomb’s-law slips are unit errors
Size then direction: use the magnitudes to get the number, then decide attract vs repel from the signs — never quote a “negative force”
Measure r centre to centre and remember to square it — the single most common lost mark
⚠ Common mistakes
Forgetting to square r, or measuring it from the surfaces instead of the centres
Not converting μC/nC → C or mm/cm → m before substituting
Quoting a negative force from the charge signs — the sign only tells you attract vs repel; state the direction in words
Using Coulomb’s law on irregular charged objects — it’s only for point charges / uniform spheres with size ≪ separation
You can now put a number on the force between charges. Up next: Electric Field Strength — instead of asking about the force between two charges, we ask what one charge does to the space around it, and define the field E = F ÷ q that any other charge then feels.
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