Chapter 1 · Section 1.7Textbook pages 15–18

Forces Between Multiple Charges – the Superposition Principle

Coulomb's law tells you the force between two charges. But a real charge is usually surrounded by many others. On this page you find the force on one charge due to several charges, drag charges around a virtual lab and watch the forces add as vectors, and solve Examples 1.6 and 1.7 in board-exam layout.

📖 What you need (pages 1 and 7 of this guide)

Coulomb's law (കൂളോം നിയമം): F₂₁ = 14πε₀ q₁q₂r₂₁² r̂₂₁, with 14πε₀ ≈ 9 × 10⁹ N m² C⁻².
Force is a vector (സദിശം): two forces add by the parallelogram law (or triangle law). Resultant of two equal forces F at an angle α: R = 2F cos(α/2).
Unit vector (യൂണിറ്റ് സദിശം) r̂ = r/r – length 1, it only gives a direction.

1. The principle of superposition (Fig 1.8)

Consider a system of n stationary point charges (ബിന്ദു ചാർജ്) q₁, q₂, q₃, …, q_n in vacuum. What is the force on q₁? Coulomb's law alone cannot answer this – it only talks about pairs. We already know that mechanical forces add by the parallelogram law. Experiments show that electric forces do the same.

⭐ Superposition principle (അധ്യാരോപണ തത്വം)

The force on any charge due to a number of other charges is the vector sum of all the forces on that charge due to the other charges, taken one at a time. The individual forces are unaffected by the presence of the other charges.

🖼 Fig 1.8 – (a) three charges, (b) many charges

For three charges (Fig 1.8 a), the force on q₁ due to q₂ is still given by Coulomb's law, even though q₃ is present:

F₁₂ = 14πε₀ q₁q₂r₁₂² r̂₁₂ F₁₃ = 14πε₀ q₁q₃r₁₃² r̂₁₃

Here r̂₁₂ is the unit vector pointing from q₂ to q₁ (so a positive product q₁q₂ gives a push away from q₂). The total force on q₁ is

F₁ = F₁₂ + F₁₃ = 14πε₀ q₁q₂r₁₂² r̂₁₂ + 14πε₀ q₁q₃r₁₃² r̂₁₃ (1.4)

For n charges (Fig 1.8 b) we simply keep adding:

F₁ = F₁₂ + F₁₃ + … + F_1n = q₁4πε₀ ∑_{i = 2}ⁿ q_ir_1i² r̂_1i (1.5)

💭 Why this matters

All of electrostatics is basically a consequence of Coulomb's law and the superposition principle. Everything later in this chapter – electric field, field lines, dipoles, Gauss's law – is built from these two ideas.

⚠️ Common mistake

Do not add the magnitudes (3 N + 4 N = 7 N). Forces are vectors: two forces of 3 N and 4 N at right angles give √(3² + 4²) = 5 N, and two equal and opposite forces give zero. Always draw the arrows first.

2. Virtual lab: drag the charges, add the forces

🧪 Vector-addition lab – force on q₁ from up to three other charges

q₁ μC q₂ μC q₃ μC q₄ μC

Drag any charge (also q₁). A charge set to 0 μC is switched off (dashed ring). Grid squares are 10 cm × 10 cm. Each coloured arrow is worked out with Coulomb's law as if the other charges were not there; the red arrow is their vector sum. Watch the sum of magnitudes in the readout – it is almost never equal to the size of the resultant.

🌍 Real life – a salt crystal

In a crystal of common salt, each Na⁺ ion is surrounded by Cl⁻ ions on all sides (and more Na⁺ ions a little farther away). The force on one ion is the vector sum of thousands of Coulomb forces. For an ion deep inside, these forces cancel by symmetry – exactly the idea of Example 1.6 below – which is why the ion sits still. At the surface the cancellation is incomplete, which is one reason surfaces behave differently.

3. Example 1.6 – charge at the centroid (Fig 1.9)

Example 1.6. Consider three charges q₁, q₂, q₃, each equal to q, at the vertices of an equilateral triangle of side l. What is the force on a charge Q (with the same sign as q) placed at the centroid of the triangle (Fig 1.9)?

Find the distance from each vertex to the centroid (AO = l/√3). All three forces then have the same size. Add two of them by the parallelogram law (they are 120° apart) and compare with the third.

🎬 Solution in exam layout – Example 1.6

Answer: each force has magnitude 34πε₀Qql², but the three forces are 120° apart, so the total force on Q is zero. Symmetry check: if the whole arrangement is turned through 120° about O it looks exactly the same, so the force must look the same too; a non-zero arrow would have turned, so the only possibility is zero. (The textbook mentions turning through 60°; a turn of 120° is the one that brings the triangle back onto itself.)

4. Example 1.7 – q, q and −q on a triangle (Fig 1.10)

Example 1.7. Consider the charges q, q and −q placed at the vertices of an equilateral triangle, as shown in Fig 1.10. What is the force on each charge?

Every pair has charges of the same size and the same separation l, so every pair force has the same magnitude F = q²/(4πε₀l²). Only the directions differ: like charges repel, unlike attract. At A and B the two forces are 120° apart; at C they are 60° apart.

🎬 Solution in exam layout – Example 1.7

Answer: with F = q²4πε₀l²: on q at A, F₁ = F along BC; on q at B, F₂ = F along AC; on −q at C, F₃ = √3 F along the bisector of ∠BCA (pointing into the triangle). And F₁ + F₂ + F₃ = 0 – as expected, because Coulomb forces obey Newton's third law, so the internal forces of an isolated system add up to zero.

Think (the book leaves this proof to you): why must F₁ + F₂ + F₃ = 0 for any three charges, not just this triangle?

F₁ + F₂ + F₃ = (F₁₂ + F₁₃) + (F₂₁ + F₂₃) + (F₃₁ + F₃₂). Regroup in pairs: (F₁₂ + F₂₁) + (F₁₃ + F₃₁) + (F₂₃ + F₃₂). By Coulomb's law F₁₂ = −F₂₁ (Newton's third law), so every bracket is zero. This works for any number of charges in any arrangement.

🌐 Explore online needs internet · opens in a new tab

🧪 Coulomb's Law (PhET simulation) PhET, University of Colorado Boulder
Change the charges and the distance and read the force.
📄 Coulomb's Law The Physics Classroom
Worked examples of the inverse-square law.
📄 Coulomb's law Wikipedia
The law, its vector form and the superposition principle.
📄 Charles-Augustin de Coulomb Wikipedia
The French physicist and his torsion balance.

5. Check yourself

🧭 Summary

Superposition principle: the force on a charge due to many charges is the vector sum of the forces due to each charge taken one at a time; each pair force is unaffected by the others.
F₁ = F₁₂ + F₁₃ + … + F_1n = (q₁/4πε₀) ∑ (q_i/r_1i²) r̂_1i – add by the parallelogram law (or components).
Example 1.6: three equal charges at the corners of an equilateral triangle → zero force at the centroid (each force 3Qq/4πε₀l², 120° apart).
Example 1.7: q, q, −q → forces F, F and √3F, which add up to zero (Newton's third law).
All of electrostatics = Coulomb's law + superposition.

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