AP Physics C: Electricity and Magnetism · Unit 8 · Topic 1 of 4 · 6 min lesson

Coulomb’s Law & Superposition

A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.

Quiz this topic · 5 questions

The big idea

Each charge exerts a force F = kq₁q₂/r² along the line joining them, and each creates a field E = kq/r². With several charges, superposition applies: add the individual forces or fields as vectors, component by component.

1. Adding fields as vectors

Symmetry does half the work: the y-components cancel, so only the x-components need adding.

F = kq₁q₂/r² (along r̂)Coulomb’s law

E = kq/r²point charge; away from +, toward −

E_net = ΣE_iadd as vectors

k = 1/(4πε₀) = 9.0 × 10⁹ N·m²/C²

2. A routine for superposition

1. Draw each field at the point, with its direction from the sign of its charge.

2. Find each magnitude kq/r², then break it into x and y components.

3. Look for symmetry: equal-and-opposite components cancel. Add the rest.

Where is E = 0? Between two like charges, closer to the smaller one. For unlike charges, outside the pair, on the side of the smaller-magnitude charge.

3. Example, then try it

Example

A +2.0 µC charge is at the origin and a −3.0 µC charge is at (3.0 m, 4.0 m). Find the force on the +2.0 µC charge.

r = 5.0 m, so F = (9.0 × 10⁹)(2.0 × 10⁻⁶)(3.0 × 10⁻⁶) ÷ 25 ≈ 2.2 × 10⁻³ N, toward the other charge (attraction).

Direction (0.60, 0.80), so F ≈ (1.3 î + 1.7 ĵ) × 10⁻³ N.

Try it · work it on paper first

In the figure below, q = 2.0 nC and a = 0.30 m. Find the electric field at P and the force on an electron placed there.

Distance to each charge: r = a√2 ≈ 0.42 m, so each field is kq/r² = (9.0 × 10⁹)(2.0 × 10⁻⁹) ÷ 0.18 = 100 N/C.

Components: each makes 45° with the axes: x-parts 100 cos 45° ≈ 70.7 N/C, both +x. y-parts: +70.7 and −70.7, which cancel.

Net field: ≈ 141 N/C in the +x direction (= kq/(√2 a²)).

Electron: F = eE = (1.6 × 10⁻¹⁹)(141) ≈ 2.3 × 10⁻¹⁷ N in the −x direction.

Answer: 141 N/C toward +x; 2.3 × 10⁻¹⁷ N toward −x on an electron.

4. Common mistakes

Adding magnitudes.Fields are vectors: add components.

Wrong direction for a negative charge’s field.It points toward a negative charge.

Using r instead of r² (or the wrong r).r is the distance from each charge to the point.

Check yourself

Two equal positive charges sit a distance d apart. What is the electric field at the midpoint between them?

At the midpoint the two fields have equal size and point in opposite directions (each away from its own charge), so they cancel.

Ready to practice?

5 questions on Coulomb’s Law & Superposition, each with a hint and an explanation for every wrong answer.

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