§ 25. Coulomb's Law & 3D Electric Dipole Field (Grade 8–10)
3D electric field lines of two point charges, Coulomb force & test charge motion. Electrostatic field lines originate on positive charges and terminate on negative charges, while E at every point is the vector sum of individual fields.
Theory Summary & Scientific Mechanism
Established by Charles-Augustin de Coulomb in 1785 using a torsion balance, Coulomb’s Law states that the electrostatic force between two stationary point charges q₁ and q₂ in vacuum is proportional to the product of their charges and inversely proportional to the square of their separation r: F = k(|q₁ q₂|)/(r²), where k = (1)/(4πε₀) ≈ 8.988× 10⁹ N·m²/C².
Electrostatic fields obey the superposition principle: at any point in space, the net electric field vector is E = ∑ᵢ (1)/(4πε₀)(qᵢ)/(rᵢ³)rᵢ and the scalar potential is the algebraic sum φ = ∑ᵢ (1)/(4πε₀)(qᵢ)/(rᵢ). The electrostatic potential energy of the pair is Wₚ = (k q₁ q₂)/(r) (positive for like charges, negative for opposite charges).
Two opposite charges +q and -q separated by distance d form an electric dipole with dipole moment p = qd. Field lines originate on positive charges, terminate on negative charges, never cross, and are everywhere orthogonal to equipotential surfaces (E = -∇φ).
Everyday Analogy
Rub a balloon on a wool sweater and it attracts bits of paper! Like charges repel and opposite charges attract, while electric field lines map the invisible highways along which a positive test charge gets pushed.
Interactive Question & Derivation
Two opposite point charges +q₁ and -|q₂| are separated by distance d. How does the attractive Coulomb force F_C between them change if the distance d is halved, and which way does the electric field E point midway between the charges?
- A: The force doubles, and the electric field at the midpoint is zero
- B: By Coulomb's inverse-square law F ∝ 1/d² the force increases 4-fold, and at the midpoint E₁ and E₂ add together pointing from +q₁ toward -|q₂|
- C: The force decreases 4-fold, and the field lines form closed loops
Study Hint: By Coulomb’s inverse-square law F ∝ |q₁ q₂| / r²: how does the electrostatic force change when the distance r between charges is halved?
Proof & Explanation: By Coulomb's Law, electrostatic force is inversely proportional to the square of distance: F = k(|q₁ q₂|)/(d²), so halving d quadruples the force. At the midpoint between +q₁ and -|q₂|, both E₁ (away from +q₁) and E₂ (toward -|q₂|) point in the same direction (from + to -), reinforcing each other by superposition!
Key Equations & 3D Simulation Parameters
F_C = k (|q₁ q₂|)/(d²), Eₘid = (4k(|q₁| + |q₂|))/(d²)
- Left point charge q₁ (+): 1 … 4.5 μC (default: 2.5 μC)
- Right point charge q₂ (−): -4.5 … -1 μC (default: -2.5 μC)
- Charge separation d: 2.8 … 6.4 m (default: 4.4 m)
- Test charge q₀: 0.5 … 2.5 μC (default: 1 μC)