Coulomb's law in vector form is expressed as F₁₂ = kq₁q₂r̂₁₂/r², where F₁₂ is the force on charge q₁ due to charge q₂, r̂₁₂ is the unit vector from q₂ to q₁, r is the distance between the charges, and k = 9 × 10⁹ N·m²/C² is Coulomb's constant. The vector form gives both the magnitude and direction of the electrostatic force.
Coulomb's law vector form: F₁₂ = kq₁q₂r̂₁₂/r².
k = 9 × 10⁹ N·m²/C² = 1/(4πε₀).
The vector form automatically accounts for attraction (unlike charges) and repulsion (like charges).
F₁₂ = −F₂₁: forces are equal and opposite (Newton's Third Law).
Superposition principle: total force = vector sum of all individual forces.
ε₀ = 8.854 × 10⁻¹² C²/N·m² (permittivity of free space).
In a medium with permittivity εᵣ, the force is reduced by factor εᵣ.
Scalar form (magnitude only): F = k|q₁||q₂|/r²
Vector form: F₁₂ = k(q₁q₂/r²) r̂₁₂
Where:
Alternative notation: F₁₂ = (1/4πε₀) × (q₁q₂/r³) × r⃗₁₂
where r⃗₁₂ = r⃗₁ − r⃗₂ (position vector from 2 to 1)
The vector form automatically gives the correct direction:
Like charges (same sign, q₁q₂ > 0): F₁₂ = positive × r̂₁₂ → Force is along r̂₁₂ (away from charge 2) → Repulsion ✓
Unlike charges (opposite sign, q₁q₂ < 0): F₁₂ = negative × r̂₁₂ → Force is opposite to r̂₁₂ (towards charge 2) → Attraction ✓
Newton's Third Law in vector form: F₁₂ = −F₂₁
The forces are equal in magnitude, opposite in direction.
Superposition Principle: For multiple charges, the total force on charge q₁ is: F₁ = F₁₂ + F₁₃ + F₁₄ + ... (vector sum)
Coulomb's constant: k = 1/(4πε₀) = 9 × 10⁹ N·m²/C²
Permittivity of free space: ε₀ = 8.854 × 10⁻¹² C²/(N·m²) = 8.854 × 10⁻¹² F/m
In a medium of relative permittivity εᵣ: F = kq₁q₂/(εᵣ r²)
For water (εᵣ ≈ 80): Force reduced to 1/80th of the value in vacuum.
Limitations of Coulomb's Law:
Coulomb's law in vector form is: F₁₂ = (1/4πε₀) × (q₁q₂/r²) × r̂₁₂, where F₁₂ is the force on charge q₁ due to q₂, r̂₁₂ is the unit vector from q₂ towards q₁, and r is the distance between the charges.
For like charges (q₁q₂ > 0), the force F₁₂ is positive along r̂₁₂ (pointing away from q₂) → repulsion. For unlike charges (q₁q₂ < 0), the force is negative along r̂₁₂ (pointing towards q₂) → attraction. The sign of q₁q₂ automatically gives the correct direction.
Coulomb's constant k = 1/(4πε₀) = 9 × 10⁹ N·m²/C², where ε₀ = 8.854 × 10⁻¹² C²/(N·m²) is the permittivity of free space.
The superposition principle states that the total electrostatic force on a charge due to multiple charges is the vector sum of the individual forces due to each charge acting independently. F_total = F₁₂ + F₁₃ + F₁₄ + ...
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