Playground
Two charges attract or repel: drag the distance slider to see the force arrow scale with 1/r².
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Electrostatic forceoutput Magnitude of the force between two point charges | N | M·L·T⁻² | 0 – 100 | |
| Charge 1 First point charge | C | I·T | -0.00001 – 0.00001 | |
| Charge 2 Second point charge | C | I·T | -0.00001 – 0.00001 | |
| Separation distance Distance between the two point charges | m | L | 0.01 – 1 |
Deep dive
Derivation
Experimentally determined using a torsion balance. The constant k_e = 1/(4πε₀) = 8.9875 × 10⁹ N·m²/C². No derivation from first principles in classical physics — it is a fundamental empirical law.
Experimental verification
Coulomb's torsion balance (1785) established the inverse-square dependence. Modern Cavendish-type experiments confirm the exponent is 2 to within ±10⁻¹⁶. Equivalently tested via Gauss's law and absence of charge inside a hollow conductor.
Common misconceptions
- Coulomb's law applies only to point charges or spherical distributions, not arbitrary shapes
- The force is along the line joining the charges — it has no transverse component
- The law assumes static charges; moving charges require the full Lorentz force
Real-world applications
- Electrostatic precipitators in smokestacks
- Xerographic printing and photocopiers
- Van de Graaff generators in physics demonstrations
- Modeling molecular bonds in computational chemistry
Worked examples
Force between two charged spheres
Given:
- q1:
- 0.000003
- q2:
- -0.000005
- r:
- 0.2
Find: F
Solution
F = k_e × |q₁q₂| / r² = 8.9875×10⁹ × |3×10⁻⁶ × 5×10⁻⁶| / 0.2² = 3.37 N (attractive)
Equilibrium position for a third charge
Given:
- q1:
- 0.000004
- q2:
- 0.000016
- d:
- 0.3
Find: x (position where net force on a test charge is zero)
Solution
x = 0.1 m from q₁ (one-third of the way from the smaller charge)
Scenarios
What if…
- scenario:
- What if the distance doubles?
- answer:
- Force drops to 1/4 of the original value. At r = 0.4 m, F = 3.37/4 = 0.843 N — inverse-square dependence.
- scenario:
- What if one charge triples?
- answer:
- Force triples. F = 3 × 3.37 = 10.1 N. Force scales linearly with each charge.
- scenario:
- What if a dielectric (κ = 4) fills the space?
- answer:
- Force reduces by factor κ: F = 3.37/4 = 0.843 N. The medium's polarization partially screens the charges.
Limiting cases
- condition:
- r → 0
- result:
- F → ∞
- explanation:
- Force diverges at zero separation — in reality, quantum effects dominate at atomic scales.
- condition:
- r → ∞
- result:
- F → 0
- explanation:
- Force vanishes at large distances due to inverse-square falloff.
- condition:
- q₁ or q₂ → 0
- result:
- F → 0
- explanation:
- No charge means no electrostatic interaction.
Context
Charles-Augustin de Coulomb · 1785
Coulomb used a torsion balance to precisely measure the force between charged spheres, proving it follows an inverse-square law analogous to gravity.
Hook
Why does a balloon stick to the wall after you rub it on your hair?
Two point charges of +3 μC and −5 μC are separated by 0.2 m. Find the magnitude and direction of the electrostatic force.
Dimensions: [F] = [k_e]·[q]²·[r]⁻² → (N·m²·C⁻²)(C²)(m⁻²) = N ✓
Validity: Valid for point charges or spherically symmetric charge distributions in vacuum. Breaks down at subatomic distances where quantum electrodynamics applies. In media, replace ε₀ with ε₀εᵣ.