Coulomb's Law and Electric Fields
Coulomb's Law describes the electrostatic force between two point charges. The electric field is the force per unit positive test charge at any point in space.
Coulomb's Law
F = k × |q₁ × q₂| / r²
k = 8.99×10⁹ N·m²/C² (Coulomb's constant)
q = charge (Coulombs)
r = separation (m)
Proton-proton at 1nm:
F = 8.99×10⁹ × (1.6×10⁻¹⁹)² / (10⁻⁹)²= 2.3×10⁻¹⁰ NElectric Field Strength
E = F/q = k × Q / r² (V/m or N/C)
V = k × Q / r (Electric potential, Volts)
Single point charge Q=1μC at r=1m:
E = 8.99×10⁹ × 10⁻⁶ / 1 = 8990 V/mKey Facts
- Like charges repel; unlike charges attract
- Field lines point from + to −
- Uniform field between parallel plates: E = V/d
- Electron charge: e = 1.602×10⁻¹⁹ C
- Earth's surface field: ~100 V/m downward (fair weather)
Calculate electric field: Free Electric Field Calculator
Electric Field Quick-Reference Table
| Source | Distance | E-field (V/m) |
|---|---|---|
| 1 μC point charge | 1 m | 8,990 |
| 1 μC point charge | 10 cm | 899,000 |
| Parallel plate capacitor (1V, 1mm gap) | — | 1,000 |
| Air breakdown (lightning) | — | ~3×10⁶ |
| Near high-voltage power line | 1 m | ~1,000–10,000 |
How Electric Fields Work
An electric field E at a point is defined as the force per unit positive test charge placed there: E = F/q. For a point charge Q at distance r, E = kQ/r² (Coulomb's constant k = 8.99×10⁹ N·m²/C²). Field lines point away from positive charges and toward negative charges. The field's direction tells a positive test charge which way it would be pushed; its magnitude tells how hard.
Electric fields are essential in capacitor design (E = V/d for parallel plates), semiconductor physics, cathode-ray tubes, mass spectrometry, ink-jet printing, and particle accelerators. The electric field inside a conductor in electrostatic equilibrium is always zero — charges redistribute on the surface until E = 0 inside (Faraday cage principle).
Common Mistakes
- Forgetting vector addition: When multiple charges create fields at a point, you must add the field vectors — not just the magnitudes. Opposite-direction fields partially cancel.
- Confusing E field and potential: V (electric potential) is a scalar; E is a vector. E = −dV/dr (gradient of potential). High potential ≠ high field if potential changes slowly.
- Using wrong distance unit: Coulomb's law uses metres. Plugging in centimetres without converting gives answers off by factors of 10² or 10⁴.
Frequently Asked Questions
In a conductor, free electrons rearrange themselves until the net force on each is zero — which means the internal E field must be zero in electrostatic equilibrium. Any external field causes charge separation until the induced field exactly cancels the external one inside. This is the principle behind Faraday cages used to shield sensitive electronics.
Electric field strength E is force per unit charge at a point (V/m or N/C). Electric flux Φ = E·A·cosθ measures the total field "flow" through a surface area A. Gauss's Law states that total flux through a closed surface equals enclosed charge ÷ ε₀ — a powerful tool for calculating E fields with high symmetry (spheres, cylinders, planes).
Energy stored in a capacitor U = ½CV² = ½ε₀E²·(volume). The energy is distributed throughout the space between the plates in the electric field itself — not in the charge on the plates. This energy density (½ε₀E²) is a fundamental concept in electromagnetic energy storage, radio waves, and light.