Electrostatics

⚡ Coulomb's Law

Two Point Charges — Force Diagram

Coulomb's Law Formula

F₁₂ = (Q₁·Q₂)/(4πε₀·R₁₂²) · â_R12 [N]

ε₀ = 8.854 × 10⁻¹² F/m (permittivity of free space)

F₂₁ = −F₁₂ (Newton's 3rd law — equal, opposite direction)

Vector Form

F₁₂ = Q₁Q₂(r₂−r₁) / [4πε₀|r₂−r₁|³]

Where r₁, r₂ are position vectors. The sign of charges is included — no need to manually flip for attraction.

💡 Always keep charge signs in the formula. If F comes out negative direction, it means attraction.

Key Constants

Quantity	Value
ε₀	8.854×10⁻¹² F/m
e (proton)	+1.6×10⁻¹⁹ C
e (electron)	−1.6×10⁻¹⁹ C
1/(4πε₀)	≈ 9×10⁹ N·m²/C²

📝 Tutorial Example — Superposition

q₁=+25nC at P₁(4,−2,7), q₂=+60nC at P₂(−3,4,−2). Find force on Q₂ due to Q₁.

R₁₂ = P₂ − P₁ = (−7, 6, −9) → |R₁₂| = √166
F = (25×10⁻⁹ × 60×10⁻⁹) / (4π×8.854×10⁻¹² × 166^(3/2)) × R₁₂
F = 6.3×10⁻⁹ × (−7â_x + 6â_y − 9â_z) N

➡️ Electric Field Intensity E

Electric Field Lines — Positive and Negative Charges

Definition of E

Electric field = Force per unit positive test charge placed at that point.

E = F/q [V/m or N/C]
E = Q/(4πε₀R²) · â_R

R = distance from source charge Q to observation point. â_R points away from Q.

Superposition of E

Total E at a point = vector sum from all source charges.

E_total = Σ Qₖ(r−rₖ) / [4πε₀|r−rₖ|³]

💡 Compute E from each charge separately using vector subtraction, then add all the vectors component by component.

Electric Field Line Rules

Start on + charges, end on − charges
Never cross each other
Closer lines = stronger field
Perpendicular to conductor surface
Number of lines ∝ magnitude of charge

📝 Example — E from two charges

Q₁=25nC at P₁(4,−2,7), Q₂=60nC at P₂(−3,4,−2). Find E at P(1,2,3).

R₁ = P−P₁ = (−3, 4, −4), |R₁| = √41
R₂ = P−P₂ = (4, −2, 5), |R₂| = √45
E_t = [25×10⁻⁹×R₁]/[4πε₀|R₁|³] + [60×10⁻⁹×R₂]/[4πε₀|R₂|³]
E_t = 4.58â_x − 0.149â_y + 5.5â_z V/m

🧪 Material Classification

Conductors

Free electrons can move through the material. Charges redistribute on the surface.

σ (Gold) ≈ 4.1×10⁷ S/m

Insulators (Dielectrics)

All electrons are bound — no free movement. Under E-field, molecules polarize (slight shift), creating an induced opposing field.

σ (Glass) ≈ 10⁻¹² S/m

Semiconductors

Conductivity between conductors and insulators. Used in electronics.

σ (Silicon) ≈ 4.4×10⁻⁴ S/m

💡 Higher σ = easier for current to flow = better conductor.

Dielectric Polarization — Without and With E-field

Permittivity in a Medium

ε = ε₀ · εᵣ [F/m]
εᵣ = relative permittivity (dimensionless)

Material	εᵣ
Vacuum	1
Air	1.0006
Glass	4.5–10
Sea water	72–80

E in a Dielectric Medium

E = Q / (4πε₀εᵣ r²) · â_r

Higher εᵣ → weaker E field for the same charge. The dielectric reduces the effective field.

💡 εᵣ = 1 + χₑ where χₑ is electric susceptibility.