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# Gauss's Law ∮ E·dA = Q_enc / ε₀ "Total electric flux through a closed surface equals enclosed charge divided by permittivity." Notation: E = electric field (N/C) | dA = outward area vector | ∮ = closed surface integral | Q_enc = charge inside | ε₀ = 8.85×10⁻¹² C²/N·m². ## 1 What It Is For any closed surface: flux Φ = ∮ E·dA = Q_enc/ε₀. Flux depends ONLY on charge inside — not on surface size, shape, or outside charges. ## 2 Visual Intuition (Field Lines) Field lines exiting → positive flux. Entering → negative. Charge inside → net outward lines. Charge outside → equal in/out → cancel. Diagram: sphere with positive charge inside, radiating lines piercing surface; outside charge lines entering and exiting equally. ## 3 Breakdown ∮ E·dA = total electric flux through entire closed surface | Q_enc = net enclosed charge | ε₀ = vacuum permittivity. Surface can be any shape — sphere, cube, balloon. ## 4 Numerical Example Point charge q=+2μC at center of sphere r=0.5m: Φ = Q_enc/ε₀ = (2×10⁻⁶)/(8.85×10⁻¹²) ≈ 2.26×10⁵ N·m²/C. Radius doesn't matter — same flux for r=0.5m or r=10m! ## 5 Applications Point charge field (Coulomb's law), infinite line (E∝1/r), infinite plane (uniform E), charged sphere (inside E=0 for conductor). ## 6 Why Powerful Choose symmetric surface (sphere, cylinder, pillbox) to solve E instantly — only when symmetry exists. ## Summary Formula: ∮ E·dA = Q_enc/ε₀ | Meaning: flux = enclosed charge | Shape doesn't matter, only what's inside. ## Extensions Maxwell's First Equation | Differential: ∇·E = ρ/ε₀ | Gravitational analog: ∮ g·dA = -4πGM_enc. One sentence: Count field lines leaving a closed surface — only the charges inside determine the net count.

Gauss's Law: Electric Flux Through Closed Surfaces Explained

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# Gauss's Law ∮ E·dA = Q_enc / ε₀ "Total electric flux through a closed surface equals enclosed charge divided by permittivity." Notation: E = electric field (N/C) | dA = outward area vector | ∮ = closed surface integral | Q_enc = charge inside | ε₀ = 8.85×10⁻¹² C²/N·m². ## 1 What It Is For any closed surface: flux Φ = ∮ E·dA = Q_enc/ε₀. Flux depends ONLY on charge inside — not on surface size, shape, or outside charges. ## 2 Visual Intuition (Field Lines) Field lines exiting → positive flux. Entering → negative. Charge inside → net outward lines. Charge outside → equal in/out → cancel. Diagram: sphere with positive charge inside, radiating lines piercing surface; outside charge lines entering and exiting equally. ## 3 Breakdown ∮ E·dA = total electric flux through entire closed surface | Q_enc = net enclosed charge | ε₀ = vacuum permittivity. Surface can be any shape — sphere, cube, balloon. ## 4 Numerical Example Point charge q=+2μC at center of sphere r=0.5m: Φ = Q_enc/ε₀ = (2×10⁻⁶)/(8.85×10⁻¹²) ≈ 2.26×10⁵ N·m²/C. Radius doesn't matter — same flux for r=0.5m or r=10m! ## 5 Applications Point charge field (Coulomb's law), infinite line (E∝1/r), infinite plane (uniform E), charged sphere (inside E=0 for conductor). ## 6 Why Powerful Choose symmetric surface (sphere, cylinder, pillbox) to solve E instantly — only when symmetry exists. ## Summary Formula: ∮ E·dA = Q_enc/ε₀ | Meaning: flux = enclosed charge | Shape doesn't matter, only what's inside. ## Extensions Maxwell's First Equation | Differential: ∇·E = ρ/ε₀ | Gravitational analog: ∮ g·dA = -4πGM_enc. One sentence: Count field lines leaving a closed surface — only the charges inside determine the net count.

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