Short on time? This page is the recap. For the derivations, Figure 1 and the worked examples, use the full study guide. This topic is in the calculus-based course only; it is not part of Physics 2.
Recap
- Gauss’s law links the net flux through a closed surface to the net charge inside it. Area vectors point outward.
- The flux does not change if you resize the surface, as long as q_enc stays the same.
- The law is always true, but it gives E directly only when symmetry makes E constant and perpendicular on parts of the surface, and parallel (zero flux) on the rest.
- The course applies it quantitatively to point charges and to spherical, cylindrical and planar symmetry.
- ε₀ = 8.85 × 10⁻¹² C²/(N·m²); 1/(4πε₀) = 8.99 × 10⁹ N·m²/C².
Key relationships
| Charge distribution | Gaussian surface | Field |
|---|---|---|
| Point charge or outside any spherical distribution | sphere, radius r | E = Q/(4πε₀r²) |
| Inside a uniform solid sphere (r < R) | sphere, radius r | E = Qr/(4πε₀R³) = ρr/(3ε₀) |
| Infinite line, charge per length λ | coaxial cylinder | E = λ/(2πε₀r) |
| Inside a uniform long cylinder (r < R) | coaxial cylinder | E = ρr/(2ε₀) |
| Infinite thin sheet, σ | pillbox | E = σ/(2ε₀), independent of distance |
| Non-uniform sphere, ρ(r) | sphere, radius r | q_enc = ∫₀ʳ ρ(r′)4πr′² dr′, then E = q_enc/(4πε₀r²) |
Assumptions behind the results
- The charge distribution has exact symmetry: “infinite” lines and sheets, and perfect spheres.
- Insulators hold their charge in place; conductors are treated in Unit 10.
- The system is in electrostatic equilibrium (charges at rest).
Mistakes to avoid
- Using total Q inside a body. Use q_enc(r).
- ρ × volume when ρ varies. Integrate ρ dV.
- Zero flux ⇒ zero field. Wrong: zero flux means zero enclosed charge.
- End caps on a line-charge cylinder. They carry no flux.
- σ/ε₀ for a thin sheet. Two faces of the pillbox carry flux, so E = σ/(2ε₀).
- Skipping the checks. E should match at r = R and approach Q/(4πε₀r²) for large r.
Quick self-check
- What is the net flux through a closed surface around +8.85 × 10⁻¹² C? (1.00 N·m²/C)
- Inside a uniform sphere, how does E change if r is doubled (still inside)? (It doubles, since E ∝ r)
- A sphere has ρ = ρ₀r/R. What is q_enc(r)? (πρ₀r⁴/R)
Next: practice questions.