Gauss’s Law

Gauss’s law links the electric flux through a closed surface to the charge enclosed. In H3, you use its integral form to compute vector E efficiently for highly symmetric charge distributions.

1. Definitions (Must Know)

• Electric flux, Φ_E (N m² C⁻¹): Φ_E = ∮ vector E · d vector A where d vector A points outward normal to the surface.
• Gauss’s law (electric, integral form): ∮ vector E · d vector A = Q_encl/ε₀
• Gauss’s law (magnetic, integral form): ∮ vector B · d vector A = 0 This zero net magnetic flux is consistent with the non-existence of magnetic monopoles in the syllabus model.
• Gaussian surface: an imaginary closed surface chosen to exploit symmetry.
• Vacuum permittivity: ε₀ ≈ 8.85 × 10⁻¹² F m⁻¹.
• Symbols used in this lesson: vector E (N C⁻¹), vector B (T), Q_encl (C), ε₀ (F m⁻¹), λ (C m⁻¹), σ (C m⁻²), ρ (C m⁻³), r (m), R (m), A (m²).

2. Key Ideas (What Earns Marks)

• Gauss’s law is always true, but it is only useful when symmetry lets you pull E out of the surface integral.
• The Gaussian surface must match the symmetry of the charge distribution (spherical / cylindrical / planar).
• Q_encl means “charge inside the surface”; charges outside do not contribute to the RHS (though they can still affect vector E on the surface).
• Magnetic version: flux through any closed surface is zero, so magnetic field lines do not begin or end in this model.

Symmetry → Gaussian surface (quick table):

3. Detailed Explanations

A. What Gauss’s law is telling you

Electric flux measures how much the electric field “pierces” a surface. Gauss’s law says: the net outward flux through a closed surface depends only on how much charge is enclosed.

B. When you can turn the integral into EA

If, on your chosen Gaussian surface:

• E has constant magnitude, and
• vector E is everywhere parallel to d vector A (so vector E · d vector A = E dA),

then: ∮ vector E · d vector A = E∮ dA = EA and you can solve for E quickly.

C. Canonical symmetries (what surfaces to choose)

• Spherical symmetry (point charge, uniformly charged sphere, conducting sphere): choose a sphere of radius r.
• Cylindrical symmetry (infinite line charge): choose a cylinder of radius r and length L.
• Planar symmetry (infinite plane sheet): choose a “pillbox” straddling the sheet.

H3 questions typically signal “infinite / very long / uniform / symmetric” to tell you Gauss’s law will be efficient.

4. Common Mistakes

• Using a Gaussian surface that does not match the symmetry, then incorrectly treating E as constant.
• Forgetting that vector E · d vector A uses the normal component of vector E.
• Mixing up A (area) with dA (infinitesimal area element).
• Treating the magnetic form as ∮ vector B · d vector A = μ₀ I (that’s Ampère’s law, different integral).

5. Exam Tips

• Start with a 3-line template:
  1. “Choose Gaussian surface: …”
  2. “By symmetry, vector E is … and has constant magnitude on the surface.”
  3. “So ∮ vector E · d vector A = EA (or 2EA, etc.).”
• Always state the enclosed charge explicitly before substituting.
• Keep ε₀ symbolic until the final step unless asked for a number.

6. Worked Examples

7. Mind Stretchers

8. Optional/Enrichment

A. Field “just outside a conductor” (pillbox)

A standard application uses a tiny pillbox Gaussian surface across a conductor surface to relate the field just outside to surface charge density. This is useful, but H3 questions may not require the full derivation.

For a deeper (university) walkthrough:

• UY1: Gauss’s Law For Conductors

B. Differential form (not required)

The syllabus does not require the differential form of Gauss’s law; stay with the integral forms unless explicitly guided.

Next step

Continue to Ampère’s Law, where the same symmetry strategy is applied to circulation around a closed path.