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UY1: Usage Of Gauss's Law

Apply Gauss's law to negative, zero, and multiple enclosed charges with clear sign interpretation.

  • University Physics Year 1
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Learning objectives

  • Construct electric-field and potential models for discrete and continuous charge distributions.
Why this matters + quick links

This page gives the UY1 working model/result for Usage Of Gauss’s Law. You reuse it when you build fields/potentials by symmetry or superposition, and when you connect fields to forces, energy, and circuits.

This lesson focuses on quick interpretation of Gauss’s law before using symmetry-heavy calculations.

1) At a glance

  • Prerequisites: flux sign convention (Gauss’s Law (Simple Version))
  • Outcomes: predict flux sign/magnitude from enclosed charge only, and avoid the “external charges” trap
  • Key result: ∮ vector E · d vector A = Q_encl/ε₀
  • Common trap: mixing up enclosed charge with nearby external charge

Motivation / intuition

Before you use symmetry to solve for vector E, Gauss’s law already gives you a fast “charge audit”: the net flux is set by Q_encl alone, no matter how complicated the external charge configuration is.

2) Setup

Choose any closed Gaussian surface. Outward normal defines positive area direction.

You will often use three immediate cases:

  • Q_encl > 0 ⇒ Φ_E > 0
  • Q_encl < 0 ⇒ Φ_E < 0
  • Q_encl = 0 ⇒ Φ_E = 0

3) Core derivation/explanation

Case A: Negative enclosed charge

If a surface encloses net negative charge, more field lines enter than leave: Φ_E = Q_encl/ε₀ < 0

Case B: No enclosed charge

If Q_encl = 0, then Φ_E = 0 This means algebraic cancellation of entering and leaving contributions, not necessarily vector E = 0 at each point.

Case C: Several enclosed charges

If enclosed charges are q₁,q₂,…, then Q_encl = ∑ᵢ qᵢ and Φ_E = 1/ε₀∑ᵢ qᵢ

This follows from superposition because vector E = ∑ᵢ vector Eᵢ so the closed-surface integral is the sum of each contribution.

Quick checks (units + limits/sign)
  • Units: Φ_E is N m²/C, so Q_encl/ε₀ must evaluate to the same units.
  • Limits/signs: Q_encl = 0 ⇒ Φ_E = 0 but vector E can still be nonzero on the surface; flipping the sign of all enclosed charges flips the sign of flux.

4) Worked example(s)

A closed surface encloses three charges: + 4 nC, -7 nC, and + 1 nC.

Net enclosed charge: Q_encl = (4-7 + 1) nC = -2 nC Flux: Φ_E = (-2 × 10⁻⁹)/(8.85 × 10⁻¹²) ≈ -2.26 × 10² N m²/C

Sign tells you inward flux dominates.

5) Practice set (with hints + answers)

  1. Surface encloses only + 6 nC. Sign of flux?
  2. Surface encloses + 3 nC and -3 nC. Net flux?
  3. Charges are outside the surface only. What is net flux?

Hints

  1. Compare sign of Q_encl.
  2. Add charges first.
  3. Gauss’s law uses enclosed charge, not external charge.

Answers

  1. Positive.
  2. Zero.
  3. Zero.

6) Summary + next steps

  • Net flux is controlled by net enclosed charge only.
  • Sign of flux tracks sign of enclosed charge.
  • Superposition makes multi-charge problems straightforward at the flux level.

Next: Gauss’s Law For Conductors Previous: Coulomb’s Law To Gauss’s Law Back To Electromagnetism (UY1)