What is Gauss's Law?
Gauss's law relates the electric flux through any closed surface to the total charge enclosed inside it. It's one of Maxwell's four equations and a powerful shortcut for finding electric fields in symmetric situations.
Gauss's law states that the net electric flux through a closed surface equals the enclosed charge divided by the permittivity of free space: Φ_E = Q_enc/ε₀.
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Step-by-step worked examples
A closed sphere encloses a charge of 2 nC (2×10⁻⁹ C). Find the electric flux through the surface.
Φ_E = Q_enc / ε₀ Φ_E = 2×10⁻⁹ / 8.854×10⁻¹² Φ_E ≈ 225.9 N·m²/C
A closed surface encloses a charge of −5 µC. Find the flux.
Φ_E = Q_enc / ε₀ Φ_E = −5×10⁻⁶ / 8.854×10⁻¹² Φ_E ≈ −5.65×10⁵ N·m²/C
A closed box contains two charges: +4 nC and −4 nC. Find the net electric flux through the box.
Q_enc = (+4 nC) + (−4 nC) = 0 Φ_E = Q_enc / ε₀ = 0 / ε₀ Φ_E = 0 N·m²/C
Flashcards
Quick quiz
Q1.A closed surface encloses 10 nC of charge. Approximately what is the flux?
Q2.What does Gauss's law say the flux depends on?
Q3.A closed surface encloses equal positive and negative charges. Net flux is:
Q4.Gauss's law is most useful for finding electric fields when:
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Common mistakes
Gauss's law only works for point charges. — Correct: It works for any closed surface and any charge distribution, though it's most useful for symmetric ones.
Charges outside the surface add to the enclosed flux. — Correct: Only charge strictly inside the closed surface contributes to Q_enc; outside charges give net zero flux.
The flux depends on the surface's shape and area. — Correct: Flux depends only on Q_enc — a bigger or oddly-shaped surface enclosing the same charge gives the same flux.
Φ_E = Q_enc × ε₀. — Correct: It's division, not multiplication: Φ_E = Q_enc / ε₀.
FAQ
What is the formula for Gauss's law?
Φ_E = Q_enc/ε₀, where Q_enc is the total charge enclosed by a closed surface and ε₀ ≈ 8.854×10⁻¹² C²/(N·m²).
What is Gauss's law used for?
It relates electric flux to enclosed charge, letting you find electric fields quickly for symmetric charge distributions.
How do you calculate electric flux with Gauss's law?
Divide the total enclosed charge by ε₀: Φ_E = Q_enc/ε₀. Charges outside the surface don't count.
Is Gauss's law one of Maxwell's equations?
Yes — Gauss's law for electric fields is the first of Maxwell's four equations of electromagnetism.