PHYC10004 Chap.2 Electric Flux and Gauss's Law
Electric Flux and Gauss's Law
Gauss's law is the Week 2 tool that turns symmetry into a one-line field calculation. It states that the electric flux through any closed surface equals the enclosed charge divided by ε₀, and it holds for every charge distribution, making it more general than Coulomb's law and one of Maxwell's equations.
The law yields the field directly only when the charge has spherical, cylindrical or planar symmetry.
Then a well-chosen Gaussian surface makes the field constant and either parallel or perpendicular to each face, so the flux integral collapses to E times an area. This chapter builds that procedure, applies it to spheres, lines, sheets and conductors, and explains why the field inside a static conductor is zero, why excess charge sits on its surface, and how a closed conducting box shields its interior as a Faraday cage.
Past papers repeatedly ask for the field in several regions of the same arrangement, so the chapter practises region-by-region reasoning.
What this chapter covers
- 01
Area Vectors and Flux
- 02
Gauss's Law Statement
- 03
Choosing a Gaussian Surface
- 04
Spherical Symmetry
- 05
Line and Cylinder Charges
- 06
Sheets and Conductor Surfaces
- 07
Conductors in Equilibrium
- 08
The Faraday Cage
Worked example · free
Field inside and outside a charged conducting tube
- 1Symmetry: the field must be radial and depend only on the distance r from the axis, so choose a coaxial cylinder of length L.
- 1Inside (r = 1.0 cm): the Gaussian cylinder encloses no charge, so E(2πrL) = 0 and E = 0.
- 1Outside: it encloses λL, so E(2πrL) = λL/ε₀ and E = λ/2πε₀r.
- 1Substitute r = 0.050 m: E = 4.0 × 10⁻⁸/(2π × 8.854 × 10⁻¹² × 0.050) = 1.4 × 10⁴ N/C.
- 1Direction: radially outward from the axis, because the charge is positive.
Key terms
- Electric Flux
- Electric flux is the integral of E · dA over a surface, a measure of how much field crosses it. It is positive where the field leaves a closed surface and negative where it enters.
- Gauss's Law
- Gauss's law states that the net electric flux through any closed surface equals the enclosed charge divided by ε₀.
- Gaussian Surface
- A Gaussian surface is an imaginary closed surface chosen so that the field is constant and parallel or perpendicular to it on each face, making the flux easy to evaluate.
- Surface Charge Density
- Surface charge density σ is the charge per unit area on a surface, measured in C/m².
- Faraday Cage
- A Faraday cage is a closed conducting enclosure whose induced surface charges cancel external electric fields inside it, shielding whatever it contains.
- Electrostatic Equilibrium
- Electrostatic equilibrium is the state in which charges in a conductor have stopped moving, so the field inside the metal is zero and any excess charge lies on the surface.
- Linear Charge Density
- Linear charge density λ is the charge per unit length along a line or wire, measured in C/m.
Electric Flux and Gauss's Law FAQ
Is Gauss's law only true for symmetric charge distributions?
No. The law holds for every distribution and every closed surface. Symmetry matters only for using it to find the field, because without symmetry the field varies over the surface and the flux integral cannot be simplified into a field strength times an area.
Why is the field from a conductor's surface twice that of an insulating sheet?
For an insulating sheet the field leaves through both faces of the pillbox, giving σ/2ε₀. For a conductor the field inside the metal is zero, so all the flux exits through the outer cap and the field there is σ/ε₀ for the same surface charge density.
What happens when a charge is placed inside a hollow conductor?
An equal and opposite charge gathers on the inner wall of the cavity so that a Gaussian surface inside the metal encloses zero net charge. A matching charge appears on the outer surface, so the field outside is as if the original charge were at the centre.
How do I earn the marks in a Gauss's law derivation?
Write the law explicitly, draw and name the Gaussian surface, state which faces contribute no flux and why, express the enclosed charge using the given density, then solve for the field and state its direction. Marking hints for past derivations credit each of these statements.
Exam move
Learn Gauss's law as a four-step routine and apply it to a deliberately varied set: a point charge, a long line, an insulating sheet, a conductor surface, a solid charged sphere and a coaxial pair of cylinders. For each, write out the routine in full, including the sentence that names the faces with zero flux.
Then practise the region questions past papers favour by finding the field inside, within and outside a conducting shell around a central charge, and plot E against r to see where it jumps. Finish by writing a half-page explanation of the Faraday cage with a labelled diagram.
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