Skip to content

Electricity & Magnetism

Flux & Gauss's Law

Count what crosses a closed surface and the answer forgets everything except the charge sitting inside it.

Nothing here is graded. Your place is kept in this browser.

Read as one page

Flux & Gauss's Law

Count what crosses a closed surface and the answer forgets everything except the charge sitting inside it.

30 min3 checkpoints4 simulation tasks

Introduction

The water passing through a net in a river depends on the flow, the net's size and how squarely it faces the stream. Electric flux asks the same of an electric field.

Predict

A point charge sits at the centre of a sphere of radius R, and of a second sphere of radius 2R. Through which does more flux pass?

Counting what crosses

Try it

One charge inside a closed surface. Make the surface twice as wide, and watch the flux and the field on the surface as you do.

GoalMake the surface twice as wide.

Loading simulation…

Flux through a small patch of surface is the field component perpendicular to the patch, times its area. Add up every patch of a closed surface and you have the flux through it:

ΦE=∮E⃗⋅dA⃗\Phi_E = \oint \vec{E}\cdot d\vec{A}

Field leaving the surface counts positive, and field entering counts negative.

Gauss's law

Try it

A third charge drifts past. Set its drift height so it passes through the surface, and watch the flux as it goes.

GoalSend the drifting charge through the surface.

Loading simulation…

That is Gauss's law:

∮E⃗⋅dA⃗=qencε0\oint \vec{E}\cdot d\vec{A} = \frac{q_{\text{enc}}}{\varepsilon_0}

The total flux out of any closed surface equals the charge it encloses, divided by ε0\varepsilon_0. Not the charge near it, and not the charge that makes the biggest field on it: the charge inside it.

Practice

Charge A, nC per metre, is inside the surface. Charge B, nC per metre, is outside. What is the flux through each metre of the surface?

Charge outside the surface

Predict

Nothing is inside the surface now: charge A and the drifting charge are set to zero. Charge B sits just outside. What does the flux integral give?

Try it

Nothing inside, and a strong charge just outside. Slide charge B closer to the surface without letting it in, and watch both bars of the ledger.

GoalBring charge B closer, but keep it outside.

Loading simulation…

Practice

A larger surface now encloses both charge A, nC per metre, and charge B, nC per metre. What is the flux through each metre of it?

Always true, only sometimes useful

Predict

Switch the surface shape to 'Blob' with one charge inside it. Which statement is true?

Try it

The ribbon's height is E·n̂, the field across the surface. Move charge A until the ribbon is the same height all the way round.

GoalMake the ribbon the same height all the way round.

Loading simulation…

Gauss's law always holds, for any closed surface and any charges. To get EE out of it you must pull EE outside the integral, and that needs symmetry: a sphere of charge, an infinite line or an infinite sheet. Choosing that surface well is the whole skill.

Practice

A long line of charge, nC per metre, runs along the axis of a cylinder of radius m. What is the field on the cylinder's wall?

What comes next

You grew a surface without changing its flux, sent a charge through one, and centred a charge to read its field off Gauss's law. Next we follow the field lines instead of counting them: what it costs to push a charge from one place to another, and why that cost defines a landscape you can draw as contour lines.

Ideas this lesson took on

Each of these is a common intuition the lesson put to the test. If one still feels right, the section that tests it is worth another look.

  • “A bigger surface catches more flux”
  • “Zero flux through a surface means zero field on it”
  • “A charge just outside the surface still adds some flux”
  • “Gauss's law only holds for symmetric situations”

Finished with this lesson?

Progress is stored on this device only. There is no account and nothing is sent anywhere.

Research behind this lesson

  • Guisasola et al. 2008: University students' trouble with Gauss's and Ampère's laws springs less from isolated misconceptions than from 'common-sense' reasoning applied to a formula. In the authors' words, students "establish a causal link between the charge … enclosed by any closed surface/path and the field at the points on that surface/path. They wrongly infer that the only sources of the field are those enclosed by the Gaussian surface or Amperian path", and from that, that a zero flux means the field on the surface is zero.
  • Maloney et al. 2001: The Conceptual Survey of Electricity and Magnetism (CSEM): a 32-question multiple-choice instrument refined over four years and administered to more than 5000 introductory students at 30 institutions. Pretest averages were 31% (calculus-based) and 25% (algebra/trigonometry-based); after instruction they reached only 47% and 44%.
  • Singh 2006: Across 541 introductory students plus advanced undergraduates and teaching assistants, students conflate electric flux with electric field: reading a vanishing net flux as a vanishing field, with 55% claiming the field inside a uniformly charged hollow cube is zero everywhere. Asked to compare two Gaussian surfaces around one charge, they choose the larger for its area or the smaller for its stronger field rather than seeing the two cancel; and they treat the symmetry needed to extract E as though it were a condition on the law itself, applying it to finite cylinders and cubes where no such symmetry exists.