Flux versus local field · Anannt Study

Zero net flux is not a zero field

You will be able to: Reject the claim that $Q_{\mathrm{enc}}=0$ implies $\vec{E}=0$ on a surface, using uniform-field and external-charge examples.

1. Model

Name the physical situation, the idealisations, and the quantities that matter.

2. Represent

Draw the diagram, choose coordinates, and write the symbols you will use.

3. Derive

Select a principle and produce a result from it, keeping the chain of reasoning visible.

4. Check

Units, signs, limiting cases, and whether the result could be true of the situation.

5. Explain

Say what the result means, what it assumed, and why a tempting alternative fails.

Reviewed 2026-09-10 · v1.0 · Anannt physics author (author) · Independent physics reviewer (reviewer) · Video is never required for mastery.

2. Explanation (text route)

Side-by-side: a cube in a uniform field with flux in and out labelled, and a sphere with an external charge whose field lines enter and leave. A caution card reads: zero net flux ≠ zero field. A transfer prompt then shows a charge outside a Gaussian cylinder.

This lesson exists because a very common error treats a surface integral as if it were a local measurement. The integral $\oint E \cdot d A$ can vanish while $E$ is large. Think of a river: net flow out of a loop of boom can be zero while water rushes past.

3. Worked example

Uniform field through a sphere

Situation. A sphere of radius $R$ is placed in uniform $E_{0}$ . Find net flux and discuss $E$ on the surface.

Assumptions. The uniform field is produced by distant sources; no charge inside the sphere.

Derivation. $Q_{enc} = 0 \Rightarrow Φ_{net} = 0$ . On the surface $E = E_{0}$ (in this empty-space idealisation), which is not zero and is not radial. You cannot pull $∣ E ∣$ out of the flux integral as a constant radial field.

Units. Flux is zero; $E_0$ in N/C.

Interpretation. Useful Gaussian surfaces require matching symmetry. This sphere is a legal surface for Gauss’s law, but it is not a useful one for finding $E_{0}$ .

Why tempting alternatives fail. Declaring $E = 0$ on the sphere contradicts the given uniform field. Declaring Gauss’s law false because $E \neq = 0$ misunderstands the law.

6. Independent questions (different situations)

These submissions are required for evidence. Watching or reading the explanation does not advance mastery.

A point charge $+ Q$ lies outside a closed Gaussian surface $S$ . Which is correct?

A.Net flux through

S

Q / ε_{0}

B.Net flux is 0, but

E

S

is generally not 0C.

E = 0

everywhere on

S

D.Gauss’s law does not apply unless

Q

is at the centre

A spherical Gaussian surface is placed in a uniform electric field in empty space. Net flux is zero. A student concludes $E = 0$ on the sphere. The error is

A.Gauss’s law is false in uniform fieldsB.treating

\oint E \cdot d A = 0

as if

E = 0

C.the sphere has the wrong radiusD.uniform fields have no Gaussian surfaces

A Gaussian cylinder is drawn around empty space next to an infinite line charge (the line is outside the cylinder). Net flux through the cylinder is

λ L / ε_{0}

B.0C.equal to

E

times the curved areaD.undefined because the line is infinite

Zero net flux is not a zero field

2. Explanation (text route)

3. Worked example

4. Faded example

5. Representation task

6. Independent questions (different situations)