A flat surface is held in a uniform electric field. The surface is then rotated until it is exactly parallel to the field lines. What is the electric flux through the surface?
AMaximum — the field passes fully along the surface area
BHalf the maximum — the surface is at 45° to the normal
CZero — no field lines pass through the surface when it is parallel to the field
DIt depends on the magnitude of E and the area of the surface
Flux measures how much field passes *through* a surface, not along it. The formula Φ = EA cos θ uses θ as the angle between E and the outward normal to the surface. When the surface is parallel to the field, the normal is perpendicular to E, so θ = 90° and cos 90° = 0 — flux is zero. Think of the wind-and-net analogy: a net held parallel to the wind catches nothing. The most common error is confusing 'angle with the surface' and 'angle with the normal' — these are complementary angles.
Question 2 Multiple Choice
A closed spherical surface surrounds a region containing a complicated, non-uniform electric field. No net charge is enclosed within the sphere. What is the net electric flux through the sphere?
APositive — because E² is always positive, field contributions never cancel
BNonzero — the complicated field means contributions do not perfectly cancel
CZero — every field line that enters the closed surface must also exit somewhere
DNegative — field lines entering the sphere dominate because the field points inward
By Gauss's law, net flux through any closed surface equals the enclosed charge divided by ε₀. With zero net enclosed charge, the net flux is exactly zero — regardless of how complex the field is inside. This is not an approximation: every field line that enters the closed surface must exit it somewhere, because there is no source (positive charge) or sink (negative charge) inside to terminate them. The field can be arbitrarily complicated; the net cancellation is exact. This is one of the most powerful consequences of Gauss's law.
Question 3 True / False
Electric flux is a scalar quantity, even though it is calculated using the dot product of two vectors (E and dA).
TTrue
FFalse
Answer: True
True. The dot product of two vectors produces a scalar — a single number with no direction. E · dA = |E| |dA| cos θ, where θ is the angle between them. The result is a signed number (positive when E has a component in the direction of the outward normal, negative when it has a component opposing the normal). Even though both E and dA are vectors, their combination through the dot product — and the integration over the surface — yields a scalar: the total flux. Many students mistakenly expect flux to have a direction since it involves vectors, but the dot product removes this.
Question 4 True / False
Increasing the area of a surface typically increases the electric flux through it, because more surface area intercepts more field lines.
TTrue
FFalse
Answer: False
False. Flux depends on both area and orientation. If you increase the area of a surface that is already parallel to the field (normal perpendicular to E), the flux remains zero regardless of how large the surface becomes — no field lines pass through it. More precisely, Φ = ∫ E · dA, and each area element contributes E cos θ dA. If θ = 90° everywhere, additional area adds nothing. It is only when the surface (or part of it) is oriented with a component perpendicular to the field that area increases flux.
Question 5 Short Answer
A closed cube is placed in a uniform electric field directed parallel to one pair of faces (so two faces are perpendicular to the field, two are parallel, and two are at right angles to both). Which faces contribute to net flux, and what is the total net flux through the cube?
Think about your answer, then reveal below.
Model answer: Only the two faces perpendicular to the field contribute nonzero flux. The field enters through one face (negative flux, since field is antiparallel to the outward normal) and exits through the opposite face (positive flux, since field is parallel to the outward normal). The four faces parallel or oblique to the field have zero flux. The entering and exiting fluxes are equal in magnitude, so the total net flux is zero — consistent with no enclosed charge.
This example illustrates both aspects of flux: the angle-dependence (faces parallel to the field contribute nothing) and the cancellation on closed surfaces with no enclosed charge (entering flux equals exiting flux). Gauss's law guarantees the net is zero. If instead a positive charge were enclosed, the exiting flux would exceed the entering flux, and the net would be positive.