A point charge +q sits inside a small sphere, and you calculate the total outward electric flux through the sphere's surface. You then replace the sphere with a much larger cube that also encloses +q. How does the flux through the cube compare to the flux through the sphere?
AThe flux through the cube is larger — the cube's surface area is greater, so more field lines pass through it.
BThe flux through the cube is smaller — the field is weaker at the greater distance from the charge, reducing flux.
CThe flux through the cube equals the flux through the sphere — only the enclosed charge determines total outward flux.
DThe flux through the cube is zero — field lines strike the flat faces at varying angles and cancel out.
Gauss's law states that total outward flux through any closed surface equals Q_enclosed/ε₀ — it depends only on the enclosed charge, not on the shape or size of the surface. Options A and B reflect a common confusion: while field strength does decrease with distance (less flux per unit area), the cube's larger surface area exactly compensates, leaving total flux unchanged. This is the topological insight: flux counts field lines originating inside, and that count does not change when you change the surface shape.
Question 2 Multiple Choice
In a region of space where ∇·E = 0 everywhere, what must be true?
AThe electric field E is zero throughout the region.
BThere are no free charges in the region — the charge density ρ = 0.
CThe electric field has constant magnitude and direction throughout the region.
DThe region is enclosed by a conducting shell that shields it from external fields.
The differential form of Gauss's law is ∇·E = ρ/ε₀. If ∇·E = 0, then ρ = 0 — there is no charge density at those points. The field can still be nonzero (field lines from external charges can thread through the region), it just has no sources or sinks inside. Option A confuses zero divergence with zero field; option C confuses it with a uniform field. Divergence measures whether field lines spread from or converge to a point, not the field's magnitude.
Question 3 True / False
The total electric flux through any closed surface depends only on the net charge enclosed within it, not on the shape or size of the surface.
TTrue
FFalse
Answer: True
True — this is Gauss's law: ∮E·dA = Q_enclosed/ε₀. The shape and size of the Gaussian surface are irrelevant. Intuitively, flux counts field lines threading through the surface, and every field line originating from a charge inside must pass through any closed surface surrounding that charge, regardless of how that surface is shaped or how large it is.
Question 4 True / False
Increasing the radius of a spherical Gaussian surface surrounding a fixed point charge will increase the total electric flux through the surface.
TTrue
FFalse
Answer: False
False. The total flux equals Q_enclosed/ε₀ and does not depend on radius. While field strength decreases as 1/r², surface area increases as 4πr², and the two effects cancel exactly: E × 4πr² = (q/4πε₀r²) × 4πr² = q/ε₀, independent of r. This constancy is precisely why Gauss's law is powerful — the Gaussian surface can be chosen for geometric convenience without affecting the result.
Question 5 Short Answer
Explain why the total outward flux through a closed surface depends only on the enclosed charge and not on the shape or size of the surface.
Think about your answer, then reveal below.
Model answer: Field lines from a charge radiate outward continuously and do not terminate in empty space. Any closed surface surrounding the charge will be threaded by all of those field lines — each one must pass through the surface to reach infinity. Reshaping or enlarging the surface does not create or destroy field lines; it only changes which patch of surface each line crosses. The total count of field lines threading the surface is therefore determined entirely by how many originate inside — that is, by the enclosed charge. The divergence theorem makes this precise: ∮E·dA = ∫∇·E dV = ∫(ρ/ε₀) dV = Q_enclosed/ε₀.
Flux is a topological count, not a local measurement. Field lines from charges outside the surface enter on one side and exit on the other, contributing zero net flux. Only sources (positive charges) and sinks (negative charges) inside the surface generate net outward or inward flux — this is why only the enclosed charge matters.