Flux through a closed surface.

A resizable closed surface (sphere, cube or cylinder) among point charges, with flux through each face and the total compared with q_enc/ε₀.

Put a +2 μC charge right beside the surface and keep the total flux at zero, then enclose it and read 2.26 × 10⁵ N m² C⁻¹.

Counting outside charges in q_enc, or thinking the flux depends on the surface's size or shape.

• For a point charge q at the centre of a sphere of radius r, E is the same everywhere on the sphere and along the normal, so the flux is E × 4πr² = q/ε₀, whatever the radius.
• Gauss's law: the total electric flux through any closed surface equals 1/ε₀ times the net charge enclosed, φ = ∮E·dS = q_enc/ε₀.
• It holds for a closed surface of any shape or size; the enclosed charges may lie anywhere inside it.
• q_enc counts only charges inside the surface, but the field E on the surface is produced by all charges, inside and outside.
• If a closed surface encloses no net charge, the total flux through it is zero, as for a closed cylinder in a uniform field: the flux in through one end equals the flux out through the other, and none crosses the curved side.
• The chosen surface is the Gaussian surface. It must not pass through a point charge (the field is undefined there), though it may pass through a continuous distribution.
• Gauss's law rests on the inverse-square form of Coulomb's law; it is most useful for finding E when the charge has symmetry.

Teaching this idea? Put the simulation on your own page or LMS with this code.