Electric Flux and Gauss's Law 3D – Sphere, Cylinder and Box
PhysicsElectric & Magnetic FieldsAges 16–17
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A rotatable 3D model: place point charges, a line charge, a charged plane or a charged spherical shell, then choose a Gaussian sphere, cylinder or box and resize, tilt and move it. The flux Φ = ∮E·dA through the surface is computed numerically and compared with q_enc/ε₀. The Flat surface screen shows Φ = EA·cos θ as a surface turns in a uniform field, and the Symmetry → E screen derives E for a point charge, a line (λ/(2πε₀r)), a sheet (σ/(2ε₀)) and a shell (zero inside, kQ/r² outside).
Lesson: Electric flux; Gauss's law; electric field of a line charge, a charged sheet and a charged spherical shell
What it shows
Gauss's law states that the total electric flux through any closed surface equals the enclosed charge divided by ε₀, whatever the shape of the surface. Here the flux is added up numerically over a sphere, a cylinder or a box, so students can see that moving or reshaping the surface changes nothing unless charge crosses it. The colour of the surface shows where flux leaves and enters. A flat surface in a uniform field shows Φ = EA·cos θ, and the symmetry screen uses Gauss's law to find E for a point charge, a line, a sheet and a shell.
How to use
On Gaussian surface, choose a charge source and a surface, then change Radius r, Tilt and Surface centre and compare Φ with q_enc/ε₀; press Sweep to slide the surface past the charges. On Flat surface, turn the angle θ and record Φ. On Symmetry → E, pick a case, vary r and press Sweep to plot E against r. Try the three tasks.
Parameters you can change
- Screen Closed Gaussian surface, Flat surface, Symmetry → E
- Charge source Point charges q₁, q₂, Line charge, Charged plane, Charged spherical shell
- Charge q₁ (at the origin O; also the charge Q of the shell) -10–10 nC
- Charge q₂ -10–10 nC
- Position x of q₂ -10–10 cm
- Linear charge density λ of the line -50–50 nC/m
- Surface charge density σ of the plane -50–50 nC/m²
- Radius R of the shell 1–8 cm
- Gaussian surface Sphere, Cylinder, Box
- Radius r (sphere, cylinder) or base side a (box) 1–16 cm
- Length h of the cylinder / height h of the box 1–24 cm
- Tilt of the Gaussian surface 0–90 °
- x coordinate of the surface centre -10–10 cm
- y coordinate of the surface centre -10–10 cm
- Show E arrows
- Uniform field strength E (Flat surface screen) 200–5000 N/C
- Side s of the square surface 2–20 cm
- Angle θ between the normal n̂ and E 0–180 °
- Symmetric case Point charge, Long line charge, Large charged sheet, Spherical shell
- Radius r of the Gaussian surface (Symmetry screen) 0.5–15 cm
Questions to explore
- Why does the flux stay the same when a Gaussian sphere around a charge is made larger or moved?
- Why is there no flux through the end caps of a cylinder around a long line charge?
- Why is the field of a large charged sheet independent of the distance from it?
