V at probe - |E| - Energy U - Flux ratio - FPS 0
Bench Top view Close up
WebGL is not available here — the numbers, charts and numerical checks on the right still work.
Drag a ball to move it · drag empty space to orbit · wheel / pinch to zoom · red = positive, blue = negative, lines show the direction of the force on a positive test charge
Charge Bench: Coulomb Fields, Equipotentials, Gauss Flux and a Grounded Conducting Sphere
Place electric charges on a bench and see their field as lines, as a potential landscape and as numbers. Drag the charges, add a metal sphere to see induced charge, and wrap an imaginary Gaussian sphere around some of them to test Gauss's law.
Educational model. Point charges and an ideal metal sphere in vacuum; the plane you see is a slice through a fully three-dimensional field (1/r² law). Static charges only: no currents, no air breakdown, no moving metal. Sizes are bench scale (1 square = 10 cm).
Pause Reset Quality: High
Setup
Charges
Selected charge
Add + charge Add − charge Remove selected
Metal sphere
Conductor None Grounded sphere Isolated neutral sphere
Gaussian sphere
Display
Live readout
Potential along the bench (z = 0) Test charge
A 1 mg dust grain with a charge you choose. Press Release, then click the bench to drop it there (at rest). It follows the force of the fixed charges; kinetic + potential energy stays constant.
Release (click bench) Clear
Experiment 1: Gauss's law
Grows the Gaussian sphere around its present centre from 0.4 to 12 squares and compares the numerically integrated flux of E through the surface with the enclosed charge divided by ε0 .
Run Gauss sweep
Experiment 2: pull of a grounded sphere
A +3 nC charge is moved towards a grounded sphere of radius 1.5 squares. The force found from the image charges is compared with the closed form F = k q² a d / (d² − a²)² and with the (wrong for a sphere) flat-plate mirror force.
Run force sweep
Independent numerical checks
Also run in Node: node src/check.js.
Controls Drag a ball (or the sphere, or the green centre marker) to move it. Drag empty space or use one finger to orbit; wheel or pinch to zoom. Choose a charge in the list and set its value from −10 to +10 nC; add or remove charges (up to 10). Hover or tap on the bench to read the potential and field at that point. Views: Bench, Top view, Close up. The science
Coulomb's law and superposition. A point charge q makes the field E = k q / r² pointing away from it (towards it if q is negative) and the potential V = k q / r, with k = 1/(4πε0 ) = 8.988×109 N m²/C². Fields and potentials of many charges simply add. Field lines are curves tangent to E; the number of lines leaving a charge is drawn proportional to |q|.
Equipotentials. A surface of constant V. They always cross field lines at right angles, and a charge moved along one gains no energy. The landscape shows V (compressed with an inverse hyperbolic sine so that the peaks at the charges stay readable).
Gauss's law. The flux of E through any closed surface equals the enclosed charge over ε0 : Φ = ∮ E·dA = Qenc /ε0 . Here the surface is a sphere and the integral is done numerically (40 Gauss-Legendre points in cosθ times 72 in φ). Charges outside contribute field but no net flux. If the sphere surface cuts through the metal ball the page shows "n/a".
Conductors. In equilibrium the field inside metal is zero, the surface is an equipotential and E is perpendicular to it. For a point charge q at distance d from the centre of a grounded sphere of radius a, the outside field equals that of q plus an image charge q' = −q a/d at distance a²/d from the centre (method of images). The induced charge on the sphere is q', the force on q is F = k q q' / (d − a²/d)² = −k q² a d /(d² − a²)² (attraction), and the stored energy is U = ½ k q q'/(d − a²/d). An isolated neutral sphere gets the same image plus +|q'| at its centre so that its total charge stays zero; it then floats to the potential k Q/d.
Test charge. Newton's second law with the field of the fixed charges, velocity-Verlet integration with a step that shrinks near a charge; the conserved energy is ½ m v² + q V. The grain does not itself induce charge in the sphere (it is treated as a tiny probe) and radiates nothing.
Simplifications and honest limits. Charges are ideal points softened at 0.2 cm; no leakage into air (real charges on a humid day leak away in minutes, and fields above about 3 MV/m ionise air); the metal sphere is perfectly conducting and the charge balls are not allowed to induce charge in each other; the bench, stands and balls are drawn only for orientation and do not alter the field. Values such as the 1 mg grain are chosen for visible motion.