Charge bench - 2D field map

Coulomb's law and the field of point charges

A point charge q produces E = k q r̂ / r2 and potential V = k q / r with k = 1/(4πε0) = 8.99×109 N m2/C2. Fields add as vectors, potentials as numbers. The map shows the plane that contains the charges: field lines are curves tangent to E, drawn with RK4 and starting from the charges (more lines from larger charges); equipotentials are contours of V that cross the field lines at right angles.

Gauss's law

The flux of E through any closed surface equals the enclosed charge over ε₀: Φ = ∮ E·dA = Qenc / ε0. The Gaussian sphere is a real 3D sphere; its flux is integrated numerically over the surface. Charges outside it contribute field but no net flux, which is why the flux graph is a staircase that jumps when the radius passes a charge.

Conducting sphere and image charges

Inside a conductor in equilibrium E = 0 and the surface is an equipotential. For a grounded sphere of radius a and a charge q at distance d from its centre, the induced charge produces exactly the field of an image charge q' = −q a / d at distance a²/d from the centre, so the real charge is attracted with F = k q² a d / (d² − a²)². An isolated neutral sphere adds a compensating charge +q a / d at its centre; the net induced charge is then zero.

Honest simplifications: point charges and a perfectly conducting sphere in vacuum, electrostatics only (no motion, no radiation, no dielectrics), several charges near one sphere are handled by superposition of images (exact for a grounded sphere), and the map shows only the plane through the charges.

• Load Gauss test: move the Gaussian sphere so a charge crosses its surface and watch the flux jump by q/ε₀ in the flux graph.
• Load Charge near a grounded sphere and drag the charge: the force readout follows the attraction curve.
• Compare grounded and isolated spheres: the zero-volt contour hugs the grounded one.
• Switch on field vectors, then equipotentials, and see them cross at right angles.