Electric Field Lab (2D)

Coulomb field and potential

Each point charge q produces E = k q r_hat / r^2 and potential V = k q / r with k = 8.99e9 N m^2/C^2. For several charges the fields add as vectors and the potentials as numbers (superposition). The field is the negative gradient of the potential, E = -grad V, so field lines always cross equipotential lines at right angles and point from high to low potential.

Field lines

Lines are traced from small circles around the positive charges (around the negative ones if no positive charge exists) with a midpoint step of 1.5 mm along the local field direction. They end on negative charges or leave the box. The number of lines is proportional to the charge, so line density is a rough measure of field strength. A small 2 mm softening length avoids a singularity when a charge sits on a trace.

Test charge

The released test charge obeys m a = q E and is advanced with the velocity-Verlet scheme. Its kinetic energy plus q V is constant because the field is conservative; the energy graph shows both parts and their sum.

Numerical checks on this page

(1) The on-axis field of a dipole at ten separations is compared with the far-field formula 2 k p / r^3. (2) E = -grad V is checked by central finite differences at a point for your current charges. (3) A 3000-step test-charge run in a fixed dipole must conserve energy.

Honest simplifications

The charges are treated as 3D point charges whose field is drawn in a plane through them, with no conductors, dielectrics or induced charge. Charges are fixed (they do not move in response to each other), the test charge does not disturb the others and radiates nothing, and a softening length is applied at very short distance. Numbers are in centimetres and nanocoulombs so that a classroom-sized set-up gives readable kilovolts.

• Start with the Dipole and drag one charge away: the potential graph shows the peak and the trough.
• Choose Two like charges and look for the saddle point between them where E_x crosses zero in the field graph.
• Switch the click action to release test charge and click near a positive charge: kinetic energy rises while potential energy falls, the total stays flat.
• Increase the number of lines per nC to see the dense bundle near the larger charge in the Unequal pair.