Electric Field of Point Charges

Every electric charge creates an electric field E around it. A second charge q placed in the field feels a force F = qE. For a point charge, Coulomb's law gives the field and the electric potential:

Fields add as vectors (superposition), so the field of several charges is the sum of the individual fields. The field always points downhill in potential: E = −∇V.

• Field lines follow the direction of E. They start on positive charges and end on negative ones, and their number is proportional to the charge. Where lines crowd together, the field is strong.
• Equipotentials (dashed) join points at the same potential. They always cross field lines at right angles, and no work is needed to move a charge along one.
• The background shows the potential (red positive, blue negative) or the field strength, on a compressed scale because both grow without limit near a point charge.
• The probe arrow shows the field direction at that point; its values appear in the live readouts.

Fields and potentials are computed exactly from equation (1) by superposition. Field lines are traced with fourth-order Runge–Kutta integration along the field direction. Equipotentials are extracted from a grid with the marching-squares algorithm. Automated tests confirm Coulomb's law, check that E = −∇V, and check that equipotentials of a single charge are circles of radius kq/V. They also verify Gauss's law: the flux of E through a sphere equals the enclosed charge divided by ε₀, to 0.1%, including when a second charge sits outside the sphere.

• Point charges in empty space (vacuum), shown in a plane through all the charges.
• Static charges: no currents, no magnetic effects, no radiation.
• Fields are not plotted within a few centimetres of a charge, where they become infinite.