Charge distributions create invisible electric fields that exert forces on other charges. Field lines reveal the field's direction and strength at a glance — they point away from positive charges, toward negative ones, and never cross each other.
The field at any point is the vector sum of each charge's contribution: E = k·q·r̂/r² (Coulomb's law). Field lines are integrated numerically in 3D using small Euler steps. Switch to arrow mode to sample the field on a spatial grid, coloured by magnitude.
Pick + or − and double-click in the scene to add a charge. Click a charge to select it, then drag to move, press F to flip its sign, or Delete to remove it. Try the dipole, two-like and quadrupole presets.
Michael Faraday introduced field lines in 1831 to visualise electromagnetic forces he could not mathematically describe. When Maxwell later provided the mathematics — Maxwell's equations — field lines became one of physics' most powerful tools. They directly predict how antennas, capacitors and particle accelerators work.
This simulation models the electric field created by one or more point charges in three-dimensional space. Each charge generates a field described by Coulomb's law (E = kq/r²), and when multiple charges are present the total field at any point is the vector sum of their individual contributions — the principle of superposition. You can observe how field lines always originate on positive charges, terminate on negative charges, and never intersect one another.
Electric fields underpin virtually every electromagnetic device humans have built, from simple capacitors and cathode-ray tubes to modern particle accelerators and microprocessor transistors. Understanding how charges shape space is foundational to both classical electrodynamics and quantum electrodynamics.
An electric field is a region of space in which a charged particle experiences a force. It is defined as the force per unit positive test charge at each point in space: E = F/q. The field exists whether or not a test charge is placed there — it is a property of the space itself, created by the source charges.
Select either the "+ Add Positive" or "- Add Negative" button, then double-click anywhere in the 3D scene to place a charge at that location. Click an existing charge to select it (it will glow brighter), then drag it to reposition it. While a charge is selected you can press F to flip its sign or Delete to remove it. The field lines update in real time as you make changes.
Like charges repel each other because the electric forces they exert point in opposite directions. In the field-line picture, both positive charges emit lines radially outward, and because field lines cannot cross (that would imply two different field directions at one point), the lines from each charge curve away from those of the other. Use the "Two +" preset to see a neutral point appear between them where the fields exactly cancel.
Coulomb's law states that the electric force between two point charges is F = k·q1·q2/r², where k is Coulomb's constant (approximately 8.99 × 10⁹ N·m²/C²), q1 and q2 are the charges, and r is the distance between them. In the simulation a scaled version (k = 1 in dimensionless units) is used to compute the field vector E = k·q·r̂/r² at each point. Field lines are then traced by numerically integrating this field in small Euler steps from seed points placed on a sphere around each positive charge.
The electric field between the plates of a capacitor stores energy in electronic circuits. In a thunderstorm the field between clouds and the ground can exceed 10⁶ V/m, triggering lightning. Cathode-ray tubes in old televisions steered electron beams using electric fields. In biology the electric field across a cell membrane (around 10⁷ V/m) drives ion channels and nerve impulses. Modern semiconductor transistors switch by controlling the field in a thin gate oxide just a few nanometres thick.
This is a common misconception. Field lines show the direction of the electric force on a positive test charge at each point, not the trajectory that charge would follow. A positive charge released from rest does travel along a field line (since its velocity and acceleration are always parallel), but if it already has a sideways velocity it will curve off the field line — just as a ball thrown sideways curves away from the vertical gravitational field line. Field lines are a static snapshot of force direction, not a kinematic path.
Charles-Augustin de Coulomb established the inverse-square force law experimentally in 1785 using a torsion balance. Michael Faraday introduced the concept of field lines in the 1830s as an intuitive way to visualise forces he could not yet mathematically formalize. James Clerk Maxwell unified Faraday's ideas into a rigorous mathematical framework in 1865 with Maxwell's equations, showing that electric and magnetic fields are two aspects of a single electromagnetic field that propagates as light.
A stationary charge creates only an electric field. A moving charge (current) also produces a magnetic field. When a charge accelerates it radiates energy as an electromagnetic wave — a coupled oscillation of electric and magnetic fields propagating at the speed of light. Maxwell's equations describe all of these relationships. In the simulation you can explore the companion Magnetic Field Lines simulator to see how a current-carrying wire or bar magnet shapes space differently from a point charge.
Electric fields are exploited in inkjet printers (charged droplets steered by deflection plates), electrostatic precipitators that remove particulates from industrial exhaust, mass spectrometers that separate ions by charge-to-mass ratio, and ion thrusters that propel spacecraft. In semiconductor fabrication, extremely precise electric fields inside MOSFET gate oxides — now only a few silicon atoms thick — switch billions of transistors per second in every modern CPU and GPU.
At the quantum level, the interaction of electrons with their own electric field leads to infinities in the bare theory that must be removed by renormalization — a procedure that works extraordinarily well but whose deeper physical meaning is still debated. Strong-field quantum electrodynamics predicts that an electric field above the Schwinger limit (about 1.3 × 10¹⁸ V/m) can spontaneously produce electron-positron pairs from vacuum; reaching this threshold with lasers is an active experimental goal. In condensed matter physics, controlling electric fields at the atomic scale is central to developing new ferroelectric memories and two-dimensional materials like graphene.