Point Charge Field Simulator
Formula: Introduction: What the simulator models
This simulator places a fixed source charge Q at the center of the canvas and releases a test charge q with mass m from the position (x₀, y₀). The blue radial lines show the electric field direction, and the orange point shows the test charge as it accelerates under Coulomb's law. The model uses SI units throughout: coulombs for charge, kilograms for mass, meters for position, seconds for time, joules for energy, and newtons per coulomb for electric field strength.
Core equations
The electric field from an ideal point charge has magnitude
where k = 8.9875517923 × 10⁹ N·m²/C² and r is the distance from the source. The force on the test charge is F = qE. In vector form, the acceleration used by the animation is
If Qq is positive, the force is repulsive and the test charge accelerates away from the source. If Qq is negative, the force is attractive and the charge accelerates inward. The simulator also tracks kinetic energy KE = 1/2 mv² and electric potential energy PE = kQq/r.
How to choose inputs
- Q (C): source charge at the origin. Positive values point the field outward; negative values point it inward.
- q (C): moving test charge. Its sign determines whether the force follows or opposes the field direction.
- m (kg): test-charge mass. Smaller masses accelerate more strongly under the same electric force.
- x₀, y₀ (m): starting position. The position cannot be exactly at the origin because the point-charge field is singular there.
- Δt (s): integration time step. Smaller values are slower but conserve energy better; this page requires 0.00001 to 0.05 seconds.
Worked example
With the default inputs Q = 1e-6 C, q = 1e-6 C, m = 1e-3 kg, and r = 0.10 m, the initial electric field is about 8.99 × 10⁵ N/C. The force on the test charge is therefore about 0.899 N, so the initial acceleration is about 899 m/s². The initial potential energy is 0.0899 J. Because both charges are positive, the orange test charge accelerates away from the center.
Comparison table
| Scenario | Q (C) | q (C) | r (m) | Initial |E| (N/C) | Initial |F| (N) | Motion |
|---|---|---|---|---|---|---|
| Default repulsion | 1e-6 | 1e-6 | 0.10 | 8.99e5 | 0.899 | Away from source |
| Opposite signs | 1e-6 | -1e-6 | 0.10 | 8.99e5 | 0.899 | Toward source |
| Twice as far away | 1e-6 | 1e-6 | 0.20 | 2.25e5 | 0.225 | Away, weaker force |
Reading the output
The result panel reports simulation time, distance from the source, and relative energy drift. The kinetic-energy bar and potential-energy bar compare the instantaneous kinetic energy with the magnitude of potential energy, so attractive cases with negative potential energy still display a useful balance. The CSV button exports time, position, velocity, kinetic energy, and potential energy for further graphing.
Limitations
This is an idealized one-source model. It ignores radiation, relativity, collisions, boundaries, and interactions with any other charges. The field becomes infinite at the exact source location, so the simulation halts near the origin. Very large charges, very small masses, or large time steps can produce fast motion and visible numerical error; reduce Δt when energy drift becomes large.
How to use this calculator
- Enter Q (C) using the unit or time period shown by the field.
- Enter m (kg) using the unit or time period shown by the field.
- Enter x₀ (m) using the unit or time period shown by the field.
- Run the calculation and compare the output with a second scenario before acting on it.
Arcade Mini-Game: Point Charge Field Simulator Calibration Run
Use this quick arcade run to practice separating useful scenario inputs from common planning mistakes before you rely on the calculator output.
Start the game, then use your pointer or arrow keys to catch useful inputs and avoid bad assumptions.