Charged Particle Motion Simulator

Physics Electricity and Magnetism • Electric Fields and Charges

Written by STEM Calculators Team Published January 10, 2026 Updated February 24, 2026

9. Charged Particle Motion Simulator

Simulate the trajectory of a charged particle in a known electric field. Uses $\mathbf{F}=q\,\mathbf{E}$ , $\mathbf{a}=\mathbf{F}/m$ , and a simple time-stepping integrator. For a uniform field inside parallel plates, the path is parabolic when $ \mathbf{v}_0 \perp \mathbf{E} $.

Inputs support: pi, e, sqrt(), sin, cos, exp, log. Use * for multiplication. In “Field expression” mode you may use variables x, y, t (SI units).

Setup

E-field type: Coulomb constant $k$ (N·m²/C²): Charge $q$ (C): Mass $m$ (kg): Uniform $E_x$ (N/C): Uniform $E_y$ (N/C): Expression $E_x(x,y,t)$ (N/C): Expression $E_y(x,y,t)$ (N/C): Initial position $x_0$ (m): Initial position $y_0$ (m): Initial velocity $v_{x0}$ (m/s): Initial velocity $v_{y0}$ (m/s): Sim duration $t_{max}$ (s): Time step $\Delta t$ (s): Auto view padding (%): Trail length (points): Animation speed: Time cursor: Show velocity arrow: Hover probe: Parallel plates (uniform mode): Field only between plates: Capacitor center: Cap center $c_x$ (m): Cap center $c_y$ (m): Plate separation $d$ (m): Plate length $L$ (m): Field lines count:

Ready

For the capacitor case: the particle moves straight until it enters the gap (field region), then curves due to constant acceleration inside.

Steps

Enter values and click Solve.

Trajectory (pan/zoom + animation)

Pan/zoom • hover to probe • time cursor + play

Hover: (x, y)=… t = 0 s

Mouse wheel zooms at the cursor. Double-click resets view. The particle dot shows the current time cursor.

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Charged Particle Motion Simulator — Theory

This tool models a charged particle moving under an electric field. The physics starts with the electric force and Newton’s second law.

1) Force and acceleration

\[ \mathbf{F} = q\,\mathbf{E},\qquad \mathbf{a} = \frac{\mathbf{F}}{m} = \frac{q}{m}\,\mathbf{E}. \]

If the field is uniform (constant $E_x,E_y$), then $\mathbf{a}$ is constant. The motion is constant-acceleration kinematics in 2D.

2) Uniform field: analytic motion (parabolic paths)

\[ a_x=\frac{qE_x}{m},\qquad a_y=\frac{qE_y}{m}. \] \[ x(t)=x_0+v_{x0}\,t+\frac{1}{2}a_x t^2,\qquad y(t)=y_0+v_{y0}\,t+\frac{1}{2}a_y t^2. \]

If $E_x=0$ and you launch horizontally ($v_{y0}=0$), then $x$ increases linearly while $y$ changes quadratically, producing a parabola (just like projectile motion, except the “acceleration” comes from $q\mathbf{E}/m$).

3) Known varying fields: numerical integration

When $\mathbf{E}$ depends on position and/or time, the acceleration changes along the path: $\mathbf{a}(t)=\frac{q}{m}\mathbf{E}(x(t),y(t),t)$. In that case we approximate the motion with small time steps $\Delta t$.

\[ \mathbf{v}_{n+1}=\mathbf{v}_n+\mathbf{a}_n\,\Delta t, \qquad \mathbf{r}_{n+1}=\mathbf{r}_n+\mathbf{v}_{n+1}\,\Delta t. \]

The simulator uses a semi-implicit Euler update (also called symplectic Euler in many contexts). If the field varies rapidly or the acceleration is huge (e.g., electrons), you typically need a smaller $\Delta t$.

4) Units and interpretation

$q$ is in coulombs (C), $m$ in kilograms (kg).
$\mathbf{E}$ is in newtons per coulomb (N/C).
$\mathbf{a}$ is in m/s² and follows $\mathbf{a}=(q/m)\mathbf{E}$.
If $q<0$, the acceleration points opposite $\mathbf{E}$.

5) Practical tips

If the path “explodes” or becomes jagged, reduce $\Delta t$ or reduce field magnitude.
For a clean parabola in uniform field, use constant $E_y$, $E_x=0$, and a horizontal $v_0$.
In expression mode, keep your formulas smooth (avoid discontinuities) or use a very small $\Delta t$.