---
title: "A Charge Bending through Fields Interactive Lab | Sandbox Physics"
description: "How can a magnetic field turn motion without changing speed? The magnetic force always points at right angles to the velocity and does no work."
canonical_url: "https://sandboxphysics.com/labs/charged-particle/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/charged-particle/"
last_modified: "2026-10-27"
---

Electromagnetism · force and energy

# Charged Particle

**How can a magnetic field turn motion without changing speed?** The magnetic force always points at right angles to the velocity and does no work. All it can do is turn the particle. An electric field is different: it can change the kinetic energy directly.

Send a particle through a cyclotron gap and tune the RF rhythm. Compare acceleration with phase slip, then switch to uniform fields to explore magnetic circles, drift and electric acceleration.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

The particle is nonrelativistic, and all quantities are in SI units. Uniform-field modes use fixed electric and magnetic fields. RF mode uses magnetic arcs and ideal gap kicks; collisions, radiation, space charge, relativity and back reaction are left out. The positive horizontal and vertical axes, together with the magnetic field normal to the plane, form a right-handed system. The view is scaled to fit the whole recorded flight.

One stored trajectory

## What happened at each gap crossing?

Replay time $2\,\mathrm{s}$

### The latest gap crossings

Each energy step and each crossing marker come from the trajectory shown above.

Model, units and sources

The RF mode uses exact nonrelativistic magnetic arcs and a zero-width gap kick. The dees are shown with their roofs removed; the gap and particle are enlarged. Charge, mass, length, time and voltage use SI units at a teaching scale. The RF clock is fixed during each recorded flight. Electrode outlines resize to enclose the saved flight, so there is no fixed-aperture loss model. Fringing fields, finite gap transit time, collisions, radiation, space charge, relativistic phase slip and extraction are omitted. A decelerating barrier that would reflect the particle ends the trajectory at the gap.

$$
\omega_c=\frac{|qB_z|}{m},\qquad \Delta K_n=qV_0\cos(\omega_{\mathrm{RF}}t_n+\phi_0)\,\mathrm{sgn}(v_{x,n})
$$

The cyclotron angular frequency is set by charge, magnetic field and mass. At each arrival time, the signed gap voltage and crossing direction determine the work. In the uniform-field modes, electric work is calculated from displacement, and no RF gap acts on the particle.

[Lawrence’s cyclotron lecture](https://www.nobelprize.org/uploads/2018/06/lawrence-lecture.pdf) [Lawrence and Livingston, 1932](https://doi.org/10.1103/PhysRev.40.19) [Read the experimental history](https://sandboxphysics.com/stories/nuclear-particle/cyclotron-resonant-acceleration/)

Physics tutorial

## Charged particles: electric fields change energy, magnetic fields turn motion

**Background** A charged particle feels both electric and magnetic fields. The electric force lies along the electric field; the magnetic force comes from the cross product of the velocity and the magnetic field.

**Why it matters** Mass spectrometers, cyclotrons, plasma confinement, and the Hall effect all use the same Lorentz force.

Start with the essentials

- **Focus question**: Why can a pure magnetic field bend a path into a circle without making the particle faster?
- **One-sentence intuition**: Magnetic force stays perpendicular to velocity and does no work; the electric field directly changes kinetic energy.

### Core mathematical model

#### Checking work against kinetic energy

$$
K(t)-K(0)=q\,\mathbf E\cdot[\mathbf r(t)-\mathbf r(0)]
$$

In a uniform field, the electric work depends only on the displacement. Compare it with the change in kinetic energy computed from the recorded velocity. The magnetic field does no work.

#### Arrive at the gap in phase

$$
\omega_c=\frac{|qB_z|}{m},\qquad\Delta K=qV_0\cos(\omega_{\mathrm{RF}}t+\phi_0)\,\mathrm{sgn}(v_x)
$$

A nonrelativistic particle takes the same time for each magnetic half-turn. Reversing the gap field at that rhythm adds energy at every crossing. Detuning eventually presents a decelerating phase. The gap supplies instantaneous work; transit time is omitted.

#### Lorentz force

$$
\mathbf F=q(\mathbf E+\mathbf v\times\mathbf B)
$$

The cross product sets both the size and the direction of the magnetic force. Flipping the sign of the charge reverses the total force.

#### Gyration radius and drift velocity

$$
r_c=\frac{mv_\perp}{|q|B},\qquad\mathbf v_D=\frac{\mathbf E\times\mathbf B}{B^2}
$$

A heavier particle or a faster perpendicular speed gives a larger orbit. The $\mathbf E\times\mathbf B$ drift does not depend on the mass or on the sign of the charge, as long as the charge and the magnetic field are nonzero.

### Common difficulties

#### A magnetic field does not increase speed

**Typical misconception** A force that keeps acting must speed the particle up.

**Better mental model** A force perpendicular to the velocity only turns it, so the power $q(\mathbf v\times\mathbf B)\cdot\mathbf v$ is always zero.

#### A negative charge reverses the right-hand rule

**Typical misconception** $\mathbf v\times\mathbf B$ is the force direction for every particle.

**Better mental model** $\mathbf v\times\mathbf B$ gives the force direction for a positive charge; when $q<0$, the force points the other way.

### Run the experiment

1. 01

   #### Compare the RF rhythm

   Compare In step, Losing the rhythm and A quarter-cycle late with the recorded gap voltages.

   **What to observe:** Resonance expands the orbit; detuning eventually removes energy; a quarter-cycle offset places the crossings at voltage zeros.

2. 02

   #### Start with pure circular motion

   Choose Pure magnetic circle and drag through the recorded time.

   **What to observe:** The speed stays the same, and the path bends into a circle.

3. 03

   #### Reverse the charge

   Press Reverse charge, then compare the recorded paths.

   **What to observe:** The particle circles the other way, with about the same radius.

4. 04

   #### Add a perpendicular electric field

   Choose Crossed-field drift, move the electric-field slider.

   **What to observe:** The center of the orbit starts to move steadily sideways.

Physical conclusion

Start by separating the forces that change energy from the forces that only change direction.

## Related experiments and articles

Particles and detectors · Reading path

### Detecting a passing particle

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Lenard lets cathode rays out of the tube**](https://sandboxphysics.com/stories/particle-evidence/lenard-cathode-ray-window/)
2. [Article **Thomson finds a common particle in cathode rays**](https://sandboxphysics.com/stories/particle-evidence/thomson-electron-charge-mass/)
3. [Article **Wilson turns ionisation into a visible track**](https://sandboxphysics.com/stories/particle-evidence/wilson-cloud-chamber/)
4. [Article **Glaser makes a particle trail start the boiling**](https://sandboxphysics.com/stories/particle-evidence/glaser-bubble-chamber/)

Particles and detectors · Reading path

### Finding a new particle

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **The lead plate that revealed a positron**](https://sandboxphysics.com/stories/particle-evidence/anderson-positron-lead-plate/)
2. [Article **Finding the antiproton among thousands of pions**](https://sandboxphysics.com/stories/particle-evidence/chamberlain-antiproton/)

Nuclei and particles · Reading path

### The nucleus and its reactions

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How a cyclotron gives the same ion another push**](https://sandboxphysics.com/stories/nuclear-particle/cyclotron-resonant-acceleration/)

Starter path · Step 2 of 6

## From a field to an electron microscope

[Next step → **Electron Wavelength & Voltage** Raise the accelerating voltage and compare the classical and relativistic wavelengths of the electron beam. Transfer · 18 min](https://sandboxphysics.com/labs/electron-wavelength/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)