Lorentz Force & Charged Particle Simulator

The fundamental law of motion is Newton's second law with the Lorentz force as the driving mechanism. This gives us the equation of motion for the charged particle.

Here, $m$ is particle mass (kg), $q$ is electric charge (C), $\vec{v}$ is velocity (m/s), $\vec{E}$ is the electric field (V/m), and $\vec{B}$ is the magnetic flux density (T). The cross product $\vec{v}\times \vec{B}$ ensures the magnetic force is perpendicular to the plane containing $\vec{v}$ and $\vec{B}$.

For the special case of a uniform magnetic field ($\vec{B}=B\hat{z}$) and no electric field ($\vec{E}=0$), the motion is purely circular in the plane perpendicular to $\vec{B}$. Two key quantities define this motion:

$r_c$ is the cyclotron (or Larmor) radius — the radius of the circular orbit. $v_\perp$ is the speed component perpendicular to $\vec{B}$. $\omega_c$ is the cyclotron angular frequency (rad/s). The rotation period is $T = 2\pi / \omega_c$. Notice how a heavier particle (like a proton) has a larger radius and slower frequency than a light electron in the same field, which you can test directly with the "Particle Type" dropdown.

Cyclotron & Synchrotron Design: These particle accelerators use a constant magnetic field to bend charged particles into circular paths while an oscillating electric field accelerates them at just the right frequency ($\omega_c$). Engineers use these exact equations to design the magnet size and RF frequency. The simulator's frequency calculation shows why heavier particles require lower frequency accelerators.

Magnetic Resonance Imaging (MRI): The core principle involves the precession (a form of gyration) of proton spins in a massive static magnetic field. The resonant frequency at which these spins absorb energy is directly proportional to the field strength ($\omega \propto B$), just like the cyclotron frequency. Adjusting the B-field changes the calculated frequency, mimicking how different MRI machines operate.

Plasma Confinement in Fusion Devices: To contain a 100-million-degree plasma, powerful magnetic fields are used. The charged particles in the plasma gyrate around magnetic field lines. The E×B drift phenomenon you can create in the simulator is critical for analyzing and controlling plasma stability in devices like tokamaks, as unwanted drifts can lead to energy loss.

Mass Spectrometry: In a magnetic sector mass spectrometer, ions are sent into a known magnetic field. Heavier ions (larger $m$) bend with a larger radius ($r_c \propto m$), while lighter ions bend more sharply. By measuring the radius of curvature, the mass-to-charge ratio ($m/q$) of an unknown ion can be determined. Try varying the mass and charge sliders independently to see how the orbit radius changes.

First, let's establish the point that "magnetic fields do no work." You can confirm this in the simulator by observing that a particle's speed doesn't change when only a magnetic field is applied. The Lorentz force $q \vec{v} \times \vec{B}$ is always perpendicular to velocity, so it does not increase or decrease the particle's kinetic energy. Only the work done by the electric field can change the energy. In practice, attempting to accelerate a particle using only a magnetic field is a fundamental error.

Next, pay attention to the relationship between the initial velocity direction and the magnetic field direction. The velocity component parallel to the magnetic field $v_{\parallel}$ is not bent at all, causing the particle to undergo helical motion. If you enter a value for "initial velocity vz" in the simulator, you'll see it moving in the Z-direction while performing circular motion. The pitch of this helix is determined by $v_{\parallel}$, so setting the initial conditions correctly is crucial for accurately guiding a particle beam.

Finally, beware of confusion between unit systems. The tesla (T) unit for magnetic field is particularly unintuitive and a common source of mistakes. For example, 0.01 T is about twice the Earth's magnetic field, 0.1–0.5 T is typical at the surface of a common permanent magnet, and MRI machines use 1.5 T or 3 T. In the simulator, setting "proton, velocity 1e6 m/s, magnetic field 1 T" yields a gyroradius of about 10 mm. Misreading this as "1 m" would severely throw off your design. Make it a habit to always check the order of magnitude.