Charged Particle in a Magnetic Field Simulator
Equations in this simulation
| q | Charge q (in units of e) | e = 1.602 × 10⁻¹⁹ C; the sign decides which way it turns | |
|---|---|---|---|
| v | Speed v (m/s, as a power of 10) | ||
| B | Magnetic field B (tesla) | the arrows; a fridge magnet is about 0.01 T, an MRI scanner 1.5 to 3 T | |
| F | Lorentz force | always at right angles to v and B, so the magnetic part bends the path but never changes the speed |
With the current values:
| m | Mass m (atomic mass units u, as a power of 10) | 1 u = 1.6605 × 10⁻²⁷ kg | |
|---|---|---|---|
| v⊥ | speed across the field | v sin(angle); the part along B is untouched, which stretches the circle into a helix | |
| γ | relativistic factor 1 ÷ √(1 − v²/c²) | close to 1 below a tenth of the speed of light | |
| r | radius of the circle (gyroradius) |
With the current values:
| f | cyclotron frequency | the same whatever the speed (until γ grows), which is what lets a cyclotron accelerate particles with a fixed-frequency voltage | |
|---|---|---|---|
| T | period of one turn |
With the current values:
| h | distance along B per turn | v cos(angle) × T | |
|---|---|---|---|
| v_drift | E × B drift speed | with an electric field across B every charged particle drifts sideways at E ÷ B, the same for all of them; a velocity selector lets through only the particles moving at exactly this speed |
With the current values:
How to use the charged particle in a magnetic field simulator
- Pick a proton, an electron, an alpha particle or your own charge and mass, and set its speed, the field strength and the angle between its velocity and the field. The blue arrows are the field; the green arrow on the particle is its velocity and the red one the Lorentz force.
- At 90° the particle goes round a circle; at smaller angles the part of the velocity along the field carries it along while it circles, tracing a helix. Read the radius, the period, the cyclotron frequency and the pitch, worked through below with your numbers. The particle is redrawn at a fixed size, so a slow electron and a fast alpha particle look alike; the radius under the picture shows how different they are.
- Add an electric field across the magnetic one: the circle now drifts sideways at E ÷ B, the same speed for every particle, which is how a velocity selector works. The Particle Collider Simulator shows the same bending used to measure the momentum of particles in a detector.
Frequently asked questions
What is the Lorentz force?
The force on a charge q moving with velocity v through an electric field E and a magnetic field B: F = q(E + v × B). The magnetic part is at right angles to both the velocity and the field, so it changes the particle's direction but never its speed or energy. A charge at rest, or moving along the field, feels no magnetic force at all.
How do I find the radius of the circle?
r = m v ÷ (|q| B), from setting the magnetic force qvB equal to the centripetal force mv²/r. Faster or heavier particles make wider circles, stronger fields and bigger charges tighter ones. A proton at 10⁶ m/s in 0.5 T circles with a radius of about 2 cm; an electron at the same speed, about 11 µm.
Why does the cyclotron frequency not depend on the speed?
Because a faster particle travels a proportionally larger circle: the radius grows with v, the circumference with it, and the time per turn, 2πm ÷ (qB), stays the same. A cyclotron uses this to accelerate particles with an alternating voltage of fixed frequency as they spiral outward. Near the speed of light the mass term γm grows and the timing drifts, which is why larger machines are synchrotrons.
What is the E × B drift?
With an electric field across the magnetic field, the particle speeds up on one half of its circle and slows down on the other, so its path is wider on one side and it creeps sideways, at right angles to both fields, at v = E ÷ B. Positive and negative particles drift the same way at the same speed. A velocity selector sets E and B so that only particles with v = E ÷ B go straight through.
It says WebGL is turned off.
The 3D view needs WebGL, which every current browser has. It can be switched off by hardware acceleration being disabled in the browser settings, or by a very old graphics driver. Turn hardware acceleration on, or try another browser.
Is anything uploaded?
No. The simulation is drawn by your own browser with WebGL; nothing is sent anywhere, and it keeps working offline once the page has loaded.