Why does a magnetic field bend a charged particle without changing its energy?

For motion perpendicular to a magnetic field, the magnetic Lorentz force is perpendicular to the velocity. It changes momentum direction but does no work, so kinetic energy and speed stay constant. Relativistically, r = p/(|q|B) with p = γmv.

What makes a velocity selector work?

With mutually perpendicular E, B and beam velocity, the electric and magnetic forces cancel at v = E/B. The selected speed is independent of mass and of the magnitude or sign of a nonzero charge; reversing charge reverses both forces together.

How does the 180° sector separate isotopes?

After the same accelerating voltage, ions with different m/q follow different radii. In the ideal semicircular geometry used here, the image position shifts by approximately twice the radius change, so the detector spacing is Δx ≈ 2Δr. Real sector instruments also depend on slit width, energy spread, angular spread and magnetic aberrations.

Why does a classical cyclotron eventually lose synchronism?

The classical frequency |q|B/(2πm) is speed-independent only in the nonrelativistic limit. As γ rises, the orbital frequency falls to |q|B/(2πγm), so a fixed RF frequency accumulates phase error. Synchrocyclotrons vary RF frequency; isochronous cyclotrons shape B with radius.

What does m/Δm mean here?

The page shows the nominal isotope mass divided by the isotope mass difference as an intuitive resolving-power comparison. Real mass-spectrometer resolving power depends on the instrument’s peak-width/separation convention and its optics, so this number is not a prediction that two peaks will be resolved by a real machine.

Drawing No. EH–GE–002 // General Engineering

Charged Particle, Mass Spectrometer & Cyclotron Simulator

Reviewed August 2026

One force law — F = q(E + v × B) — produces a straight beam, a circle, a mass separation and a particle accelerator, depending only on how you arrange the fields. Change the field, the accelerating voltage, the mass and the charge, and watch the radius, the frequency, the separation and the energy respond.

Educational model. Ideal uniform fields, no space charge, no collisions, no fringing at the pole edges, and ideal 180° sector geometry. Relativistic correction is applied to the free magnetic orbit and cyclotron. The sector, selector-energy and time-of-flight teaching equations use their nonrelativistic forms, with the page identifying that scope explicitly.

A charged particle in a uniform field

the force is always perpendicular to the motion

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Visual variant A · refined technical / 2.5D apparatus view

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Parameters

Live readout

What to watch for

The magnetic force never changes the speed

On the first tab, watch the speed readout while the particle loops. It does not move. q v × B is always perpendicular to v, so it does no work — the field can steer a beam arbitrarily hard without adding a single electron-volt of energy.

The selector does not care about mass

v = E/B contains no m and no q. Change the particle on the velocity-selector tab and the transmitted speed does not move. That is exactly why it goes before the analyser: it fixes the speed so the following magnet sorts purely on mass-to-charge.

Radius sorts momentum, not mass

r = p/(|q|B). Two ions with the same momentum and the same charge magnitude follow the same radius. After acceleration through a known voltage, the nonrelativistic sector relation makes radius a measure of m/q.

Frequency is independent of energy — until it is not

On the cyclotron tab, raise the dee radius and watch f_c hold constant while energy climbs, then watch the relativistic frequency start to fall away from it. The phase slip readout shows how quickly a fixed-frequency machine loses step.

Separation is a square-root problem

Because r ∝ √m, heavy isotopes are harder to separate than light ones. Compare ¹²C against ¹³C with ²³⁵U against ²³⁸U on the sector tab: a 1 u difference buys far less at mass 238 than at mass 12.

Time of flight favours the light

t ∝ √m, so light ions arrive first and the peaks crowd together as mass rises. Lengthening the drift tube spreads them linearly; raising the accelerating voltage compresses them all.

Where this is used

Application	What the field is doing	The limiting parameter
Analytical mass spectrometry	Sorting ionised fragments by m/z to identify a compound	Resolution, set by slit width, beam divergence and field uniformity
Isotope separation	Physically routing two isotopes to different collectors	The √m scaling: heavier elements need larger machines or more stages
Medical cyclotrons	Accelerating protons or deuterons to make short-lived PET isotopes	Magnet field, extraction radius, RF synchronism and target/beam requirements
Proton therapy	Delivering a beam whose range in tissue is set by its energy	Energy stability and beam steering precision
Residual gas analysis and leak detection	Watching for a specific m/z, usually helium	Sensitivity rather than resolution
Cathode-ray and electron-beam devices	Deflecting an electron beam with magnetic or electric fields	Deflection sensitivity against beam energy
Space weather and radiation belts	Particles gyrate around and follow Earth's magnetic-field lines; nonuniform mirror geometry helps form radiation-belt trapping	Field geometry rather than engineering

The common thread. Every one of these is the same equation with a different constraint held fixed — speed in the selector, momentum in the sector, frequency in the cyclotron, and time in a drift tube.

Frequently asked questions

Key interpretation points behind the five simulator modes.

Sources and technical basis

Motion follows the Lorentz force law. Fundamental constants use the 2022 CODATA recommended values; isotope presets are educational reference values as noted below. Technical content reviewed 16 August 2026.

OpenStax — University Physics Volume 2Magnetic forces and fields: motion of a charged particle, velocity selectors, mass spectrometers and the cyclotron.NIST — CODATA fundamental constantsElementary charge, atomic mass unit, electron and proton mass, and the speed of light used throughout this page.NIST — atomic weights and isotopic compositionsReference relative isotope masses used for the preset ion pairs; the NIST compilation cites AME2012 for isotope masses, so these presets are illustrative rather than a current precision-mass database.CERN — SynchrocyclotronAccelerator background, including CERN’s synchrocyclotron; CERN describes a synchrocyclotron as varying RF frequency to compensate relativistic effects.