Microsecond proton-like timing
A 1 u, +1 e particle in 200 mT cycles at about 3.07 MHz, completing roughly 30.7 turns in a 10 microsecond gate.
Physics and electromagnetism
Magnetic Force Rate Calculator
Calculate nonrelativistic cyclotron angular rate, frequency, period, and turns during an observation interval from particle mass, charge state, and magnetic field.
Charged-particle turning rate
Convert magnetic curvature into a cyclotron clock
Here rate means the angular and cyclic rate of ideal nonrelativistic gyromotion in a uniform magnetic field. It is not force per time or changing field strength; the frequency follows the particle charge-to-mass ratio and B.
Cyclotron frequency-
Angular rate-
Cyclotron period-
Turns in interval-
Current model evidence
Cyclotron-rate conversion ledger
Compare period with instrument timing and use turns only within the entered observation window.
| Rate stage | Numerator / input | Denominator / conversion | Current value | Scope / unit |
|---|
DETAILED CALCULATION PROCESS
Formula, units, default substitution, and reconciliation
1. Governing relation
omega_c = |q|B/m; f_c = omega_c/(2 pi); T_c = 1/f_c; N = f_c t_obs
Normalize mass, charge, B, and observation time to SI; evaluate angular frequency; convert radians per second to cycles per second; invert for period; then count cycles in the window.
2. Symbols and default basis
| Symbol | Meaning | Unit | Default basis |
|---|---|---|---|
| m | Particle mass | kg | 1 u |
| q | Signed charge | C | +1 e |
| B | Uniform magnetic-field magnitude | T | 200 mT |
| omega_c | Cyclotron angular rate | rad/s | Solved |
| f_c | Cyclotron cycle frequency | Hz | Solved |
| t_obs | Observation interval | s | 10 microseconds |
3. Unit and sign normalization
- Mass in u uses the CODATA atomic mass constant.
- Charge number uses the exact elementary charge and an absolute value in rate magnitude.
- B is divided by 1000 and microseconds by 1e6 before calculating rate and turns.
4. Current numerical substitution
5. Independent reconciliation
HOW TO USE THIS CALCULATOR
Five steps from measured inputs to a defensible result
- Enter the actual particle or ion mass rather than its mass number when precision matters.
- Enter the signed charge state; keep the sign as evidence even though frequency uses its magnitude.
- Use the local uniform B magnitude in millitesla.
- Set an observation window that represents the detector gate, transit interval, or simulation step being assessed.
- Compare both f and T with the sampling system and use fT = 1 as the independent check.
ELECTROMAGNETIC FOUNDATIONS
Concepts that control this specific model
- Rate is independent of speed classically
- For ideal nonrelativistic circular motion, faster speed increases radius but not cyclotron frequency.
- Charge-to-mass ratio controls response
- Larger |q| or B increases rate; larger mass decreases it.
- Angular rate differs from frequency
- omega is radians per second, while f counts complete cycles per second.
- Period is the reciprocal clock
- T is the time for one ideal gyration and must satisfy fT = 1.
- Charge sign selects handedness
- Positive and negative charges rotate oppositely in the same B while sharing the same rate magnitude.
DEEP ANALYSIS 1
Why speed cancels out
Equating magnetic force |q|vB with centripetal force mv^2/r gives r = mv/(|q|B); dividing circumference by speed removes v from the period.
DEEP ANALYSIS 2
Observation turns need not be an integer
A detector gate can end partway through a cycle. The decimal turn count is phase advance divided by 2 pi, not a count of completed events.
DEEP ANALYSIS 3
Relativity breaks the speed independence
At high speed the momentum factor gamma raises the effective orbital inertia and lowers the observed cyclotron rate. This page intentionally stops at the classical formula.
RESULT INTERPRETATION
What the current output does and does not decide
Frequency and period characterize ideal local gyromotion, not collision rate, detector count rate, or magnetic-field switching rate.
Turns estimate accumulated phase only if mass, charge state, and B remain constant throughout the observation interval.
REAL USE CASES
Two decisions with different boundary conditions
Heavy-ion timing contrast
At the same B and charge magnitude, a 40 u ion rotates forty times more slowly, which may move its period outside an instrument’s sampling window.
EVIDENCE AND DATA QUALITY
What to retain with the exported result
Retain species and charge-state assignment, local B calibration and polarity, observation-window clock source, nonrelativistic justification, and exported reciprocal check. Document any field nonuniformity across the expected orbit.
LIMITS AND EXCLUSIONS
Where this physical model stops
- Uses a nonrelativistic single-particle cyclotron formula.
- Assumes uniform, static B and constant mass and charge state.
- Excludes electric fields, collisions, radiation losses, field gradients, and relativistic mass effects.
- Turns represent phase accumulation, not guaranteed detector events.
- Does not calculate radius because perpendicular speed is not an input.
TERMS USED HERE
Six terms that keep the calculation unambiguous
- Cyclotron frequency
- Number of ideal gyromotion cycles per second.
- Angular rate
- Phase advance in radians per second.
- Cyclotron period
- Time required for one complete ideal cycle.
- Charge state
- Signed multiple of elementary charge carried by a particle or ion.
- Gyromotion
- Circular velocity component about a magnetic-field line.
- Observation window
- Specified time interval over which phase or turns are accumulated.
RELIABLE SOURCES
References supporting the equation and units
FREQUENTLY ASKED QUESTIONS
Questions specific to this calculation
Why is particle speed not an input?
It cancels from the classical uniform-field period derivation; speed changes radius instead.
Why does a negative charge give a positive frequency?
Frequency is a nonnegative rate magnitude; the sign is reported separately as rotation sense.
Can the charge be zero?
No. An uncharged particle has no cyclotron motion, so frequency and period are not defined by this model.
What is the difference between Hz and rad/s?
Hz counts cycles per second; rad/s counts angular phase, with omega = 2 pi f.
Do 30.7 turns mean 30 detected particles?
No. It describes one particle’s phase accumulation, not an event count.
Can this model a relativistic cyclotron?
No. Relativistic momentum changes the rate and requires a different equation.
IMPORTANT BOUNDARY
Use the result as analysis, not certification
This ideal cyclotron timing estimate is not a detector calibration, accelerator synchronization design, plasma collision model, or relativistic frequency calculation.