---
title: Magnetic Trajectories
module: Magnetic Field
moduleNumber: 6
lessonNumber: 1
order: 601
summary: >
  A charged particle in a magnetic field never speeds up or slows down, yet its
  path curves relentlessly. We work out why: the magnetic force is always
  perpendicular to velocity, so it does no work and bends the transverse motion into
  a circle of radius $r=mv_\perp/(|q|B)$ while leaving the parallel motion untouched,
  producing a helix. We derive the cyclotron frequency, show why it is independent of
  speed until relativity intervenes, and turn the geometry around: a measured curvature
  reads back a particle's momentum, which is how tracking detectors weigh what they
  cannot see.
topics: [Magnetic Field]
draft: false
sources:
  - book: Tipler & Mosca
    ref: "Ch. 26 — The Magnetic Field; §§26-1–26-2"
---

## Transverse motion and the gyroradius

In a uniform magnetic field, resolve a particle velocity into components parallel
and perpendicular to the field:

$$
\vec v=\vec v_\parallel+\vec v_\perp,
\qquad
\vec v_\parallel\parallel\vec B,
\qquad
\vec v_\perp\mathbin{\cdot}\vec B=0.
$$

Only the transverse component appears in the magnetic-force magnitude. The force
has magnitude $|q|v_\perp B$ and remains perpendicular to the instantaneous
transverse velocity. For a nonrelativistic particle, equating that magnitude to
the required centripetal force gives

$$
|q|v_\perp B=\frac{mv_\perp^2}{r_L},
\qquad
r_L=\frac{mv_\perp}{|q|B}.
$$

The radius $r_L$ is often called the gyroradius or Larmor radius. It grows with
transverse momentum and shrinks with charge magnitude or field strength. A sign
change of charge reverses the sense of rotation without changing the radius. The
field direction fixes the orbit plane: a magnetic field along the page normal gives
a circular projection in the page, whereas a field tilted in space gives a circle
in the plane perpendicular to that tilted direction.

$$
% caption: Circular orbit of a positive charge in a uniform magnetic field pointing out of the page (dot symbols). The velocity is tangent to the circle and the magnetic force points to the centre, so the orbit is traced clockwise; a charge of opposite sign circulates the other way around a circle of the same radius $r_L=mv_\perp/(|q|B)$.
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$$

The force has no component along the velocity, so it does no mechanical work:

$$
\frac{\d K}{\d t}=\vec F_B\mathbin{\cdot}\vec v
=q(\vec v\times\vec B)\mathbin{\cdot}\vec v=0.
$$

The transverse speed stays constant, and the circle is traversed at a constant
angular speed. This conservation statement holds at each point of the path. Momentum
direction rotates continuously while its magnitude and kinetic energy remain
constant.

**Angular frequency, period, and orientation.**

Dividing the centripetal-force relation by $mv_\perp$ gives the angular frequency

$$
\omega_c=\frac{|q|B}{m},
\qquad
T_c=\frac{2\pi m}{|q|B},
\qquad
f_c=\frac{|q|B}{2\pi m}.
$$

In the nonrelativistic uniform-field model, neither frequency nor period depends on
orbit radius or speed. A faster particle traces a larger circle in the same period.
This feature permits frequency-based mass-to-charge measurements, while the radius
gives a separate momentum measurement. Relativistic momentum changes both
relations at high speed and is treated by replacing $m$ with the energy-dependent
factor $\gamma m$.

> **Worked example (Electron in a 0.10 T field).** An electron moves at
> $v=1.0\times10^7\ \mathrm{m\,s^{-1}}$ perpendicular to a uniform
> $B=0.10\ \mathrm{T}$ field. Find its orbit radius and revolution frequency. With
> $m_e=9.11\times10^{-31}\ \mathrm{kg}$ and $e=1.60\times10^{-19}\ \mathrm{C}$,
>
> $$
> r_L=\frac{m_ev}{eB}
> =\frac{(9.11\times10^{-31})(1.0\times10^7)}{(1.60\times10^{-19})(0.10)}
> =5.7\times10^{-4}\ \mathrm{m},
> $$
>
> a radius under a millimetre. The frequency is set by the field alone, not by the
> speed,
>
> $$
> f_c=\frac{eB}{2\pi m_e}
> =\frac{(1.60\times10^{-19})(0.10)}{2\pi(9.11\times10^{-31})}
> =2.8\times10^{9}\ \mathrm{Hz}.
> $$
>
> Doubling the speed would double the radius but leave $f_c$ unchanged; the electron
> simply covers a bigger circle in the same $3.6\times10^{-10}\ \mathrm{s}$ period.

Direction questions need a defined coordinate system. Put $\vec B$ along $+z$.
Take a positive particle initially moving along $+x$. The cross product
$\vec v\times\vec B$ points along $-y$, so the orbit begins by turning toward
negative $y$. An electron with the same velocity turns toward positive $y$. Apply
the right-hand rule to the vector product first, then multiply by the signed charge;
combining those two steps mentally is a frequent source of reversed tracks.

## Helical trajectories and pitch angle

The component parallel to $\vec B$ experiences zero magnetic force and therefore
remains constant. Combining uniform parallel translation with transverse circular
motion gives a helix. If $\alpha$ is the pitch angle between velocity and field,

$$
v_\perp=v\sin\alpha,
\qquad
v_\parallel=v\cos\alpha,
\qquad
p=v_\parallel T_c.
$$

Here $p$ is the axial advance during one complete turn, conventionally called the
pitch of the helix. The radius depends on $v\sin\alpha$, whereas the pitch depends
on $v\cos\alpha$. A trajectory nearly parallel to the field has small radius and
large pitch; a trajectory nearly perpendicular to the field approaches a circle.
Both limiting cases follow from the same component equations.

$$
% caption: Helical path of a charge in a uniform magnetic field. The velocity component along $\vec B$ is unchanged and advances the orbit by one pitch $p$ each period, while the transverse component drives the circular motion of radius $r_L$.
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\draw[<->,thick] (2.06,0.58)--(3.11,0.58) node[midway,below] {$p$};
\end{tikzpicture}
$$

The uniform-field helix is an ideal local model. A particle in a slowly changing
magnetic field can be followed by a guiding-centre description, but gradients and
curvature modify the orbit centre and may produce drifts. Collisions randomize
velocity components, and an electric field parallel to the magnetic field changes
particle energy. Each effect has a distinct signature in a measured track, so a
helix fit should report the fitted interval and the field-uniformity range.

## Reconstructing a trajectory from measurements

A cloud chamber, bubble chamber, tracking detector, or plasma diagnostic records
positions rather than force vectors. In a known uniform field, fit a circle to the
projection perpendicular to the field. The fitted curvature gives

$$
p_\perp=|q|Br_L.
$$

If the instrument also measures the advance along the field, the ratio of pitch to
circumference gives $v_\parallel/v_\perp$. A sequence of three-dimensional points
can then distinguish a low-momentum tightly curved track from a high-momentum broad
track. The charge sign follows from curvature only after the magnetic-field
orientation and the particle travel direction have both been established.

Finite spatial resolution biases a small-radius fit because only a few detector
samples span the orbit. Multiple scattering can make a high-momentum trajectory
appear more curved, while energy loss changes radius along a long path. A robust
analysis fits short segments, compares curvature across adjacent segments, and
propagates the field-calibration uncertainty into momentum. The radius formula
therefore provides the physical relation; it does not replace the measurement model.

## Equations of motion and phase-space description

Take a uniform, time-independent magnetic field in the positive z direction and
write the transverse velocity as $\vec v_\perp=(v_x,v_y)$. The Lorentz force has
no z component, so the transverse Newton equations are

$$
m\frac{\d v_x}{\d t}=qBv_y,
\qquad
m\frac{\d v_y}{\d t}=-qBv_x.
$$

With the signed cyclotron frequency $\Omega=qB/m$, differentiating either equation
once gives a simple harmonic equation: $\d ^2v_x/\d t^2+\Omega^2v_x=0$, and likewise
for $v_y$. The two components are not independent oscillators. One is shifted by a
quarter cycle relative to the other because each component's rate of change is set by
the other component. A convenient phase choice is

$$
v_x=v_\perp\cos(\Omega t+\phi),
\qquad
v_y=-v_\perp\sin(\Omega t+\phi).
$$

The signed frequency $\Omega=qB/m$ determines the rotation direction for any
field and coordinate orientation. Changing the sign of charge reverses that
rotation without changing its speed or radius.

The transverse velocity components differ by one quarter cycle, which keeps
$v_\perp$ constant. When $v_x$ has its positive maximum, $v_y=0$ and
$\d v_y/\d t=-\Omega v_x$. For positive charge in a positive z-directed field, the
velocity turns from positive x toward negative y; negative charge reverses that
rotation.

Initial transverse position and velocity determine four constants: two centre
coordinates, the speed $v_\perp$, and phase $\phi$. For example, measured values
at a reference time give $x_c=x+v_y/\Omega$ and $y_c=y-v_x/\Omega$, after which
the phase follows from the normalized velocity components. There is no independent
radial acceleration to solve for; the apparent centripetal acceleration is already
contained in the coupled Cartesian equations. A numerical integrator should preserve
the circular velocity locus and the fixed centre over many periods. A scheme that
updates one component with a new value and the other with an old value may introduce
an artificial growth or decay of speed, even though the exact magnetic force has no
such effect.

Integrating the velocity components gives the transverse position in a form that
makes the orbit centre explicit:

$$
x=x_c+\frac{v_\perp}{\Omega}\sin(\Omega t+\phi),
\qquad
y=y_c+\frac{v_\perp}{\Omega}\cos(\Omega t+\phi).
$$

Equivalently, the combinations $x+v_y/\Omega$ and $y-v_x/\Omega$ remain constant.
They are the centre coordinates $x_c$ and $y_c$ for the chosen sign convention.
The geometric radius is $r_L=v_\perp/|\Omega|=mv_\perp/(|q|B)$. These relations
separate a particle's instantaneous position from the fixed centre about which it
rotates, a distinction needed when comparing tracks with different starting phases.

In velocity phase space, the point $(v_x,v_y)$ moves around a circle of radius
$v_\perp$ at angular rate $|\Omega|$. The phase-space circle is conserved because
the magnetic force does no work: $\vec v\mathbin{\cdot}q(\vec v\mathbin{\times}\vec B)=0$.
Thus transverse kinetic energy, total speed, and the parallel velocity are constant
in this ideal field. A correct numerical integration preserves the velocity-phase
circle, guiding centre, and speed.

## Relativistic motion and the limits of the cyclotron period

At relativistic speed, the momentum rather than the velocity is the natural
quantity in the magnetic equation of motion:

$$
\frac{\d \vec p}{\d t}=q\vec v\mathbin{\times}\vec B,
\qquad
\vec p=\gamma m\vec v,
\qquad
\gamma=\frac{1}{\sqrt{1-v^2/c^2}}.
$$

A static magnetic field remains perpendicular to velocity, so it does no work and
keeps $\gamma$ constant during an individual orbit. For motion perpendicular to a
uniform field, the momentum magnitude is constant and the curvature relation becomes
$p_\perp=|q|Br$. Replacing $mv_\perp$ by $p_\perp=\gamma mv_\perp$ gives the
relativistic angular frequency and period:

$$
\omega_c=\frac{|q|B}{\gamma m},
\qquad
T_c=\frac{2\pi\gamma m}{|q|B}.
$$

The orbit radius grows with momentum, while the revolution frequency falls as the
Lorentz factor grows. The familiar mass-independent nonrelativistic frequency is
therefore an approximation valid only when kinetic energy is small compared with
rest energy. Its failure is not a change in the magnetic force law; it follows from
the increasing relation between momentum and velocity.

The momentum substitution also clarifies what remains unchanged. The transverse
curvature equation still reads force equals rate of momentum-direction change, so a
measured radius continues to determine transverse momentum through $p_\perp=|q|Br$
without a separate velocity estimate. What changes is the conversion from that
momentum to speed and orbital rate. For a particle with a parallel velocity
component, the transverse rotation frequency has the same $1/\gamma$ reduction
because the total energy fixes one common Lorentz factor. The particle follows a
helix whose pitch is enlarged by the longer period, while its transverse radius is
set by transverse momentum. These distinctions matter in track reconstruction: a
large radius can represent high momentum, but timing information is needed to test
the accompanying relativistic frequency shift.

$$
% caption: Relativistic fall of the cyclotron frequency. A fixed drive frequency (dashed) stays flat, while the true revolution rate $|q|B/(\gamma m)$ decreases as energy and the Lorentz factor rise; the widening gap is the phase slip a fixed-frequency cyclotron must correct.
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$$

A conventional cyclotron drives particles across an accelerating gap with a fixed
radio-frequency voltage. At low energy, the orbital period matches the RF period,
so a particle reaches the gap when the electric field has the accelerating polarity
on every turn. As energy rises, $\gamma$ increases, the particle takes longer to
return, and its arrival phase lags the fixed RF waveform. This phase slip eventually
places the particle in a weakly accelerating or decelerating part of the cycle.
Increasing the magnetic field alone changes the frequency in the opposite direction
only if it is programmed to compensate for the increase in $\gamma$.

A synchrocyclotron makes that correction by sweeping the RF frequency downward as
the orbit frequency falls. The drive is synchronized to an accelerating packet over
a programmed energy range rather than held at one frequency for every radius. The
method restores phase coherence for particles in the selected packet, although its
repeated frequency sweep gives lower average beam current than a fixed-frequency
machine. More elaborate synchrotrons also change magnetic field and RF frequency
together, keeping a chosen closed orbit while momentum increases.

Measure the revolution interval, track radius, and RF arrival phase to test
relativistic phase slip. Track radius in a calibrated field gives momentum, so
plotting revolution frequency against $p$ tests
$\omega_c=|q|B/(\gamma m)$. At an RF gap, phase-sensitive beam monitors compare
arrival time with drive phase. A steadily drifting phase, broadened arrival-time
distribution, or loss of energy gain indicates inadequate frequency programming.

## Magnetic rigidity and bending measurements

A uniform magnetic field converts transverse momentum into a geometric curvature.
The relation

$$
p_\perp=|q|Br_L
$$

is often written in accelerator and detector work as magnetic rigidity. A particle
with large momentum requires either a large radius or a stronger field to bend by a
specified amount. The relation contains no kinetic-energy approximation when
$p_\perp$ is interpreted as relativistic momentum. It is therefore a practical
bridge between a measured track shape and a particle momentum scale.

> **Worked example (Radius of a 1.0 MeV proton).** A proton with kinetic energy
> $K=1.00\ \mathrm{MeV}=1.60\times10^{-13}\ \mathrm{J}$ moves perpendicular to a
> uniform $B=1.00\ \mathrm{T}$ field. Because $K\ll m_pc^2\approx938\ \mathrm{MeV}$,
> nonrelativistic momentum is accurate. Get the momentum from the energy,
>
> $$
> p=\sqrt{2m_pK}
> =\sqrt{2(1.67\times10^{-27})(1.60\times10^{-13})}
> =2.31\times10^{-20}\ \mathrm{kg\,m\,s^{-1}},
> $$
>
> then read the radius straight off the rigidity relation,
>
> $$
> r_L=\frac{p}{eB}=\frac{2.31\times10^{-20}}{(1.60\times10^{-19})(1.00)}
> =0.144\ \mathrm{m}.
> $$
>
> A 14 cm orbit for a 1 MeV proton in a 1 T field. Note that the radius came from
> momentum alone; had the proton instead been relativistic, the same fitted radius
> would still give $p=eBr_L$ directly, only the conversion of $p$ back to speed would
> change.

Suppose a charged particle crosses a uniform-field region of path length $\ell$ and
changes direction through a small bend angle $\beta$. The circular-arc geometry gives

$$
\beta=\frac{\ell}{r_L}=\frac{|q|B\ell}{p_\perp}.
$$

The field-length product $B\ell$ sets the momentum kick. Doubling field strength,
magnet length, or charge magnitude doubles the bend angle at fixed momentum.
Doubling transverse momentum halves it. A real dipole magnet has fringe regions at
its entrance and exit, so an experiment uses the calibrated integral
$\int B\d \ell$ along the reference path rather than a nominal central
field times a mechanical length.

$$
% caption: Momentum analysis in a uniform bending field. The track enters straight, curves through the magnet gap, and leaves deflected by the bend angle $\beta=|q|\int B\,\d\ell/p_\perp$; a wider gap or stronger field bends a given momentum more.
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$$

Track measurements usually use sagitta instead of an explicitly measured bend angle.
A circular arc observed between two outer measurement points separated by chord
length $L$ has sagitta $s$, the maximum distance from chord to arc. When
$L$ is much smaller than the radius,

$$
s\simeq\frac{L^2}{8r_L},
\qquad
p_\perp\simeq\frac{|q|BL^2}{8s}.
$$

The inverse dependence on sagitta makes high-momentum measurements demanding.
Doubling momentum halves the deflection and halves the small sagitta. Detector
point resolution, mechanical alignment, multiple scattering, and the uncertainty in
the magnetic map can then dominate the fitted curvature. A broad track with a
nearly straight chord is often a less precise momentum measurement than a tightly
curved low-momentum track, even when the broad track crosses more detector layers.

$$
% caption: Sagitta of a curved track. Three measured points fix a chord of length $L$ and a mid-chord offset $s\simeq L^2/(8r_L)$; because $s$ falls as $1/p_\perp$, nearly straight high-momentum tracks give the smallest, hardest-to-measure offset.
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$$

The sign of curvature is as valuable as the magnitude. A field-map convention
specifies the direction of $\vec B$ in the detector coordinate system. The
ordered hit sequence gives the direction of travel. The Lorentz-force direction
then determines the charge sign. Reversing the hit order or using a field map with
the opposite polarity reverses the inferred sign. Reconstruction software should
retain those conventions with the fitted parameters throughout the analysis.

An uncertainty estimate begins with the differential form of the rigidity relation.
With small independent fractional uncertainties, the magnitude estimate is

$$
\left(\frac{\delta p_\perp}{p_\perp}\right)^2
\simeq
\left(\frac{\delta B}{B}\right)^2+
\left(\frac{\delta r_L}{r_L}\right)^2.
$$

The radius uncertainty contains statistical point resolution and systematic
alignment error. A scale error in the magnetic field shifts every momentum in the
same direction; random hit errors broaden individual fits. Keeping those components
separate helps a calibration team decide whether additional events, tighter
alignment, or a better field survey will improve the result.

## Departures from the ideal uniform orbit

The circular and helical formulas assume a static, spatially uniform magnetic field
and no force except the magnetic Lorentz force. A uniform electric field changes
that equation of motion. Its component parallel to velocity can change kinetic
energy, while a transverse component shifts the path away from a circle. A velocity
selector deliberately uses this competition. In an accelerator gap, the electric
field transfers energy and the magnetic field bends the growing momentum; treating
either region as a magnetic-only orbit would miss the actual
energy transfer.

Collisions change a track in a different way. Elastic scattering changes direction
through localized deflections, producing a sequence of circular segments with
slightly different fitted centres. Ionization and radiation lower momentum, so a
particle can curve more tightly later in a detector. A fit that enforces one radius
over a long material path can then average away physical energy loss and underestimate
the uncertainty. Segment-by-segment curvature and a material map distinguish such
effects from a magnetic-field calibration error.

Radiation from accelerated charge is usually tiny for slow heavy particles, yet it
sets an important high-energy limit for light particles in circular machines.
Synchrotron radiation carries energy away while the magnetic force itself remains
perpendicular to instantaneous velocity. The energy loss arises through the emitted
electromagnetic field, not from a direct magnetic-work term. Radio-frequency systems
replace that loss in storage rings, and the required power rises strongly with
particle energy and with tighter bending radius.

Field nonuniformity can be handled locally when the field changes little over one
gyroradius. The instantaneous force remains perpendicular to velocity, but the
circle centre and radius evolve as the particle samples different field strength.
The resulting guiding-centre drifts belong to a broader nonuniform-field treatment.
In a uniform-field measurement, map boundaries and fringe fields should be included
in the propagated trajectory; discarding them because the central region is uniform
changes the effective bending integral.

> **Worked example (Proton helix in a 0.5 T field).** A proton enters a uniform
> $0.500\ \mathrm{T}$ field at speed $v=3.00\times10^6\ \mathrm{m\,s^{-1}}$, its
> velocity making a $30.0^\circ$ pitch angle with the field. Only the transverse
> component sets the radius, so resolve the speed first:
>
> $$
> v_\perp=v\sin30.0^\circ=1.50\times10^6\ \mathrm{m\,s^{-1}},
> \qquad
> v_\parallel=v\cos30.0^\circ=2.60\times10^6\ \mathrm{m\,s^{-1}}.
> $$
>
> With $m_p=1.67\times10^{-27}\ \mathrm{kg}$ and $e=1.60\times10^{-19}\ \mathrm{C}$,
>
> $$
> r_L=\frac{m_pv_\perp}{eB}
> =\frac{(1.67\times10^{-27})(1.50\times10^6)}{(1.60\times10^{-19})(0.500)}
> =3.13\times10^{-2}\ \mathrm{m}.
> $$
>
> The period comes from the cyclotron rate, which is independent of speed and pitch
> angle,
>
> $$
> T_c=\frac{2\pi m_p}{eB}=1.31\times10^{-7}\ \mathrm{s},
> $$
>
> and the axial advance in one turn is the pitch,
>
> $$
> p=v_\parallel T_c=(2.60\times10^6)(1.31\times10^{-7})=3.40\times10^{-1}\ \mathrm{m}.
> $$
>
> Radius and pitch differ by more than an order of magnitude because the proton moves
> mostly along the field. Substituting the full speed $v$ for $v_\perp$ would double
> the radius, the most common error here. Both limits check: $p\to0$ as
> $v_\parallel\to0$ (a circle), and $r_L\to0$ as $v_\perp\to0$ (straight drift along
> the field).

$$
% caption: Two projections of the worked proton helix. Seen along the field the orbit is a circle of radius $r_L=3.13\ \mathrm{cm}$; seen across the field it advances one pitch $p=34.0\ \mathrm{cm}$ per turn. Keeping the projections separate prevents substituting the full speed for the transverse speed in $r_L$.
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$$

Suppose the measured radius is $3.13\ \mathrm{cm}$ with an uncertainty of
$0.20\ \mathrm{mm}$ and the field calibration has relative uncertainty
$0.30\%$. The radius term contributes about $0.64\%$ to transverse-momentum
uncertainty. Combining independent radius and field terms in quadrature gives a
relative uncertainty near $0.71\%$. That estimate presumes the fitted circle is
unbiased. A detector offset that shifts every hit in one direction can change
curvature systematically and must be constrained by alignment data, charge-sign
reversal, or tracks entering from the opposite side.

The proton speed here is only about one percent of the speed of light, so
$\gamma-1$ is roughly $5\times10^{-5}$. The nonrelativistic period is adequate at
this precision. Repeating the same calculation for an electron at high kinetic
energy requires relativistic momentum even when the observed path still resembles a
circle. The visual shape of a track does not determine which momentum model is
appropriate; the energy scale does.

## Numerical propagation and validation

A field map sampled at discrete points gives the coupled trajectory equations
equations

$$
\frac{\d \vec r}{\d t}=\vec v,
\qquad
\frac{\d \vec p}{\d t}=q\vec v\times\vec B(\vec r).
$$

A time step should resolve both the cyclotron period and the spatial scale over
which the magnetic field changes. In a uniform field, first propagate many periods
with no electric field and compare the numerical result with
an exact circle or helix. The fitted speed, radius, and orbit centre should remain
constant to the intended tolerance. A trajectory that slowly spirals outward or
inward is a numerical artifact unless another physical force has been included.

The Boris rotation method is widely used because it rotates momentum under a
magnetic update while preserving its magnitude accurately in the absence of electric
field. Runge--Kutta methods can also produce accurate short trajectories, but their
step size must be checked against long-time energy drift. The choice of integrator
does not remove the need for a field-map interpolation test: a coarse or
discontinuous map can introduce artificial kicks even when the time integrator is
stable.

Measure step-size convergence with physical outputs and the solver's internal error
estimate. Halve the time step, refit curvature and arrival phase, then compare those
quantities with the measurement precision. Smooth-looking coordinates can still
accumulate an unacceptable phase error over many turns. In a spectrometer, a small
phase error may matter little if only curvature is used; in a cyclotron, the same
error can move a particle out of the accelerating RF phase.

Energy and momentum checks need the correct scope. Magnetic force alone preserves
speed, yet a field map that includes fringe regions can change the direction of the
parallel and transverse components as the local field direction changes. A validation
log records total speed, local field magnitude, fitted radius in a
uniform reference region, phase error relative to an analytic orbit, and any work
done by electric fields. Those quantities separate physical departures from uniform
motion from an integration failure.

## Detector coordinates, projections, and uncertainty

Track curvature is measured in the plane perpendicular to the local magnetic field.
In a solenoidal detector with field approximately parallel to the laboratory
$z$ axis, that plane is the transverse $x$--$y$ plane. A helix fit returns
transverse momentum from its projected circle and returns the longitudinal momentum
from the advance in $z$ per turn. The total momentum is

$$
p=\sqrt{p_\perp^2+p_\parallel^2},
\qquad
\tan\alpha=\frac{p_\perp}{p_\parallel}.
$$

Viewing a helical track from the wrong direction can conceal its curvature. A side
view along a transverse axis shows an oscillating projection whose wavelength is the
pitch. A view parallel to the field shows a circle. The same three-dimensional
trajectory can look nearly straight in one projection and tightly curved in another.
Detector displays, fit diagnostics, and written solutions should identify the
projection before quoting a radius or a bend sign.

A curvature fit minimizes the signed distance between measured hit positions and a
model circle or helix. The residual pattern matters as much as the residual size.
Random measurement errors produce residuals that change sign without a spatial
pattern. A detector layer shifted outward produces a common residual sign at that
layer. A field-scale error changes curvature consistently across tracks with
different radii. Energy loss produces a gradual change in curvature with path
length. These signatures guide calibration because they identify which part of the
measurement model needs correction.

Point resolution is only one term in the uncertainty budget. A field map has a
scale uncertainty and local interpolation uncertainty. Mechanical survey establishes
the positions and rotations of detector layers. Material before or between layers
causes multiple Coulomb scattering, which adds a random bend that is largest for low
momentum particles. A standard estimate for the rms scattering angle in thickness
$x$ is

$$
\theta_0\simeq
\frac{13.6\ \mathrm{MeV}}{\beta pc}\,|z|\,
\sqrt{\frac{x}{X_0}}
\left[1+0.038\ln\!\left(\frac{x}{X_0}\right)\right],
$$

where $X_0$ is radiation length. The expression estimates a distribution width; an
individual event can scatter more or less strongly. Curvature fitting must allow
these random changes instead of interpreting every departure from a circle as an
incorrect field polarity.

In a short track with hit-position uncertainty $\sigma_x$, the sagitta uncertainty
often scales with $\sigma_x$ divided by a geometry factor. Extending the lever arm
$L$ improves curvature resolution rapidly because the sagitta grows as $L^2$.
Adding material simply to add more layers can work against that gain when scattering
becomes large. Detector design balances magnetic field, lever arm, point resolution,
and material budget rather than maximizing any one quantity alone.

Calibration tracks provide a direct test. A particle species with known momentum can
be sent through the detector at both charge signs or both field polarities. The
fitted curvature magnitude should agree after the sign convention is reversed.
Cosmic-ray tracks crossing the detector from above and below offer a complementary
alignment test because the same physical path can be reconstructed in opposite
directions. Disagreement between these reversals identifies an offset that a single
sample of tracks would hide.

**Momentum, energy, and charge-state interpretation.**

Magnetic curvature measures momentum divided by charge magnitude. A measured radius
alone cannot identify mass, kinetic energy, and charge state simultaneously. For
nonrelativistic motion, a separate speed measurement gives

$$
\frac{m}{|q|}=\frac{Br_L}{v_\perp}.
$$

At relativistic energy, combine curvature with a time-of-flight, calorimetric, or
Cherenkov speed measurement and use $p=\gamma mv$. A mass spectrometer specializes
this combination with a velocity selector and a known charge state. A tracking
detector often measures momentum first and identifies particle species by additional
detector responses.

Charge-changing interactions illustrate the same limitation. If an ion loses an
electron in material, its charge magnitude rises and its later curvature changes
even at nearly unchanged momentum. A radius change can therefore signal energy
loss, a charge-state change, a field change, or a combination. Independent energy
deposition, timing, and field-map data determine which explanation is physically
consistent. The track geometry provides a strong constraint within the full
particle-identification measurement.

Magnetic fields also establish an acceptance range. A particle with gyroradius much
larger than the detector aperture bends too little to remain inside enough layers
for a reliable fit. A particle with very small radius can curl repeatedly and cross
the same layers many times, complicating hit association. The usable momentum range
is set by detector size, field strength, readout spacing, and the material budget.
Stating those limits prevents a curvature formula from being applied outside the
region in which a track can actually be reconstructed.

**Analysis protocol for an unknown track.**

An orbit reconstruction begins with the coordinate and field conventions. Record
the magnetic-field direction in the detector axes, the definition of positive
curvature, the ordered direction of the hit sequence, and the unit system used by
the field map. These choices determine the charge-sign result. A curvature number
without an orientation convention carries only a magnitude.

Select a track segment that lies inside the mapped uniform region and contains enough
separated hit positions to constrain a circle. Exclude entrance and exit fringe
regions when the goal is a simple uniform-field radius. Fit the transverse
projection, inspect residuals by layer and by arc length, then repeat the fit after
removing one hit at a time. A radius that changes substantially when one hit is
removed is dominated by leverage or an outlier; its nominal least-squares error
understates the measurement fragility.

Convert the fitted radius to transverse momentum with the signed charge hypothesis
kept explicit:

$$
\vec p_\perp=q\,r_L\,\hat n\times\vec B,
\qquad
|\vec p_\perp|=|q|Br_L.
$$

The cross product fixes the curvature orientation from the field direction and the
charge sign. The magnitude expression supports a quick scale check. In SI units, tesla times
coulomb times metre has units of momentum. In laboratory units, any numerical
conversion factor must accompany its declared momentum, field, and length units;
copying a factor from a different unit convention changes a track momentum by a
large scale factor without visibly changing the fit.

Add longitudinal information only after the transverse fit has passed its checks.
Time-of-flight gives speed through a known path length. Stereo hit positions or
multiple planes give the axial advance per revolution. A calorimeter measures energy
after material interactions. Combining two independent quantities exposes invalid
assumptions: a radius and time-of-flight pair can reveal a charge-state change,
while a radius and calorimetric energy pair can distinguish a relativistic particle
from a slow high-mass particle.

The fitted covariance matrix must follow the reported momentum. Statistical
uncertainty from hit resolution can be small while the total uncertainty is set by
field calibration or alignment. Quote these components separately when they have
different correlation structures. A common field-scale shift affects an entire
dataset coherently and can move a resonance mass or beam-energy result. Point
resolution broadens individual tracks and often averages down in an ensemble. A
single combined error bar conceals the correction strategy.

Repeated reversal tests expose sign and alignment mistakes. Reverse the magnetic
field while holding the detector geometry fixed, or compare oppositely charged
particles with matched momentum. The physical curvature direction reverses. A
fitted offset that remains unchanged under reversal belongs to geometry or readout;
an offset that reverses with the field can arise from field-map or charge-sign
conventions. This symmetry test is stronger than visual agreement with one expected
arc because it changes the underlying Lorentz-force sign.

**Dimensional, limiting, and physical checks.**

Each result has limiting cases that should be inspected before it is used. As
$B\rightarrow0$, the gyroradius tends to infinity and the local path becomes
straight. As $v_\perp\rightarrow0$, the radius tends to zero while the particle
moves along the field with finite parallel speed. As $v_\parallel\rightarrow0$, the
pitch tends to zero and the helix becomes a circle. Reversing either charge sign or
field direction reverses the rotation sense; reversing both restores it. These
checks catch most sign and component errors without requiring numerical values.

The magnetic force has units of newtons:

$$
[qvB]=\mathrm C\,\frac{\mathrm m}{\mathrm s}\,
\frac{\mathrm{kg}}{\mathrm{C\,s}}=\mathrm N.
$$

The radius has units of metres and the cyclotron frequency has units of inverse
seconds. An expression for period that increases with field strength, or an orbit
radius that uses total speed when a parallel component is present, fails these
physical checks. A result that predicts changing speed in a static magnetic-only
region has included an electric field, a collision, numerical energy drift, or an
incorrect force direction.

The charge-state assumption deserves an explicit check for ions. A spectrometer may
accelerate singly charged ions, yet a collision or stripping foil can produce a
different charge state before the analyzing magnet. The measured curvature then
corresponds to a different momentum-to-charge ratio. Isotope separation follows this
momentum-to-charge dependence: two ions with equal momentum and different mass-to-charge ratios
follow different radii, while two charge states of the same isotope can also follow
different radii. The apparatus must establish charge state through source
preparation, energy selection, or complementary detector signals.

The word uniform refers to the modeled field over the fitted orbit segment. A large
instrument can contain central and fringe regions with different fields. Carry a
radius from a central calibration region into a fringe region only after trajectory
propagation through the measured field map. Field-integral methods handle that
transition directly. The local circle remains a diagnostic, while the full path uses
the actual spatial field.

Separate measurement uncertainty from model inadequacy. Random hit noise, thermal
sensor drift, and finite field-probe resolution broaden a valid result and enter a
covariance estimate. Electric stray fields, energy loss, charge exchange, and
unmodelled material change the equation of motion. The second group requires an
extended trajectory model rather than a larger error bar.

**Uniform-field model boundaries in practice.**

The uniform-field fit is local. State the fitted spatial interval, time interval,
field map, and charge-state assumption beside every quoted radius or momentum.

| departure from the local model | measurement signature | required treatment |
|---|---|---|
| fringe or field-gradient region | curvature changes along the path | propagate through the measured field map |
| electric potential or accelerating gap | speed changes across the segment | include the electric term in the Lorentz equation |
| gas or material interaction | kink, energy loss, or scattering tail | split the track or apply a material model |
| pulsed field | bend depends on arrival phase | pair the track timestamp with the field waveform |
| field-map tilt | bias between parallel and transverse momentum | apply alignment survey and reference-track constraints |

- **Field extent.** A particle can follow a near-circular arc in the central field
  and then receive additional bend in a fringe region. A central radius is a valid
  local diagnostic; downstream transport still requires the field integral along
  the complete path.
- **Electric contribution.** Patch potentials, drift electrodes, and accelerating
  gaps add the electric term to the Lorentz force. Compare measured speed before and
  after the fit segment whenever an energy change is plausible. A changing speed
  excludes a magnetic-only model even when a projected arc appears circular.
- **Matter interaction.** Collisions in gas produce random scattering. Solid
  material produces average ionization loss and occasional hard kinks. Fit separate
  uniform-field segments on either side of a kink; one circle through the kink
  replaces a physical event with a spurious intermediate curvature.
- **Time and alignment.** In a pulsed magnet, match the particle arrival timestamp
  to the field waveform and field-probe record. Define parallel velocity relative
  to the local field direction, not to a mechanical ruler. A map tilt can transfer
  reconstructed momentum between parallel and transverse components while leaving
  total momentum nearly unchanged.
- **Report and reversal test.** Record charge convention, field-map uncertainty,
  fit segment, coordinate axes, material corrections, and relativistic treatment.
  Send a reference beam of known momentum through the prescribed central arc and
  compare its inferred field integral with an independent probe map. Repeat after
  reversing field polarity. A polarity-dependent difference indicates a convention,
  hysteresis, or timing problem; a polarity-independent scale difference indicates
  the map, geometry, or reference-momentum calibration.
- **Period check.** At nonrelativistic energy, equivalent orbital phases are
  separated by $2\pi m/(|q|B)$, independent of radius. A measured radius
  dependence can indicate gradients, electric acceleration, relativistic momentum,
  or phase-pickup bias. Radius and period together test more of the model than a
  circular projection alone.

These checks validate a stated field region, charge state, and energy range. A track
through a large instrument can cross several local models. Connect those segments
with the Lorentz equation and measured boundaries before assigning one global
momentum or uncertainty.

**Segment acceptance record.**

- **Geometry selection.** Retain the hit coordinates, the chosen fit interval, the
  local field direction, and the entrance and exit boundaries. A radius fitted from
  hits spanning a fringe region is not interchangeable with a radius fitted wholly
  inside the central map region.
- **Field variation.** Evaluate the mapped field at the hit positions and report
  its variation across one fitted gyroradius and along the fitted arc. The average
  field is sufficient only when that variation is smaller than the measurement
  precision required for the reported momentum.
- **Residual structure.** Store signed residuals versus arc length and detector
  layer. Random signs are consistent with point resolution. A smooth drift can
  indicate field-scale error or energy loss; one localized displacement can indicate
  scattering or a shifted layer. Quote a goodness-of-fit statistic together with the
  residual pattern rather than accepting a circle from its visual appearance.
- **Independent momentum check.** Convert the fitted transverse radius to
  $p_\perp=|q|Br_L$, then compare with timing, calorimetry, or a reference beam
  when available. The comparison separates field-scale error from a charge-state or
  material correction. A radius alone measures momentum divided by charge magnitude.
- **Boundary handoff.** Pass the fitted position, direction, covariance, charge
  hypothesis, and local field to the next transport region. Replacing that state by
  a single global radius discards the information needed to propagate through a
  nonuniform boundary or to combine several detector regions.

A reference-track run should use the same reconstruction window, hit selection, and
field interpolation as the unknown-track analysis. Fit the reference under both field
polarities and compare the inferred $p_\perp/|q|$ with its independently known
value. Retain the signed curvature, residual distribution, and fitted period when
timing is available. This control constrains field scale, coordinate handedness, and
charge-sign convention simultaneously. Arc shape alone cannot establish those three
calibration conditions.