Lorentz force, beam rigidity and bending magnets

Use SI electromagnetic fields, mechanical momentum and a right-handed laboratory coordinate system. The Lorentz equation applies to relativistic momentum as well as to its low-speed limit. Taking the scalar product with velocity gives the rate of change of total relativistic energy. The magnetic contribution vanishes because a cross product is perpendicular to velocity. A static magnetic field can therefore rotate momentum while preserving its magnitude when energy losses and electric fields are negligible.

For motion perpendicular to a uniform field, the rate of turning of the momentum vector is momentum times speed divided by the circular radius. Equating its magnitude to the magnetic force cancels the speed. With a component parallel to the field, the orbit becomes a helix: the measured transverse radius determines transverse momentum, while the pitch supplies the parallel component. Using the helical radius as though it measured total momentum underestimates the latter.

Define the positive magnetic rigidity as transverse momentum divided by charge magnitude for a perpendicular reference orbit. A signed rigidity can also be used, but its sign must then be carried consistently through lattice equations. The practical conversion between gigaelectronvolt-per-c momentum and tesla-meter rigidity contains the speed of light and elementary charge. We use a rounded coefficient for classroom arithmetic, while a precision spectrometer uses the more accurate conversion and a calibrated field map.

In a uniform dipole, the integrated bending angle is arc length divided by radius. For a short magnet at approximately constant momentum, the same result follows from its transverse field integral. Magnet length in that formula follows the reference path through the field; it is not automatically the straight chord between endpoints. Fringing fields, misalignment and a varying field require an effective integral or trajectory calculation. Reversing either charge or field reverses curvature without changing the positive radius.

The sagitta is the maximum displacement of a circular arc from a straight chord joining two measured endpoints. Expanding the exact circle geometry for a short chord produces an inverse-radius estimate. This geometry explains why a longer tracking lever arm improves curvature sensitivity: at fixed radius the sagitta grows quadratically with chord length. It also explains why a tiny sagitta is vulnerable to alignment offsets, which can imitate a change in momentum or even obscure the charge sign.

A real fit combines several hit positions and their covariance, material scattering, energy loss and field uncertainty. Multiple scattering particularly degrades low-momentum tracks, whereas a fixed spatial measurement error makes relative momentum precision worsen at high momentum. The simple circle inference assumes the relevant projection and a sufficiently small bend. It is a diagnostic estimate, not a full detector likelihood or a universal resolution formula, and it does not turn every visible curve into a unique particle identification.