Equations of Motion

  Lorentz force causes a particle to bend in a magnetic and/or electric field. In most cases, the electric field is negligible; then the magnetic field is time independent, . In this case, the energy E and momentum are constants of motion. Let s be the path length, the position of the particle, t the time, q the particle charge, c the speed of light; then the equation of motion can be written in various different ways, e.g.,

the form which is most often used,

or in terms of , etc.,

where and

For singly charged particles, |q| is the elementary charge e, which can be expressed in the usual hybrid units ( Units) m or cm, s, GeV/c, T = tesla or G = gauss, as

These are three versions of the equation of motion, two of which do not involve time. The three versions are equivalent, even though the number of equations is apparently different; in each version, there are two independent second-order differential equations. For example, the three equations of the first version satisfy the identity

The second version has a simple geometrical interpretation. In fact, is the unit vector tangent to the track, while is a normal vector of length , where is the radius of curvature. Hence, if is the angle between and , then

is the momentum component normal to , while is the radius of curvature of the projection of the track onto a plane normal to .

The same equation can be given a different interpretation. When the particle travels an infinitesimal distance , its track is deflected by an angle

where is the component of normal to the track. Integration of this formula gives

Note that so defined is usually only approximately equal to the total deflection angle, because rotation angles are only additive if the rotation axis is fixed.

The effect of a homogeneous field inside a rectangular box, say

is to give a particle passing through at any angle a transverse momentum kick

The homogeneous field in a box is a good model for many spectrometer magnets, if one replaces B(x₂-x₁) by the field integral along a straight line perpendicular to the box containing the field. For example, if  Tm (tesla metre), the transverse momentum kick of a proton is  GeV/c. The variation of the integral determines the precision of the model. With a formula containing only one parameter, one may get a precision of 10-20% in the momentum determination.

For detectors inside a homogeneous field B, one often uses the approximate relation between measured sagitta s, track length l, radius of curvature and momentum (all quantities projected into the plane perpendicular to B):

The precise solution of the Lorentz equations of motion ( Trajectory of a Charged Particle) is found using methods of numerical integration (see [Bock98]).