PHY6040 Nuclear and Particle Detectors Dr C N Booth

Measurement in a Magnetic Field

Magnetic fields are frequently applied to bend the trajectory of a charged particle, and so determine its momentum.  A particle of charge Q and velocity v subject to a uniform magnetic field B experiences a force F given by
If we consider the case of the velocity being perpendicular to the magnetic field, since the force is also perpendicular to v, the magnitude of velocity remains unchanged and the particle follows a circular path.  We can determine the radius, r, of the path as follows.
Since and 
centripetal acceleration is
then

If the particle's velocity makes an angle θ to the direction of the magnetic field, then the trajectory is a helix, with radius

.
Frequently with high momentum particles, only an arc corresponding to a small part of the circle is observed.  Consider a particle of momentum p passing through a region, of length L, with a magnetic field B.  The deviation from a straight line, s, is known as the sagitta of the track (from the Latin word for a bow).
This can be calculated as follows:
We previously showed that 
or

It is also important to consider the error on any measurement of momentum.  This is related to the error on the measured sagitta, δs, as follows.

If we define the "maximum measurable momentum", p0, as the value of momentum for which the error is as big as the momentum itself, then
So we can write:

The important result here is that the fractional error on the momentum is proportional to the momentum of the particle.

For a more detailed discussion of the trajectory of a charged particle in a magnetic fiels, see the entry in the BriefBook.

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