Measurement of particle momentum (from curvature of its path in a
magnetic field)
A particle of charge q travelling at right‐angles to a magnetic field B with a speed v experiences
a force Bqv at right angles to its motion.
This makes the particle follow a circular path of radius r and the motion is described by
Bqv = mv2/r → p = (Bq) r
This tells us that for a fixed field B, and charge q, the momentum p is proportional to the radius
of curvature r. Here, nature has been kind: all charged particles that live long enough to travel
a measurable distance have a charge equal or opposite to the charge on the electron
e=1.6x10‐19 C.
Measuring the momentum of moving charged particles
velocity v





force F


circular path,
radius r

magnetic field B
into screen








charge q
velocity v
motion in circle
magnetic force
m v2
= force F
r
force F = qvB
m v2
= qvB
r
p=
p =mv
mv==(Bq)
qrB r
at relativistic speed
p = qrB
is still true but
p > mv
Momentum of particle proportional to radius of curvature of path
Exercise 1
The radius of curvature of the curved sections of the LHC is 2804 m. Show that, whit a
magnetic field of 8.33 T, the momentum of the circling protons is  7000 Gev/c (or 7 Tev/c).
[Note: 1 eV= 1.6022 x 10-19J]
Exercise 2
Find the ratio of the proton velocity to the velocity of light in a vacuum.