We have seen that the force exerted on a charged particle by a magnetic
field is always perpendicular to its instantaneous direction of motion.
Does this mean that the field causes the particle to execute a circular
orbit? Consider the case shown in Fig. 24. Suppose that a
particle of positive charge and mass moves in a plane perpendicular
to a uniform magnetic field . In the figure, the field points into
the plane of the paper. Suppose that the particle moves, in an
anti-clockwise manner, with constant
speed (remember that the magnetic field cannot do work on the
particle, so it cannot affect its speed), in a circular orbit of radius .
The magnetic force acting on the particle is
of magnitude and, according to Eq. (158), this force is always
directed towards the centre of the orbit. Thus, if

(166) |

The angular frequency of rotation of the particle (

Note that this frequency, which is known as the

It is clear, from Eq. (168), that the angular frequency of gyration of a charged particle in a known magnetic field can be used to determine its charge to mass ratio. Furthermore, if the speed of the particle is known, then the radius of the orbit can also be used to determine , via Eq. (167). This method is employed in High Energy Physics to identify particles from photographs of the tracks which they leave in magnetized cloud chambers or bubble chambers. It is, of course, easy to differentiate positively charged particles from negatively charged ones using the direction of deflection of the particles in the magnetic field.

We have seen that a charged particle placed in a magnetic field executes a
circular orbit in the plane perpendicular to the direction of the field.
Is this the most general motion of a charged particle in a magnetic field?
Not quite. We can also add an arbitrary drift along the direction
of the magnetic field. This follows because the force
acting on the particle only depends on the component of the particle's velocity
which is *perpendicular* to the direction of magnetic field (the cross
product of two parallel vectors is always zero because the angle
they subtend is zero). The combination of circular motion in the
plane perpendicular to the magnetic field, and uniform motion along the
direction of the
field, gives rise to a *spiral* trajectory of a charged particle in
a magnetic field, where the field forms the axis of the spiral--see Fig. 25.