Charged Particle Motion in a Magnetic Field

Suppose that a particle of mass $m$ moves in a circular orbit of radius $\rho$ with a constant speed $v$. As is well known, the acceleration of the particle is of magnitude $v^2/\rho$, and is always directed toward the center of the orbit. It follows that the acceleration is always perpendicular to the particle's instantaneous direction of motion.

Figure 2.17: Circular motion of a charged particle in a magnetic field.

We have seen that the force exerted on an electrically charged particle by a magnetic field is always perpendicular to its instantaneous direction of motion. Does this imply that the field causes the particle to execute a circular orbit? Consider the case shown in Figure 2.17. Suppose that a particle of positive charge $q$ and mass $m$ moves in a plane perpendicular to a uniform magnetic field $B$. In the figure, the field is directed into the plane of the paper. Suppose that the particle moves, in a counter-clockwise manner, with constant speed $v$ (recall that the magnetic field cannot do work on the particle, so it cannot affect its speed), in a circular orbit of radius $\rho$. The magnetic force acting on the particle is of magnitude $f=q\,v\,B$ and, according to Equation (2.211), this force is always directed toward the center of the orbit. Thus, if

$$f = q\,v\,B = \frac{m\,v^2}{\rho},$$ (2.219)

then we have a self-consistent picture. It follows that

$$\rho = \frac{m\,v}{q\,B}.$$ (2.220)

The angular frequency of rotation of the particle (i.e., the number of radians the particle rotates through in one second) is

$$\omega = \frac{v}{\rho} = \frac{q\,B}{m}.$$ (2.221)

Note that this frequency, which is known as the Larmor frequency, does not depend on the velocity of the particle. For a negatively charged particle, the picture is exactly the same as described previously, except that the particle moves in a clockwise orbit.

It is clear, from Equation (2.221), 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, $q/m$. Furthermore, if the speed of the particle is known then the radius of the orbit can also be used to determine $q/m$, via Equation (2.220). In the past, this method was used extensively in high energy physics experiments to identify particles from photographs of the tracks that they left 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. However, we can also add an arbitrary drift along the direction of the magnetic field. This follows because the force $q\,{\bf v}\times{\bf B}$ acting on the particle only depends on the component of the particle's velocity that is perpendicular to the direction of magnetic field (the vector product of two parallel vectors is always zero because the angle $\theta$ they subtend is zero). (See Section A.8.) 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 Figure 2.18.

Figure 2.18: Spiral trajectory of a charged particle in a uniform magnetic field.