Motion of a Charged Particle in a Magnetic Field

Let us consider a charge q with velocity $\vec{v}$ entering a region of space with a constant magnetic field $\vec{B}$ . We assume that initially $\vec{v}$ and $\vec{B}$ are at right angles. The charge will experience the force of Eq.(1.2) which, by definition, is perpendicular to the velocity $\vec{v}$ . Because of this, the force does no work on the charge (recall W = Fdcos $\theta$ = 0 if $\theta$ = 90 ^o ), and because W = $\Delta$K = $\Delta$(${\frac{1}{2}}$mv ²) the speed of the charge will not change. It turns out that the charge will move in a circular motion, with a (centripetal) acceleration $\vec{a}_{c}^{}$ directed toward the center of the circle, as illustrated in Fig. 1.3.

  

Figure 1.3: Motion of a charged particle in a magnetic field

For such a motion we have the following relation:

$${\frac{v^2}{r}} \Rightarrow {\frac{mv}{qB}}$$ (3)

This behaviour of a charged particle in a magnetic field is the principle behind machines such as mass spectrometers, which can be used to measure the masses of charged particles by measuring their radii of curvature in a magnetic field.