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# Motion of a Charged Particle in a Magnetic Field

Let us consider a charge q with velocity entering a region of space with a constant magnetic field . We assume that initially and are at right angles. The charge will experience the force of Eq.(1.2) which, by definition, is perpendicular to the velocity . Because of this, the force does no work on the charge (recall W = Fdcos = 0 if = 90 o ), and because W = K = (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 directed toward the center of the circle, as illustrated in Fig. 1.3.

For such a motion we have the following relation:

 F = qvB = m r = . (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.

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