The Lorentz force

Let be the
(uniform) cross-sectional area of the wire, and let be the number density
of mobile charges in the conductor. Suppose that the
mobile charges each have charge and velocity . We must assume that
the conductor also contains stationary charges, of charge and number density
(say), so that the net charge density in the wire is zero. In most conductors, the
mobile charges are electrons and the stationary charges are atomic nuclei.
The magnitude of the electric current flowing through the wire is simply the
number of coulombs per second which flow past a given point. In one second,
a mobile charge moves a distance , so all of the charges contained in a
cylinder of cross-sectional area and length flow past a given point.
Thus, the magnitude of the current is . The direction of the
current is the same as the direction of motion of the charges, so the
vector current is
.
According to Eq. (229), the force per unit length acting on the wire is

(232) |

(233) |

This is called the

The
equation of motion of a free particle of charge and
mass moving in electric and
magnetic fields is

Let us analyze Thompson's experiment. Suppose that
the rays are originally traveling in the -direction, and are subject to
a uniform electric field in the -direction and a uniform magnetic
field in the -direction. Let us assume, as Thompson did, that cathode
rays are a stream of particles of mass and charge . The
equation of motion of the particles in the -direction is

where the ``time of flight'' is replaced by . This formula is only valid if , which is assumed to be the case. Next, Thompson turned on the magnetic field in his apparatus, and adjusted it so that the cathode ray was no longer deflected. The lack of deflection implies that the net force on the particles in the -direction was zero. In other words, the electric and magnetic forces balanced exactly. It follows from Eq. (236) that with a properly adjusted magnetic field strength

Thus, Eqs. (237) and (238) and can be combined and rearranged to give the charge to mass ratio of the particles in terms of measured quantities:

(239) |

Consider, now, a particle of mass and charge moving in a uniform
magnetic field,
. According, to
Eq. (235), the particle's equation of motion can be written:

(240) |

(241) | |||

(242) | |||

(243) |

Here, is called the

(244) | |||

(245) | |||

(246) |

and

(247) | |||

(248) | |||

(249) |

According to these equations, the particle trajectory is a

Finally, if a particle is subject to a force and moves a distance
in a time interval , then the work done on the particle by the
force is

(251) |

(252) |