Page 249 - Engineering Electromagnetics, 8th Edition

CHAPTER 8 Magnetic Forces, Materials, and Inductance 231

The force is in the same direction as the electric field intensity (for a positive charge)
and is directly proportional to both E and Q.If the charge is in motion, the force at
any point in its trajectory is then given by (1).
A charged particle in motion in a magnetic field of flux density B is found
experimentally to experience a force whose magnitude is proportional to the product
of the magnitudes of the charge Q, its velocity v, and the flux density B, and to the sine
of the angle between the vectors v and B. The direction of the force is perpendicular
to both v and B and is given by a unit vector in the direction of v × B. The force may
therefore be expressed as

F = Qv × B (2)

A fundamental difference in the effect of the electric and magnetic fields on
charged particles is now apparent, for a force which is always applied in a direc-
tion at right angles to the direction in which the particle is proceeding can never
change the magnitude of the particle velocity. In other words, the acceleration vector
is always normal to the velocity vector. The kinetic energy of the particle remains
unchanged, and it follows that the steady magnetic field is incapable of transfer-
ring energy to the moving charge. The electric field, on the other hand, exerts a
force on the particle which is independent of the direction in which the particle is
progressing and therefore effects an energy transfer between field and particle in
general.
The first two problems at the end of this chapter illustrate the different effects of
electric and magnetic fields on the kinetic energy of a charged particle moving in free
space.
The force on a moving particle arising from combined electric and magnetic
fields is obtained easily by superposition,

F = Q(E + v × B) (3)

This equation is known as the Lorentz force equation, and its solution is required in
determining electron orbits in the magnetron, proton paths in the cyclotron, plasma
characteristics in a magnetohydrodynamic (MHD) generator, or, in general, charged-
particle motion in combined electric and magnetic fields.

6
D8.1. Thepointcharge Q = 18nChasavelocityof5×10 m/sinthedirection
a ν = 0.60a x +0.75a y +0.30a z . Calculate the magnitude of the force exerted on
the charge by the field: (a) B =−3a x + 4a y + 6a z mT; (b) E =−3a x + 4a y +
6a z kV/m; (c)B and E acting together.

Ans. 660 µN; 140 µN; 670 µN