Page 251 - Engineering Electromagnetics, 8th Edition

CHAPTER 8 Magnetic Forces, Materials, and Inductance 233

Figure 8.1 Equal currents directed into the material are provided by positive charges
moving inward in (a) and negative charges moving outward in (b). The two cases can be
distinguished by oppositely directed Hall voltages, as shown.

magnetic field across it, to serve as electronic wattmeters, squaring elements, and so
forth.
Returning to (4), we may therefore say that if we are considering an element
of moving charge in an electron beam, the force is merely the sum of the forces on
the individual electrons in that small volume element, but if we are considering an
element of moving charge within a conductor, the total force is applied to the solid
conductor itself. We will now limit our attention to the forces on current-carrying
conductors.
In Chapter 5 we defined convection current density in terms of the velocity of
the volume charge density,

J = ρ ν v

The differential element of charge in (4) may also be expressed in terms of volume
charge density, 1

dQ = ρ ν dν
Thus
dF = ρ ν dν v × B
or

dF = J × B dν (5)

We saw in Chapter 7 that J dν may be interpreted as a differential current element;
that is,
J dν = K dS = IdL

1 Remember that dν is a differential volume element and not a differential increase in velocity.