Page 307 - Engineering Electromagnetics, 8th Edition

CHAPTER 9 Time-Varying Fields and Maxwell’s Equations 289

their sources, charge and current density. The auxiliary equations relating D and E,

D = E (24)

relating B and H,

B = µH (25)

defining conduction current density,

J = σE (26)

and defining convection current density in terms of the volume charge density ρ ν ,

J = ρ ν v (27)

are also required to define and relate the quantities appearing in Maxwell’s equations.
The potentials V and A have not been included because they are not strictly
necessary, although they are extremely useful. They will be discussed at the end of
this chapter.
If we do not have “nice” materials to work with, then we should replace (24) and
(25) with the relationships involving the polarization and magnetization fields,

D = 0 E + P (28)

B = µ 0 (H + M) (29)

For linear materials we may relate P to E

P = χ e 0 E (30)

and M to H

M = χ m H (31)

Finally, because of its fundamental importance we should include the Lorentz
force equation, written in point form as the force per unit volume,

f = ρ ν (E + v × B) (32)

The following chapters are devoted to the application of Maxwell’s equations to
several simple problems.