Page 308 - Engineering Electromagnetics, 8th Edition

290 ENGINEERING ELECTROMAGNETICS

D9.4. Let µ = 10 −5 H/m, = 4 × 10 −9 F/m, σ = 0, and ρ ν = 0. Find k
(including units) so that each of the following pairs of fields satisfies Maxwell’s
2
equations: (a) D = 6a x − 2ya y + 2za z nC/m , H = kxa x + 10ya y − 25za z
6
A/m; (b) E = (20y − kt)a x V/m, H = (y + 2 × 10 t)a z A/m.
8
2
Ans. 15 A/m ; −2.5 × 10 V/(m · s)
9.4 MAXWELL’S EQUATIONS
IN INTEGRAL FORM
The integral forms of Maxwell’s equations are usually easier to recognize in terms of
theexperimentallawsfromwhichtheyhavebeenobtainedbyageneralizationprocess.
Experiments must treat physical macroscopic quantities, and their results therefore
are expressed in terms of integral relationships. A differential equation always rep-
resents a theory. Let us now collect the integral forms of Maxwell’s equations from
Section 9.3.
Integrating (20) over a surface and applying Stokes’ theorem, we obtain Faraday’s
law,
∂B

E · dL =− · dS (33)
S ∂t
and the same process applied to (21) yields Amp`ere’s circuital law,

∂D

H · dL = I + · dS (34)
S ∂t
Gauss’s laws for the electric and magnetic fields are obtained by integrating (22)
and (23) throughout a volume and using the divergence theorem:

ρ ν dv (35)
D · dS =
S vol

B · dS = 0 (36)
S

These four integral equations enable us to find the boundary conditions on B, D, H,
and E, which are necessary to evaluate the constants obtained in solving Maxwell’s
equations in partial differential form. These boundary conditions are in general un-
changed from their forms for static or steady fields, and the same methods may be
used to obtain them. Between any two real physical media (where K must be zero on
the boundary surface), (33) enables us to relate the tangential E-field components,
E t1 = E t2 (37)