Page 318 - Engineering Electromagnetics, 8th Edition

300 ENGINEERING ELECTROMAGNETICS

equations to help find (a) β,if β> 0; (b) the displacement current density
at z = 0; (c) the total displacement current crossing the surface x = 0.5d,
0 < y < b,0 < z < 0.1minthe a x direction.
9.19 In Section 9.1, Faraday’s law was used to show that the field
1
kt
kt
E =− kB 0 e ρa φ results from the changing magnetic field B = B 0 e a z .
2
(a) Show that these fields do not satisfy Maxwell’s other curl equation.
6 −1
(b)Ifwelet B 0 = 1T and k = 10 s ,we are establishing a fairly large
magnetic flux density in 1 µs. Use the ∇× H equation to show that the rate
at which B z should (but does not) change with ρ is only about 5 × 10 −6 T
per meter in free space at t = 0.
9.20 Given Maxwell’s equations in point form, assume that all fields vary as e st
and write the equations without explicitly involving time.
9.21 (a) Show that under static field conditions, Eq. (55) reduces to Amp`ere’s
circuital law. (b)Verify that Eq. (51) becomes Faraday’s law when we take
the curl.
9.22 In a sourceless medium in which J = 0 and ρ ν = 0, assume a rectangular
coordinate system in which E and H are functions only of z and t. The
medium has permittivity and permeability µ.(a)If E = E x a x and
H = H y a y ,begin with Maxwell’s equations and determine the second-order
partial differential equation that E x must satisfy. (b) Show that
E x = E 0 cos(ωt − βz)isa solution of that equation for a particular value of
β.(c) Find β as a function of given parameters.
9.23 In region 1, z < 0, 1 = 2 × 10 −11 F/m, µ 1 = 2 × 10 −6 H/m, and σ 1 =
4 × 10 −3 S/m; in region 2, z > 0, 2 = 1 /2, µ 2 = 2µ 1 , and σ 2 = σ 1 /4. It is
9
known that E 1 = (30a x + 20a y + 10a z ) cos 10 t V/m at P(0, 0, 0 ). (a)
−
Find E N1 , E t1 , D N1 , and D t1 at P 1 .(b) Find J N1 and J t1 at P 1 .(c) Find E t2 ,
D t2 , and J t2 at P 2 (0, 0, 0 ). (d) (Harder) Use the continuity equation to help
+
show that J N1 − J N2 = ∂ D N2 /∂t − ∂ D N1 /∂t, and then determine D N2 ,
J N2 , and E N2 .
9.24 Avector potential is given as A = A 0 cos(ωt − kz) a y .(a) Assuming as
many components as possible are zero, find H, E, and V .(b) Specify k in
terms of A 0 , ω, and the constants of the lossless medium, and µ.
9.25 In a region where µ r = r = 1 and σ = 0, the retarded potentials are given
z √
by V = x(z − ct)V and A = x c − t a z Wb/m, where c = 1 µ 0 0 .
∂V
(a) Show that ∇ · A =−µ .(b) Find B, H, E, and D.(c) Show that
∂t
these results satisfy Maxwell’s equations if J and ρ ν are zero.
9.26 Write Maxwell’s equations in point form in terms of E and H as they apply
to a sourceless medium, where J and ρ v are both zero. Replace by µ, µ by
, E by H, and H by −E, and show that the equations are unchanged. This
is a more general expression of the duality principle in circuit theory.