Page 92 - Engineering Electromagnetics, 8th Edition

74 ENGINEERING ELECTROMAGNETICS

3.24 In a region in free space, electric flux density is found to be
ρ 0 (z + 2d) a z C/m (−2d ≤ z ≤ 0)

2
D =
−ρ 0 (z − 2d) a z C/m 2 (0 ≤ z ≤ 2d)
Everywhere else, D = 0. (a) Using ∇· D = ρ v , find the volume charge
density as a function of position everywhere. (b) Determine the electric flux
that passes through the surface defined by z = 0, −a ≤ x ≤ a, −b ≤ y ≤ b.
(c) Determine the total charge contained within the region −a ≤ x ≤ a,
−b ≤ y ≤ b, −d ≤ z ≤ d.(d) Determine the total charge contained within
the region −a ≤ x ≤ a, −b ≤ y ≤ b,0 ≤ z ≤ 2d.

3.25 Within the spherical shell, 3 < r < 4m, the electric flux density is given as
3
2
D = 5(r − 3) a r C/m .(a) What is the volume charge density at r = 4?
(b) What is the electric flux density at r = 4? (c)How much electric flux
leaves the sphere r = 4? (d)How much charge is contained within the sphere
r = 4?
3
3.26 If we have a perfect gas of mass density ρ m kg/m , and we assign a
velocity U m/s to each differential element, then the mass flow rate is
2
ρ m U kg/(m − s). Physical reasoning then leads to the continuity equation,
∇ · (ρ m U) =−∂ρ m /∂t.(a) Explain in words the physical interpretation of
this equation. (b) Show that s ρ m U · dS =−dM/dt, where M is the total
mass of the gas within the constant closed surface S, and explain the physical
significance of the equation.
2
2
2
2
3.27 Let D = 5.00r a r mC/m for r ≤ 0.08 m and D = 0.205 a r /r µC/m for
r ≥ 0.08 m. (a) Find ρ ν for r = 0.06 m. (b) Find ρ ν for r = 0.1m.(c) What
surface charge density could be located at r = 0.08 m to cause D = 0 for
r > 0.08 m?
3.28 Repeat Problem 3.8, but use ∇ · D = ρ ν and take an appropriate volume
integral.
3.29 In the region of free space that includes the volume 2 < x, y, z < 3, D =
2 2 (yz a x + xz a y − 2xy a z ) C/m .(a)Evaluate the volume integral side of
2
z
the divergence theorem for the volume defined here. (b)Evaluate the surface
integral side for the corresponding closed surface.
3.30 (a) Use Maxwell’s first equation, ∇· D = ρ v ,to describe the variation of the
electric field intensity with x in a region in which no charge density exists
and in which a nonhomogeneous dielectric has a permittivity that increases
exponentially with x. The field has an x component only; (b) repeat part (a),
but with a radially directed electric field (spherical coordinates), in which
again ρ v = 0, but in which the permittivity decreases exponentially with r.
16 2
3.31 Given the flux density D = r cos(2θ) a θ C/m , use two different methods to
find the total charge within the region 1 < r < 2m,1 <θ < 2 rad,
1 <φ < 2 rad.