Page 158 - Engineering Electromagnetics, 8th Edition

140 ENGINEERING ELECTROMAGNETICS

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5.15 Let V = 10(ρ + 1)z cos φ Vin free space. (a) Let the equipotential surface
V = 20 V define a conductor surface. Find the equation of the conductor
surface. (b) Find ρ and E at that point on the conductor surface where φ =
0.2π and z = 1.5. (c) Find |ρ S | at that point.
5.16 A coaxial transmission line has inner and outer conductor radii a and b.
Between conductors (a <ρ < b) lies a conductive medium whose
conductivity is σ(ρ) = σ 0 /ρ, where σ 0 is a constant. The inner conductor is
charged to potential V 0 , and the outer conductor is grounded. (a) Assuming
dc radial current I per unit length in z, determine the radial current density
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field J in A/m .(b) Determine the electric field intensity E in terms of I and
other parameters, given or known. (c)By taking an appropriate line integral
of E as found in part (b), find an expression that relates V 0 to I.(d) Find an
expression for the conductance of the line per unit length, G.
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5.17 Given the potential field V = 100xz/(x + 4) V in free space: (a) Find D at
the surface z = 0. (b) Show that the z = 0 surface is an equipotential surface.
(c) Assume that the z = 0 surface is a conductor and find the total charge on
that portion of the conductor defined by 0 < x < 2, −3 < y < 0.
5.18 Two parallel circular plates of radius a are located at z = 0 and z = d. The
top plate (z = d)is raised to potential V 0 ; the bottom plate is grounded.
Between the plates is a conducting material having radial-dependent
conductivity, σ(ρ) = σ 0 ρ, where σ 0 is a constant. (a) Find the ρ-independent
electric field strength, E, between plates. (b) Find the current density, J
between plates. (c) Find the total current, I,in the structure. (d) Find the
resistance between plates.
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5.19 Let V = 20x yz − 10z Vin free space. (a) Determine the equations of the
equipotential surfaces on which V = 0 and 60 V. (b) Assume these are
conducting surfaces and find the surface charge density at that point on the
V = 60 V surface where x = 2 and z = 1. It is known that 0 ≤ V ≤ 60Vis
the field-containing region. (c)Give the unit vector at this point that is
normal to the conducting surface and directed toward the V = 0 surface.
5.20 Two point charges of −100πµC are located at (2, −1, 0) and (2, 1, 0). The
surface x = 0isa conducting plane. (a) Determine the surface charge
density at the origin. (b) Determine ρ S at P(0, h, 0).
5.21 Let the surface y = 0bea perfect conductor in free space. Two uniform
infinite line charges of 30 nC/m each are located at x = 0, y = 1, and
x = 0, y = 2. (a) Let V = 0at the plane y = 0, and find V at P(1, 2, 0).
(b) Find E at P.
5.22 The line segment x = 0, −1 ≤ y ≤ 1, z = 1, carries a linear charge density
ρ L = π|y| µC/m. Let z = 0bea conducting plane and determine the surface
charge density at: (a)(0, 0, 0); (b)(0, 1, 0).