Page 195 - Engineering Electromagnetics, 8th Edition

CHAPTER 6 Capacitance 177

6.28 Show that in a homogeneous medium of conductivity σ, the potential field
V satisfies Laplace’s equation if any volume charge density present does
not vary with time.
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6.29 Given the potential field V = (Aρ + Bρ −4 ) sin 4φ:(a) Show that ∇ V = 0.
(b) Select A and B so that V = 100 V and |E|= 500 V/m at P(ρ = 1,
φ = 22.5 , z = 2).
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6.30 A parallel-plate capacitor has plates located at z = 0 and z = d. The region
between plates is filled with a material that contains volume charge of uniform
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density ρ 0 C/m and has permittivity . Both plates are held at ground
potential. (a) Determine the potential field between plates. (b) Determine the
electric field intensity E between plates. (c) Repeat parts (a) and (b) for the
case of the plate at z = d raised to potential V 0 , with the z = 0 plate grounded.
6.31 Let V = (cos 2φ)/ρ in free space. (a) Find the volume charge density at
point A(0.5, 60 , 1). (b) Find the surface charge density on a conductor
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surface passing through the point B(2, 30 , 1).
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6.32 A uniform volume charge has constant density ρ ν = ρ 0 C/m and fills the
region r < a,in which permittivity is assumed. A conducting spherical
shell is located at r = a and is held at ground potential. Find (a) the
potential everywhere; (b) the electric field intensity, E,everywhere.
6.33 The functions V 1 (ρ, φ, z) and V 2 (ρ, φ, z) both satisfy Laplace’s equation
in the region a <ρ < b,0 ≤ φ< 2π, −L < z < L; each is zero on
the surfaces ρ = b for −L < z < L; z =−L for a <ρ < b; and z = L for
a <ρ < b; and each is 100 V on the surface ρ = a for −L < z < L. (a)In
the region specified, is Laplace’s equation satisfied by the functions V 1 + V 2 ,
V 1 − V 2 , V 1 + 3, and V 1 V 2 ?(b)On the boundary surfaces specified, are the
potential values given in this problem obtained from the functions V 1 + V 2 ,
V 1 − V 2 , V 1 + 3, and V 1 V 2 ?(c) Are the functions V 1 + V 2 , V 1 − V 2 ,
V 1 + 3, and V 1 V 2 identical with V 1 ?
6.34 Consider the parallel-plate capacitor of Problem 6.30, but this time the
charged dielectric exists only between z = 0 and z = b, where b < d.
Free space fills the region b < z < d. Both plates are at ground
potential. By solving Laplace’s and Poisson’s equations, find (a) V (z)
for 0 < z < d;(b) the electric field intensity for 0 < z < d.
No surface charge exists at z = b,so both V and D are continuous there.
6.35 The conducting planes 2x + 3y = 12 and 2x + 3y = 18 are at potentials
of 100 V and 0, respectively. Let = 0 and find (a) V at P(5, 2, 6); (b) E
at P.
6.36 The derivation of Laplace’s and Poisson’s equations assumed constant
permittivity, but there are cases of spatially varying permittivity in which the
equations will still apply. Consider the vector identity, ∇ · (ψG) = G · ∇ψ +
ψ∇ · G, where ψ and G are scalar and vector functions, respectively.