Page 91 - Engineering Electromagnetics, 8th Edition

CHAPTER 3 Electric Flux Density, Gauss’s Law, and Divergence 73

total charge is contained within a cylinder of radius a and height b, where the
cylinder axis is the z axis?
2 2
2
3.17 A cube is defined by 1 < x, y, z < 1.2. If D = 2x ya x + 3x y a y C/m 2
(a) Apply Gauss’s law to find the total flux leaving the closed surface of the
cube. (b)Evaluate ∇ · D at the center of the cube. (c) Estimate the total
charge enclosed within the cube by using Eq. (8).
3.18 State whether the divergence of the following vector fields is positive,
2
negative, or zero: (a) the thermal energy flow in J/(m − s) at any point in a
2
freezing ice cube; (b) the current density in A/m in a bus bar carrying direct
2
current; (c) the mass flow rate in kg/(m − s) below the surface of water in a
basin, in which the water is circulating clockwise as viewed from above.
3.19 A spherical surface of radius 3 mm is centered at P(4, 1, 5) in free space. Let
2
D = xa x C/m . Use the results of Section 3.4 to estimate the net electric flux
leaving the spherical surface.
3.20 A radial electric field distribution in free space is given in spherical
coordinates as:
rρ 0
E 1 = a r (r ≤ a)
3 0
3 3
(2a − r )ρ 0
E 2 = a r (a ≤ r ≤ b)
3 0 r 2
3 3
(2a − b )ρ 0
E 3 = a r (r ≥ b)
3 0 r 2
where ρ 0 , a, and b are constants. (a) Determine the volume charge density in
the entire region (0 ≤ r ≤∞)by the appropriate use of ∇· D = ρ v .(b)In
terms of given parameters, find the total charge, Q, within a sphere of radius
r where r > b.
2
3.21 Calculate ∇ · D at the point specified if (a) D = (1/z )[10xyz a x +
3
2
2
2
5x z a y + (2z − 5x y) a z ]at P(−2, 3, 5); (b) D = 5z a ρ + 10ρz a z at
P(3, −45 , 5); (c) D = 2r sin θ sin φ a r + r cos θ sin φ a θ + r cos φ a φ at
◦
P(3, 45 , −45 ).
◦
◦
3.22 (a)A flux density field is given as F 1 = 5a z .Evaluate the outward flux of F 1
through the hemispherical surface, r = a, 0 <θ <π/2, 0 <φ < 2π.
(b) What simple observation would have saved a lot of work in part a?
(c)Now suppose the field is given by F 2 = 5za z . Using the appropriate
surface integrals, evaluate the net outward flux of F 2 through the closed
surface consisting of the hemisphere of part a and its circular base in the xy
plane. (d) Repeat part c by using the divergence theorem and an appropriate
volume integral.
3.23 (a)A point charge Q lies at the origin. Show that div D is zero everywhere
except at the origin. (b) Replace the point charge with a uniform volume
charge density ρ v0 for 0 < r < a. Relate ρ v0 to Q and a so that the total
charge is the same. Find div D everywhere.