Page 156 - Engineering Electromagnetics, 8th Edition

138 ENGINEERING ELECTROMAGNETICS

3. Fink, D. G., and H. W. Beaty. Standard Handbook for Electrical Engineers. 15th ed. New
York: McGraw-Hill, 2006.
4. Maxwell, J. C. ATreatise on Electricity and Magnetism.New York: Cambridge
University Press, 2010.
5. Wert, C. A., and R. M. Thomson. Physics of Solids. 2nd ed. New York: McGraw-Hill,
1970. This is an advanced undergraduate-level text that covers metals, semiconductors,
and dielectrics.

CHAPTER 5 PROBLEMS
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5.1 Given the current density J =−10 [sin(2x)e −2y a x + cos(2x)e −2y a y ] kA/m 2
(a) Find the total current crossing the plane y = 1inthe a y direction in the
region 0 < x < 1, 0 < z < 2. (b) Find the total current leaving the region
0 < x, y < 1, 2 < z < 3by integrating J · dS over the surface of the cube.
(c) Repeat part (b), but use the divergence theorem.
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5.2 Given J =−10 (ya x + xa y )A/m , find the current crossing the y = 0
−4
plane in the −a y direction between z = 0 and 1, and x = 0 and 2.
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2
5.3 Let J = 400 sin θ/(r + 4) a r A/m .(a) Find the total current flowing
through that portion of the spherical surface r = 0.8, bounded by
0.1π< θ< 0.3π, 0 <φ < 2π. (b) Find the average value of J over the
defined area.
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2
5.4 If volume charge density is given as ρ v = (cos ωt)/r C/m in spherical
coordinates, find J.Itis reasonable to assume that J is not a function of θ or φ.
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2
5.5 Let J = 25/ρa ρ − 20/(ρ + 0.01) a z A/m .(a) Find the total current
crossing the plane z = 0.2inthe a z direction for ρ< 0.4. (b) Calculate
∂ρ ν /∂t. (c) Find the outward current crossing the closed surface defined by
ρ = 0.01,ρ = 0.4, z = 0, and z = 0.2. (d) Show that the divergence
theorem is satisified for J and the surface specified in part (c).
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5.6 In spherical coordinates, a current density J =−k/(r sin θ) a θ A/m exists in
a conducting medium, where k is a constant. Determine the total current in
the a z direction that crosses a circular disk of radius R, centered on the z axis
and located at (a) z = 0; (b) z = h.
5.7 Assuming that there is no transformation of mass to energy or vice versa, it is
possible to write a continuity equation for mass. (a)Ifwe use the continuity
equation for charge as our model, what quantities correspond to J and ρ ν ?
(b)Given a cube 1 cm on a side, experimental data show that the rates at
which mass is leaving each of the six faces are 10.25, −9.85, 1.75, −2.00,
−4.05, and 4.45 mg/s. If we assume that the cube is an incremental volume
element, determine an approximate value for the time rate of change of
density at its center.
5.8 A truncated cone has a height of 16 cm. The circular faces on the top and
bottom have radii of 2 mm and 0.1 mm, respectively. If the material from