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Unformatted text preview: PROBLEMS (a) (b) (C) (d) (D Figure 5.17 For Review Question 5.8. SEBTIDN 5.3 BUNVEETIDN AND BUNDUCTIDN BURRENTS 5.1 5.2 5.3 5.4 In a certain region, J = 3r2 cos 0 a, — r2 sin 6 as A/m. Find the current crossing the
surface deﬁned by 6 = 30°, 0 < (1) < 27r, 0 < r < 2 m. Determine the total current in a wire of radius 1.6 mm placed along the z—axis if
J 500 az p A/mz. The current density in a cylindrical conductor of radius a placed along the z—axis is
J 10e_(1_’°/“)az A/mz Find the current through the cross section of the conductor. The charge 10‘4e_3’ C is removed from a sphere through a wire. Find the current in the
wire att = 0 andt = 2.5 s. 204 CHAPTER 5 ELECTRlE FIELDS IN MATERIAL SPACE SEBTIDN 5.4 EDNDUBTDRE 5.5 5.6 5.7 5.8 5.9 5.10 A l M0 resistor is formed by a cylinder of graphite—clay mixture having a length of 2 cm
and a radius of 4 mm. Determine the conductivity of the resistor. If the ends of a cylindrical bar of carbon (0 = 3 X 104 S/m) of radius 5 mm and length
8 cm are maintained at a potential difference of 9 V, ﬁnd (a) the resistance of the bar,
(b) the current through the bar, (0) the power dissipated in the bar. The resistance of a round long wire of diameter 3 mm is 4.04 (Z/km. If a current of 40 A
ﬂows through the wire, ﬁnd (a) The conductivity of the wire and identify the material of the wire (b) The electric current density in the wire A coil is made of 150 turns of copper wire wound on a cylindrical core. If the mean radius
of the turns is 6.5 mm and the diameter of the wire is 0.4 mm, calculate the resistance of the coil. A composite conductor 10 m long consists of an inner core of steel of radius 1.5 cm and
an outer sheath of copper_whose thickness is 0.5 cm. Take the resistivities of copper and
steel as 1.77 X 10—8 and 11.8 X 10"8 0  m, respectively. (a) Determine the resistance of the conductor.
(b) If the total current in the conductor is 60 A, what current ﬂows in each metal? (c) Find the resistance of a solid copper conductor of the same length and crosssectional
areas as the sheath. A hollow cylinder of length 2 m has its cross section as shown in Figure 5.18. If the cylin
der is made of carbon (0 = 3 X 104 S/m), determine the resistance between the ends of the cylinder. Take a = 3 cm, 19 = 5 cm. SEBTIDNS 5.5—‘5.'7 PDLARIZATlDN AND DIELECTRIB CONSTANT 5.11 At a particular temperature and pressure, a helium gas contains 5 X 1025 atoms/m3. If a
10 kV/m ﬁeld applied to the gas causes an average electron cloud shift of 10—18 m, ﬁnd
the dielectric constant of helium. Figure 5.18 For Problems 5.10 and 5.19. PROBLEMS 205 5.12 A dielectric material contains 2 X 1019 polar molecules/m3, each of dipole moment
1.8 X 10—27 C ' m. Assuming that all the dipoles are aligned in the direction of the elec
tric ﬁeld E = lOSax V/m, ﬁnd P and 8,. " 5.13 In a slab of dielectric material for which a = 2.480 and V = 300z2 V, ﬁnd (a) D and pv,
(b) P and ppv. 5.14 (a) Let V = xzyzz in a region (a = 280) deﬁned by ~l < x, y, z < 1. Find the charge
density pv in the region. (b) If the charge travels at 104yay m/s, determine the current crossing surface
0<x,z<0.5,y=1. 5.15 In a slab of Teﬂon (8 = 2, 180), E = 6ax + l2ay — 20az V/m, ﬁnd D and P. 5.16 In a dielectric material (a = 580), the potential ﬁeld V = lezyz ~ 5z2 V, determine
(a) E, (b) D, (C) P, (d) pv 5.17 A point charge Q is located at the center of a spherical dielectric (a = 8,30) shell of inner
radius a and outer radius 1). Determine E, D, P, and V. 5.18 For x < 0, P = 5 sin(ozy)ax, Where oz is a constant. Find pm and ppv. 5.19 Consider Figure 5.18 as a spherical dielectric shell so that a = soar for a < r <“b and
8 = so for O < r < a. If a charge Q is placed at the center of the shell, ﬁnd (a) Pfora < r< b
(b) ppvfora < r< b
(c) ppsatr = aandr= b 5.20 Two point charges in free space are separated by distance d and exert a force 2.6 nN on
each other. The force becomes 1.5 nN when the free space is replaced by a homogeneous
dielectric material. Calculate the dielectric constant of the material. 5.21 A conducting sphere of radius 10 cm is centered at the origin and embedded in a dielectric
material with 8 = 2.580. If the sphere carries a surface charge of 4 nC/mz, ﬁnd E at
(*3 cm,4cm, 12 cm).  5.22 A solid sphere of radius a and dielectric constants, has a uniform volume charge density
' of po.
(a) At the center of the sphere, Show that 2
pea V .—_
6808,. (28, + l)
(b) Find the potential at the surface of the sphere. 5.23 In an anisotropic medium, D is related to E as Dx 41112,.
Dy=80131Ey
D 112E Z Find D due to E = Eo(ax + a),  az) V/m. 206 CHAPTER 5 ELECTRIC FIELDS IN MATERIAL SPACE SEBTIDN 5.8 CDNTINLIITY EQUATION AND RELAXATIDN TIME 5.24 5.25 5.26 5.27 5.28 5.29 For static (time—independent) ﬁelds, which of the following current densities are possible? (a) J = 2x3yax + 4x2z2ay — 6x2yzaz (b) J = xyax + y(z + 1)ay + 2yaz 2
(c) J : Zzap + zcos<j>aZ sin 6
(d) J = 2 a. r 100 2 . . . .
If J = —2— ap A/m , ﬁnd (a) the t1me rate of 1ncrease 1n the volume charge den51ty, (b) the
. p
total current passing through surface deﬁned by p = 2, 0 < z < 1, 0 < qb < 21r.
56—104! Given that J = ——r—a,.A/m2, at t: 0.1 ms, ﬁnd (a) the current passing surface r = 2 m, (b) the charge density p, on that surface. Determine the relaxation time for each of the following media: (a) Hard rubber (a = 10~15 S/m, s = 3.180)
(b) Mica (a = 10‘15 S/m, 8 630)
(c) Distilled water (a = 10—4 S/m, 8 = 8080) The excess charge in a certain medium decreases to one—third of its initial value in 20 ,us. (a) If
the conductivity of the medium is 10—4 S/m, what is the dielectric constant of the medium?
(b) What is the relaxation time? (c) After 30 us, what fraction of the charge will remain? Lightning strikes a dielectric sphere of radius 20 mm for which 8,. = 2.5, a =
5 X 10—6 S/m and deposits uniformly a charge of l C. Determine the initial volume
charge density and the volume charge density 2 us later. SECTION 5.9 BUUNDARY BBNDITIDNE‘: 5.30 5.31 5.32 5.33 5.34 The plane z = 4 is the interface between two dielectrics. The dielectric in region 2 > 4
has dielectric constant of 5 and E = 63,, — 1221y + 83Z V/m in that region. If the dielec
tric constant is 2 in region z < 4, ﬁnd the electric ﬁeld intensity in that region. A dielectric interface is deﬁned by 4x + 3y = 10 m. The region including the origin is
free space, where D1 = 23,,  4ay + 6.53Z nC/mz. In the other region, 8,2 = 2.5. Find
D2 and the angle 62 that D2 makes with the normal. Given that E1 103,,  6ay + 1232 V/m in Figure 5.19, ﬁnd (a) P1, (b) E2 and the angle
E2 makes with the y—axis, (c) the energy density in each region. Two homogeneous dielectric regions 1 (p S 4 cm) and 2 (p 2 4 cm) have dielectric con—
stants 3.5 and 1.5, respectively. If D2 = 12ap — 6a¢ + 9az nC/mz, calculate (a) E1 and
D1, (b) P2 and ppvz, (c) the energy density for each region. A conducting sphere of radius a is halfembedded in a liquid dielectric medium of per’
mittivity 81 as in Figure 5.20. The region above the liquid is a gas of permittivity 82. If the
total free charge on the sphere is Q, determine the electric ﬁeld intensity everywhere. PRDBLEMS 207 5"] = 380 Figure 5.19 For Problem 5.32. Figure 5.20 For Problem 5.34. *5.35 Two parallel sheets of glass (8,. = 85) mounted vertically are separated by a uniform air
gap between their inner surface. The sheets, properly sealed, are immersed in oil
(8,. = 3 .0) as shown in Figure 5.21. A uniform electric ﬁeld of strength 2 kV/m in the hor—
izontal direction exists in the oil. Calculate the magnitude and direction of the electric
ﬁeld in the glass and in the enclosed air gap when (a) the ﬁeld is normal to the glass sur
faces and (b) the ﬁeld in the oil makes an angle of 75° with a normal to the glass surfaces.
Ignore edge effects. 5.36 (a) Given that E = 153x ~ 8az V/m at a point on a conductor surface, what is the sur—
face charge density at that point? Assume 8 = 80. (b) Region y 2 2 is occupied by a conductor. If the surface charge on the conductor is
— 20 nC/mz, ﬁnd D just outside the conductor. 5.37 A silver—coated sphere of radius 5 cm carries a total charge of 12 nC uniformly distributed
on its surface in free space. Calculate (a) D on the surface of the sphere, (b) D external
to the sphere, (c) the total energy stored in the ﬁeld. 5.38 The magnitude of the electric ﬁeld just above the surface of a conducting sphere of radius
10 cm is 4 MV/m. Determine the charge on the surface of the sphere. Glass Figure 5.21 For Problem 5.35. /\ Oil Oil ...
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This note was uploaded on 02/04/2012 for the course ECE 311 taught by Professor Peroulis during the Fall '08 term at Purdue.
 Fall '08
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