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Unformatted text preview: PROBLEMS 261 SECTION 6.2 POISSDN’S AND LAF‘LAGE’S EQUATIDNS 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 Given V = 5x3y2z and e = 2.2580, ﬁnd (a) E at point P(—3, 1, 2), (b) p, at P. Conducting sheets are located at y = 1 and y = 3 planes. The space between them is ﬁlled with a nonuniform charge distribution pv = f" nC/m3 and a = 480. Assuming that
7r V(y=1)=OandV(y=3)=50V,findV(y=2). The region between x = 0 and x = d is free space and has pv = p0(x  d)/d. If
V(x = 0) = O and V(x = d) = V0, ﬁnd (a) V and E, (b) the surface charge densities at
x = 0 and x = d. A certain material occupies the space between two conducting slabs located at y =
i 2 cm. When heated, the material emits electrons such that pv = 50(1 — yz) ,uC/m3. If
the slabs are both held at 30 kV, ﬁnd the potential distribution within the slabs. Take 8 = 380. An inﬁnitely long coaxial cylindrical structure has an inner conductor of radius 2 mm and
an outer conductor of radius 4.5 mm. The space between the conductor is ﬁlled with an 1080 p
the outer conductor is maintained at 40 V, determine the potential distribution for 2 mm < p < 4.5 mm. electron cloud with pv = C/m3 and a = so. If the inner conductor is grounded, and Determine which of the following potential ﬁeld distributions satisfy Laplace’s equation. (a) V1=x2+y22z2+10 1
b V = “—
( ) 2 (x2 + yz + 22)1/2
(0) V3 = pz sind) + p2
10 sin6 sinqb
(d) V. = ———~—~——r, Show that each of the following potentials satisﬁes Laplace’s equation. (a) V1 = e'ysinx (b) V2 = Vo sin<ﬂ> sinh<ﬂ>
a a cos <15
(C) V3 = 10 cos 0
(C) V4 = T Show that E = (Ex, Ey, Ez) satisﬁes Laplace’s equation. Let V = (A cos nx + B sin nx)(Ce"y + De_"y), where A, B, C, and D are constants.
Show that V satisﬁes Laplace’s equation. ‘ 262 CHAPTER 6 ELECTROSTATIC BOUNDARYVALUE PROBLEMS 6.10 The potential ﬁeld V = 2x2yz ~ y3z exists in a dielectric medium having 8 = 280. 6.11 6.12 6.13 6.14 6.15 *6.16 2 Figure 6.28 For Problem 6.11. V(z=0)=0 (a) Does V satisfy Laplace’s equation? (b) Calculate the total charge within the unit cube
0<x< lm,0<y<1m,0<z<lm. Consider the conducting plates shown in Figure 6.28. If V(z = 0) = 0 and
V(z = 2 mm) = 50 V, determine V, E, and D in the dielectric region (8,. = 1.5) between
the plates and p5 on the plates. The cylindrical capacitor whose cross section is in Figure 6.29 has inner and outer radii of
5 mm and 15 mm, respectively. If V(p = 5 mm) = 100 V and V(p = 15 mm) = O V,
calculate V, E, and D at p = 10 mm and p3 on each plate. Take 8,. = 2.0. Two conducting cylinders are located in free space at p = 1 cm and p = 3 cm and main—
tained at potentials 20 mV and — 10 mV, respectively. Find V and E. The region between concentric spherical conductingshells r = 0.5 m and r = l m is __
charge free. If V(r = 0.5) = —50 V and V(r = 1) = 50 V, determine the potential
distribution and the electric ﬁeld strength in the region between the shells. Find V and E at (3, 0, 4) due to the two conducting cones of inﬁnite extent shown in
Figure 6.30. The inner and outer electrodes of a diode are coaxial cylinders of radii a = 0.6 mm and ___g
b = 30 mm, respectively. The inner electrode is maintained at 70 V, while the outer elec—
trode is grounded. (a) Assuming that the length of the electrodes {3 >> a, b and ignoring the effects of space charge, calculate the potential at p = 15 mm. (b) If an electron iS__,j
injected radially through a small hole in the inner electrode with velocity 107 m/s, ﬁnd its :
velocity at p = 15 mm. Figure 6.29 Cylindrical capacitor of Problem 6.12. PROBLEMS 263 Figure 6.30 Conducting cones of Problem 6.15. 6.17 Another method of ﬁnding the capacitance of a capacitor is by using energy considera
tions, that is, 2W 1
= 2E = *3 [ elElzdv
V0 V0 C Using this approach, derive eqs. (6.22), (6.28), and (6.32). 6.18 An electrode with a hyperbolic shape (xy = 4) is placed above a grounded right—angle corner as in Figure 6.31. Calculate V and E at point (1, 2, 0) when the electrode is con
nected to a 20 V source. *6.19 Solve Laplace’s equation for the twodimensional electrostatic systems of Figure 6.32 and
ﬁnd the potential V(x, y). *6.20 Find the potential V(x, y) due to the twodimensional systems of Figure 6.33. 6.21 By letting V(p, qb) = R(p)¢(¢) be the solution of Laplace’s equation in a region where
p # 0, show that the separated differential equations for R and <25 are R’ )x
R"+—*——2R=0
P P
and
¢”+)\¢=0 where A is the separation constant. 6.22 A potential in spherical coordinates is a function of r and 0 but not <1). Assuming that V(r, 0) = R(r)F(0), obtain the separated differential equations for R and F in a region for
which pv = 0. CHAPTER 6 ELECTROSTATIC BOUNDARY—VALUE PROBLEMS y Figure 6.31 For Problem 6.18. (a) (b) (C) Figure 6.33 For Problem 6.20. PROBLEMS 265 Figure 6.34 For Problem 6.24. SECTION 6.5 RESISTANCE AND CAPACITANCE 6.23 Show that the resistance of the bar of Figure 6.17 between the vertical ends located at
¢ = Oanqu = 7r/2is ___7_’..._
b
Zat 1n 
a *6.24 Show that the resistance of the sector of a spherical shell of conductivity 0, with cross sec— tion shown in Figure 6.34 (where O 5 ¢ < 2%), between its base (i.e., from r = a to
r = b) is R____L__{i_q 27ro(l ~— cos 01) a b *6.25 A hollow conducting hemisphere of radius a is buried with its ﬂat face lying ﬂush with the
earth’s surface, thereby serving as an earthing electrode. If the conductivity of earth is a, show that the leakage conductance between the electrode and earth is 27raa. 6.26 The cross section of an electric fuse is shown in Figure 6.35. If the fuse is made of copper
and is 1.5 mm thick, calculate its resistance. 4cm 4cm +———————><——————> 3cm 4cm Figure 6.35 For Problem 6.26. CHAPTER 6 ELECTROSTATIC BOUNDARYVALUE PROBLEMS Depth = 15 cm 20 cm Figure 6.36 For Problem 6.28. 1mm lmm 1mm Area = 80 cm2 Figure 6.37 For Problem 6.29. 6.27 In an integrated circuit, a capacitor is formed by growing a silicon dioxide layer (8,. = 4)
of thickness 1 am over the conducting silicon substrate and covering it with a metal elec—
trode of area S. Determine S if a capacitance of 2 nF is desired. 6.28 Calculate the capacitance of the parallel—plate capacitor shown in Figure 6.36.
6.29 Evaluate the capacitance of the parallel—plate capacitor shown in Figure 6.37. 6.30 The parallel—plate capacitor of Figure 6.38 is quarter—ﬁlled with mica (8,. = 6). Find the 
capacitance of the capacitor. 6.31 Determine the force between the plates of a parallelplate capacitor. Assume that sepal’ﬂ' _
tion distance is d and the area of each plate is S. *6.32 An air—ﬁlled parallel plate capacitor of length L, width a, and plate separation d has its 
plates maintained at constant potential difference V0. If a dielectric slab of dielectric Figure 6.38 For Problem 6.30. 2mm 6.33 6.34 6.35 6.36 6.37 6.38 6.39 6.40 PRDBLEMS 267 Figure 6.39 For Problem 6.32. ‘——x—> constant a, is slid between the plates and is withdravVn until only a length x remains be—
tween the plates as in Figure 6.39, show that the force tending to restore the slab to its orig—
inal position is
are. — 1) a V3 2d A parallelplate capacitor has plate area 200 cm2 and plate separation of 3 mm. The charge
density is l ,uC/mz and air is the dielectric. Find (a) The capacitance of the capacitor
(b) The voltage between the plates
(c) The force with which the plates attract each other Two conducting plates are placed at z = ~2 cm and z = 2 cm and are, respectively,
maintained at potentials 0 and 200 V. Assume that the plates are separated by a polypropy—
lene dielectric (8 = 2.2580). Calculate (a) the potential at the middle of the plates,
(b) the surface charge densities at the plates. Two conducting parallel plates are separated by a dielectric material with a = 5.6.90 and
thickness of 0.64 mm. Assume that each plate has an area of 80 cm2. If the potential ﬁeld
distribution between the plates is V = 3x + 4y — 12z + 6 kV, determine (a) the capac—
itance of the capacitor, (b) the potential difference between the plates. The space between spherical conducting shells r = 5 cm and r = 10 cm is ﬁlled with a
dielectric material for which 8 = 2.2580. The two shells are maintained at a potential dif—
ference of 80 V. (a) Find the capacitance of the system. (b) Calculate the charge density on
shell r = 5 cm. Concentric shells r = 20 cm and r = 30 cm are held at V = 0 and V = 50, respectively.
If the space between them is ﬁlled with dielectric material (8 = 3.180, a = lO~12 S/m),
ﬁnd (a) V, E, and D, (b) the charge densities on the shells, (c) the leakage resistance. A spherical capacitor has inner radius d and outer radius a. Concentric with the spherical
conductors and lying between them is a spherical shell of outer radius c and inner radius
b. If the regions d < r < c, c < r < b, and b < r < a are ﬁlled with materials with per—
mittivites 81, 82, and 83, respectively, determine the capacitance of the system. Determine the capacitance of a conducting sphere of radius 5 cm deeply immersed in sea—
water (8, = 80). A conducting sphere of radius 2 cm is surrounded by a concentric conducting sphere of
radius 5 cm. If the space between the spheres is ﬁlled with sodium chloride (8, = 5.9),
calculate the capacitance of the system. 268 CHAPTER 6 ELECTROSTATIC BUUNDARY‘VALUE PROBLEMS *6.41 6.42 6.43 *6.44 Liquid
reservoir V0 Figure 6.40 Simpliﬁed geometry of an inkjet
printer; for Problem 6.41. Liquid jet In an ink—jet printer the drops are charged by surrounding the jet of radius 20 pm with a
concentric cylinder of radius 600 ,um as in Figure 6.40. Calculate the minimum voltage
required to generate a charge 50 fC on the drop if the length of the jet inside the cylinder is 100 um. Take 8 = 80. A given length of a cable has capacitance of 10 ,uF, and the resistance of the insulation is
100 M0. The cable, is charged to a voltage of 100 V. How long does it take the voltage to drop to 50 V? The capacitance per unit length of a two—wire transmission line shown in Figure 6.41 is
given by
7T8 d
h—1 — [2a] Determine the conductance per unit length. C: A spherical capacitor has an inner conductor of radius a carrying charge Q and is main r
tained at zero potential. If the outer conductor contracts from a radius 19 to c under internal
forces, prove that the work performed by the electric ﬁeld as a result of the contraction iS = 9% — c) W
Swebc 4 Figure 6.41 For Problem 6.43. *6.45 6.46 6.47 6.48 6.49 PROBLEMS 269 A parallel—plate capacitor has its plates at x = 0, d and the space between the plates is
ﬁlled with an inhomogeneous material with permittivity 8 = 80<l + If the plate at
x = d is maintained at V0 while the plate at x = 0 is grounded, ﬁnd: (a) Vand E
(b) P
(c) ppsatx = 0,d A spherical capacitor has inner radius a and outer radius 1) and is ﬁlled with an inhomoge—
neous dielectric with 8 = sok/rz. Show that the capacitance of the capacitor is
C = 47r80k
b — a A cylindrical capacitor with inner radius a and outer radius [9 is ﬁlled with an inhomoge—
neous dielectric having a = eok/p, where k is a constant. Calculate the capacitance per
unit length of the capacitor. If the earth is regarded a spherical capacitor, what is its capacitance? Assume the radius of
the earth to be approximately 6370 km. The space between two conducting concentric cylinders of radii a and b (b > a) is ﬁlled
with an inhomogeneous material with conductivity 0 = k/p, where k is a constant. Deter
mine the resistance of the system. BEGTIDN 6.6 METHDD BF IMAGES 6.50 6.51 6.52 6.53 *6.54 A grounded metal sheet is located in the z = 0 plane, while a point charge Q is located at
(0, 0, a). Find the force acting on a point charge —Q placed at (a, 0, a). Grounded conducting sheets are situated at x = O and y = 0, while a point charge Q is
placed at (a, a, 0). Determine the potential for x > 0, y > 0. Two point charges of 3 nC and —4 nC are placed, respectively, at (0,0, 1m) and
(0, 0, 2 m) while an inﬁnite conducting plane is at z = 0. Determine (a) The total charge induced on the plane
(b) The magnitude of the force of attraction between the charges and the plane Two point charges of 50 nC and  20 nC are located at ( 3, 2, 4) and (l, 0, 5) above the conducting ground plane z = 2. Calculate (a) the surface charge density at (7, —2, 2),
(b) D at (3, 4, 8), and (c) D at (1, 1, 1). A point charge of 10 ,uC is located at (1, l, 1), and the positive portions of the coordinate
planes are occupied by three mutually perpendicular plane conductors maintained at zero
potential. Find the force on the charge due to the conductors. 270 CHAPTER 6 ELECTROSTATIC BOUNDARYVALUE PROBLEMS 6.55 A point charge Q is placed between two earthed intersecting conducting planes that are in—
clined at 45° to each other. Determine the number of image charges and their locations. 6.56 Inﬁnite line x = 3, z = 4 carries 16 nC/m and is located in free space above the conduct
ing plane z = 0. (a) Find E at (2,  2, 3). (b) Calculate the induced surface charge density
on the conducting plane at (5, —6, 0). 6.57 In free space, inﬁnite planes y = 4 and y = 8 carry charges 20 nC/m2 and 30 nC/mz,
respectively. If plane y = 2 is grounded, calculate E at P(O, 0, 0) and Q(—4, 6, 2). ...
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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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