Livro Hayt
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CHAPTER 1
1.1. Given the vectors M = −10ax + 4ay − 8az and N = 8ax + 7ay − 2az, find:
a) a unit vector in the direction of −M+ 2N.
−M+ 2N = 10ax − 4ay + 8az + 16ax + 14ay − 4az = (26, 10, 4)
Thus
a = (26, 10, 4)
|(26, 10, 4)|
= (0.92, 0.36, 0.14)
b) the magnitude of 5ax + N − 3M:
(5, 0, 0) + (8, 7,−2) − (−30, 12,−24) = (43,−5, 22), and |(43,−5, 22)| = 48.6.
c) |M||2N|(M+ N):
|(−10, 4,−8)||(16, 14,−4)|(−2, 11,−10) = (13.4)(21.6)(−2, 11,−10)
= (−580.5, 3193,−2902)
1.2. Given three points, A(4, 3, 2), B(−2, 0, 5), and C(7,−2, 1):
a) Specify the vector A extending from the origin to the point A.
A = (4, 3, 2) = 4ax + 3ay + 2az
b) Give a unit vector extending from the origin to the midpoint of line AB.
The vector from the origin to the midpoint is given by
M = (1/2)(A + B) = (1/2)(4 − 2, 3 + 0, 2 + 5) = (1, 1.5, 3.5)
The unit vector will be
m = (1, 1.5, 3.5)
|(1, 1.5, 3.5)|
= (0.25, 0.38, 0.89)
c) Calculate the length of the perimeter of triangle ABC:
Begin with AB = (−6,−3, 3), BC = (9,−2,−4), CA = (3,−5,−1).
Then
|AB| + |BC| + |CA| = 7.35 + 10.05 + 5.91 = 23.32
1.3. The vector from the origin to the point A is given as (6,−2,−4), and the unit vector directed from the
origin toward point B is (2,−2, 1)/3. If points A and B are ten units apart, find the coordinates of point
B.
With A = (6,−2,−4) and B = 13
B(2,−2, 1), we use the fact that |B − A| = 10, or
|(6 − 2
3B)ax − (2 − 2
3B)ay − (4 + 13
B)az| = 10
Expanding, obtain
36 − 8B + 4
9B2 + 4 − 83
B + 4
9B2 + 16 + 8
3B + 1
9B2 = 100
or B2 − 8B − 44 = 0. Thus B = 8±
√
64−176
2
= 11.75 (taking positive option) and so
B = 2
3
(11.75)ax − 2
3
(11.75)ay + 1
3
(11.75)az = 7.83ax − 7.83ay + 3.92az
1
1.4. given points A(8,−5, 4) and B(−2, 3, 2), find:
a) the distance from A to B.
|B − A| = |(−10, 8,−2)| = 12.96
b) a unit vector directed from A towards B. This is found through
aAB = B − A
|B − A|
= (−0.77, 0.62,−0.15)
c) a unit vector directed from the origin to the midpoint of the line AB.
a0M = (A + B)/2
|(A + B)/2|
= (3,− √ 1, 3)
19
= (0.69,−0.23, 0.69)
d) the coordinates of the point on the line connecting A to B at which the line intersects the plane z = 3.
Note that the midpoint, (3,−1, 3), as determined from part c happens to have z coordinate of 3. This
is the point we are looking for.
1.5. A vector field is specified as G = 24xyax + 12(x2 + 2)ay + 18z2az. Given two points, P(1, 2,−1) and
Q(−2, 1, 3), find:
a) G at P: G(1, 2,−1) = (48, 36, 18)
b) a unit vector in the direction of G at Q: G(−2, 1, 3) = (−48, 72, 162), so
aG = (−48, 72, 162)
|(−48, 72, 162)|
= (−0.26, 0.39, 0.88)
c) a unit vector directed from Q toward P:
aQP = P − Q
|P − Q|
= (3,√−1, 4)
26
= (0.59, 0.20,−0.78)
d) the equation of the surface on which |G| = 60: We write 60 = |(24xy, 12(x2 + 2), 18z2)|, or
10 = |(4xy, 2x2 + 4, 3z2)|, so the equation is
100 = 16x2y2 + 4x4 + 16x2 + 16 + 9z4
2
1.6. For the G field in Problem 1.5, make sketches of Gx , Gy , Gz and |G| along the line y = 1, z = 1, for
0 ≤ x ≤ 2. We find G(x, 1, 1) = (24x, 12x2 + 24, 18), from which Gx = 24x, Gy = 12x2 + 24,
Gz = 18, and |G| = 6
√
4x4 + 32x2 + 25. Plots are shown below.
1.7. Given the vector field E = 4zy2 cos 2xax + 2zy sin 2xay + y2 sin 2xaz for the region |x|, |y|, and |z| less
than 2, find:
a) the surfaces on which Ey = 0. With Ey = 2zy sin 2x = 0, the surfaces are 1) the plane z = 0, with
|x| < 2, |y| < 2; 2) the plane y = 0, with |x| < 2, |z| < 2; 3) the plane x = 0, with |y| < 2, |z| < 2;
4) the plane x = π/2, with |y| < 2, |z| < 2.
b) the region in which Ey = Ez: This occurs when 2zy sin 2x = y2 sin 2x, or on the plane 2z = y, with
|x| < 2, |y| < 2, |z| < 1.
c) the region in which E = 0: We would have Ex = Ey = Ez = 0, or zy2 cos 2x = zy sin 2x =
y2 sin 2x = 0. This condition is met on the plane y = 0, with |x| < 2, |z| < 2.
1.8. Two vector fields are F = −10ax +20x(y−1)ay andG = 2x2yax −4ay +zaz. For the point P(2, 3,−4),
find:
a) |F|: F at (2, 3,−4) = (−10, 80, 0), so |F| = 80.6.
b) |G|: G at (2, 3,−4) = (24,−4,−4), so |G| = 24.7.
c) a unit vector in the direction of F − G: F − G = (−10, 80, 0) − (24,−4,−4) = (−34, 84, 4). So
a = F − G
|F − G|
= (−34, 84, 4)
90.7
= (−0.37, 0.92, 0.04)
d) a unit vector in the direction of F + G: F + G = (−10, 80, 0) + (24,−4,−4) = (14, 76,−4). So
a = F + G
|F + G|
= (14, 76,−4)
77.4
= (0.18, 0.98,−0.05)
3
1.9. A field is given as
G = 25
(x2 + y2)
(xax + yay )
Find:
a) a unit vector in the direction of G at P(3, 4,−2): Have Gp = 25/(9 + 16) × (3, 4, 0) = 3ax + 4ay ,
and |Gp| = 5. Thus aG = (0.6, 0.8, 0).
b) the angle between G and ax at P: The angle is found through aG · ax = cos θ. So cos θ =
(0.6, 0.8, 0) · (1, 0, 0) = 0.6. Thus θ = 53◦.
c) the value of the following double integral on the plane y = 7:
4
0
2
0
G · aydzdx
4
0
2
0
25
x2 + y2 (xax + yay ) · aydzdx =
4
0
2
0
25
x2 + 49
× 7 dzdx =
4
0
350
x2 + 49
dx
= 350 × 1
7
tan−1
4
7
− 0
= 26
1.10. Use the definition of the dot product to find the interior angles at A and B of the triangle defined by the
three points A(1, 3,−2), B(−2, 4, 5), and C(0,−2, 1):
a) Use RAB = (−3, 1, 7) and RAC = (−1,−5, 3) to form RAB · RAC = |RAB||RAC| cos θA. Obtain
3 + 5 + 21 =
√
59
√
35 cos θA. Solve to find θA = 65.3◦.
b) Use RBA = (3,−1,−7) and RBC = (2,−6,−4) to form RBA · RBC = |RBA||RBC| cos θB. Obtain
6 + 6 + 28 =
√
59
√
56 cos θB. Solve to find θB = 45.9◦.
1.11. Given the points M(0.1,−0.2,−0.1), N(−0.2, 0.1, 0.3), and P(0.4, 0, 0.1), find:
a) the vector RMN: RMN = (−0.2, 0.1, 0.3) − (0.1,−0.2,−0.1) = (−0.3, 0.3, 0.4).
b) the dot product RMN · RMP : RMP = (0.4, 0, 0.1) − (0.1,−0.2,−0.1) = (0.3, 0.2, 0.2). RMN ·
RMP = (−0.3, 0.3, 0.4) · (0.3, 0.2, 0.2) = −0.09 + 0.06 + 0.08 = 0.05.
c) the scalar projection of RMN on RMP :
RMN · aRMP = (−0.3, 0.3, 0.4) · √ (0.3, 0.2, 0.2)
0.09 + 0.04 + 0.04
= 0.05 √
0.17
= 0.12
d) the angle between RMN and RMP :
θM = cos−1
RMN · RMP
|RMN||RMP |
= cos−1
0.05 √
0.34
√
0.17
= 78◦
4
1.12. Given points A(10, 12,−6), B(16, 8,−2), C(8, 1,−4), and D(−2,−5, 8), determine:
a) the vector projection of RAB + RBC on RAD: RAB + RBC = RAC = (8, 1, 4) − (10, 12,−6) =
(−2,−11, 10) Then RAD = (−2,−5, 8) − (10, 12,−6) = (−12,−17, 14). So the projection will
be:
(RAC · aRAD)aRAD =
(−2,−11, 10) · (−12,− √ 17, 14)
629
(−12√,−17, 14)
629
= (−6.7,−9.5, 7.8)
b) the vector projection of RAB +RBC on RDC: RDC = (8,−1, 4)−(−2,−5, 8) = (10, 6,−4). The
projection is:
(RAC · aRDC)aRDC =
(−2,−11, 10) · (10√, 6,−4)
152
(10√, 6,−4)
152
= (−8.3,−5.0, 3.3)
c) the angle between RDA and RDC: Use RDA = −RAD = (12, 17,−14) and RDC = (10, 6,−4).
The angle is found through the dot product of the associated unit vectors, or:
θD = cos−1(aRDA · aRDC) = cos−1
(12, 17,−14) · (10, 6,−4) √
629
√
152
= 26◦
1.13. a) Find the vector component of F = (10,−6, 5) that is parallel to G = (0.1, 0.2, 0.3):
F||G = F · G
|G|2 G = (10,−6, 5) · (0.1, 0.2, 0.3)
0.01 + 0.04 + 0.09
(0.1, 0.2, 0.3) = (0.93, 1.86, 2.79)
b) Find the vector component of F that is perpendicular to G:
FpG = F − F||G = (10,−6, 5) − (0.93, 1.86, 2.79) = (9.07,−7.86, 2.21)
c) Find the vector component of G that is perpendicular to F:
GpF = G−G||F = G− G · F
|F|2 F = (0.1, 0.2, 0.3)− 1.3
100 + 36 + 25
(10,−6, 5) = (0.02, 0.25, 0.26)
1.14. The four vertices of a regular tetrahedron are located at O(0, 0, 0), A(0, 1, 0), B(0.5
√
3, 0.5, 0), and
C(
√
3/6, 0.5,
√
2/3).
a) Find a unit vector perpendicular (outward) to the face ABC: First find
RBA × RBC = [(0, 1, 0) − (0.5
√
3, 0.5, 0)] × [(
√
3/6, 0.5,
2/3) − (0.5
√
3, 0.5, 0)]
= (−0.5
√
3, 0.5, 0) × (−
√
3/3, 0,
2/3) = (0.41, 0.71, 0.29)
The required unit vector will then be:
RBA × RBC
|RBA × RBC|
= (0.47, 0.82, 0.33)
b) Find the area of the face ABC:
Area = 1
2
|RBA × RBC| = 0.43
5
1.15. Three vectors extending from the origin are given as r1 = (7, 3,−2), r2 = (−2, 7,−3), and r3 = (0, 2, 3).
Find:
a) a unit vector perpendicular to both r1 and r2:
ap12 = r1 × r2
|r1 × r2|
= (5, 25, 55)
60.6
= (0.08, 0.41, 0.91)
b) a unit vector perpendicular to the vectors r1 − r2 and r2 − r3: r1 − r2 = (9,−4, 1) and r2 − r3 =
(−2, 5,−6). So r1 − r2 × r2 − r3 = (19, 52, 32). Then
ap = (19, 52, 32)
|(19, 52, 32)|
= (19, 52, 32)
63.95
= (0.30, 0.81, 0.50)
c) the area of the triangle defined by r1 and r2:
Area = 1
2
|r1 × r2| = 30.3
d) the area of the triangle defined by the heads of r1, r2, and r3:
Area = 1
2
|(r2 − r1) × (r2 − r3)| = 1
2
|(−9, 4,−1) × (−2, 5,−6)| = 32.0
1.16. Describe the surfaces defined by the equations:
a) r · ax = 2, where r = (x, y, z): This will be the plane x = 2.
b) |r ×ax| = 2: r ×ax = (0, z,−y), and |r ×ax| =
z2 + y2 = 2. This is the equation of a cylinder,
centered on the x axis, and of radius 2.
1.17. Point A(−4, 2, 5) and the two vectors, RAM = (20, 18,−10) and RAN = (−10, 8, 15), define a triangle.
a) Find a unit vector perpendicular to the triangle: Use
ap = RAM × RAN
|RAM × RAN|
= (350,−200, 340)
527.35
= (0.664,−0.379, 0.645)
The vector in the opposite direction to this one is also a valid answer.
b) Find a unit vector in the plane of the triangle and perpendicular to RAN:
aAN = (−1√0, 8, 15)
389
= (−0.507, 0.406, 0.761)
Then
apAN = ap ×aAN = (0.664,−0.379, 0.645)×(−0.507, 0.406, 0.761) = (−0.550,−0.832, 0.077)
The vector in the opposite direction to this one is also a valid answer.
c) Find a unit vector in the plane of the triangle that bisects the interior angle at A: A non-unit vector
in the required direction is (1/2)(aAM + aAN), where
aAM = (20, 18,−10)
|(20, 18,−10)|
= (0.697, 0.627,−0.348)
6
1.17c. (continued) Now
1
2
(aAM + aAN) = 1
2
[(0.697, 0.627,−0.348) + (−0.507, 0.406, 0.761)] = (0.095, 0.516, 0.207)
Finally,
abis = (0.095, 0.516, 0.207)
|(0.095, 0.516, 0.207)|
= (0.168, 0.915, 0.367)
1.18. Given points A(ρ = 5, φ = 70◦
, z = −3) and B(ρ = 2, φ = −30◦
, z = 1), find:
a) unit vector in cartesian coordinates at A toward B: A(5 cos 70◦
, 5 sin 70◦
,−3) = A(1.71, 4.70,−3), In
the same manner, B(1.73,−1, 1). So RAB = (1.73,−1, 1) − (1.71, 4.70,−3) = (0.02,−5.70, 4) and
therefore
aAB = (0.02,−5.70, 4)
|(0.02,−5.70, 4)|
= (0.003,−0.82, 0.57)
b) a vector in cylindrical coordinates at A directed toward B: aAB · aρ = 0.003 cos 70◦ − 0.82 sin 70◦ =
−0.77. aAB · aφ = −0.003 sin 70◦ − 0.82 cos 70◦ = −0.28. Thus
aAB = −0.77aρ − 0.28aφ + 0.57az
.
c) a unit vector in cylindrical coordinates at B directed toward A:
Use aBA = (−0, 003, 0.82,−0.57). Then aBA · aρ = −0.003 cos(−30◦
)+0.82 sin(−30◦
) = −0.43, and
aBA · aφ = 0.003 sin(−30◦
) + 0.82 cos(−30◦
) = 0.71. Finally,
aBA = −0.43aρ + 0.71aφ − 0.57az
1.19 a) Express the field D = (x2 + y2)
−1(xax + yay ) in cylindrical components and cylindrical variables:
Have x = ρ cos φ, y = ρ sin φ, and x2 + y2 = ρ2. Therefore
D = 1
ρ
(cos φax + sin φay )
Then
Dρ = D · aρ = 1
ρ
cos φ(ax · aρ) + sin φ(ay · aρ)
= 1
ρ
cos2 φ + sin2 φ
= 1
ρ
and
Dφ = D · aφ = 1
ρ
cos φ(ax · aφ) + sin φ(ay · aφ)
= 1
ρ
[cos φ(−sin φ) + sin φ cos φ] = 0
Therefore
D = 1
ρ
aρ
7
1.19b. Evaluate D at the point where ρ = 2, φ = 0.2π, and z = 5, expressing the result in cylindrical and
cartesian coordinates: At the given point, and in cylindrical coordinates, D = 0.5aρ. To express this in
cartesian, we use
D = 0.5(aρ · ax )ax + 0.5(aρ · ay )ay = 0.5 cos 36◦ax + 0.5 sin 36◦ay = 0.41ax + 0.29ay
1.20. Express in cartesian components:
a) the vector at A(ρ = 4, φ = 40◦
, z = −2) that extends to B(ρ = 5, φ = −110◦
, z = 2): We
have A(4 cos 40◦
, 4 sin 40◦
,−2) = A(3.06, 2.57,−2), and B(5 cos(−110◦
), 5 sin(−110◦
), 2) =
B(−1.71,−4.70, 2) in cartesian. Thus RAB = (−4.77,−7.30, 4).
b) a unit vector at B directed toward A: HaveRBA = (4.77, 7.30,−4), and so
aBA = (4.77, 7.30,−4)
|(4.77, 7.30,−4)|
= (0.50, 0.76,−0.42)
c) a unit vector at B directed toward the origin: Have rB = (−1.71,−4.70, 2), and so −rB =
(1.71, 4.70,−2). Thus
a = (1.71, 4.70,−2)
|(1.71, 4.70,−2)|
= (0.32, 0.87,−0.37)
1.21. Express in cylindrical components:
a) the vector from C(3, 2,−7) to D(−1,−4, 2):
C(3, 2,−7) → C(ρ = 3.61, φ = 33.7◦
, z = −7) and
D(−1,−4, 2) → D(ρ = 4.12, φ = −104.0◦
, z = 2).
Now RCD = (−4,−6, 9) and Rρ = RCD · aρ = −4 cos(33.7) − 6 sin(33.7) = −6.66. Then
Rφ = RCD · aφ = 4 sin(33.7) − 6 cos(33.7) = −2.77. So RCD = −6.66aρ − 2.77aφ + 9az
b) a unit vector at D directed toward C:
RCD = (4, 6,−9) and Rρ = RDC · aρ = 4 cos(−104.0) + 6 sin(−104.0) = −6.79. Then Rφ =
RDC · aφ = 4[−sin(−104.0)] + 6 cos(−104.0) = 2.43. So RDC = −6.79aρ + 2.43aφ − 9az
Thus aDC = −0.59aρ + 0.21aφ − 0.78az
c) a unit vector at D directed toward the origin: Start with rD = (−1,−4, 2), and so the vector toward
the origin will be −rD = (1, 4,−2). Thus in cartesian the unit vector is a = (0.22, 0.87,−0.44).
Convert to cylindrical:
aρ = (0.22, 0.87,−0.44) · aρ = 0.22 cos(−104.0) + 0.87 sin(−104.0) = −0.90, and
aφ = (0.22, 0.87,−0.44) · aφ = 0.22[−sin(−104.0)] + 0.87 cos(−104.0) = 0, so that finally,
a = −0.90aρ − 0.44az.
1.22. A field is given in cylindrical coordinates as
F =
40
ρ2 + 1
+ 3(cos φ + sin φ)
aρ + 3(cos φ − sin φ)aφ − 2az
where the magnitude of F is found to be:
|F| =
√
F · F =
1600
(ρ2 + 1)2
+ 240
ρ2 + 1
(cos φ + sin φ) + 22
1/2
8
Sketch |F|:
a) vs. φ with ρ = 3: in this case the above simplifies to
|F(ρ = 3)| = |Fa| = [38 + 24(cos φ + sin φ)]1/2
b) vs. ρ with φ = 0, in which:
|F(φ = 0)| = |Fb| =
1600
(ρ2 + 1)2
+ 240
ρ2 + 1
+ 22
1/2
c) vs. ρ with φ = 45◦, in which
|F(φ = 45◦
)| = |Fc| =
1600
(ρ2 + 1)2
+ 240
√
2
ρ2 + 1
+ 22
1/2
9
1.23. The surfaces ρ = 3, ρ = 5, φ = 100◦, φ = 130◦, z = 3, and z = 4.5 define a closed surface.
a) Find the enclosed volume:
Vol =
4.5
3
130◦
100◦
5
3
ρ dρ dφ dz = 6.28
NOTE: The limits on the φ integration must be converted to radians (as was done here, but not shown).
b) Find the total area of the enclosing surface:
Area = 2
130◦
100◦
5
3
ρ dρ dφ +
4.5
3
130◦
100◦
3 dφdz
+
4.5
3
130◦
100◦
5 dφdz + 2
4.5
3
5
3
dρ dz = 20.7
c) Find the total length of the twelve edges of the surfaces:
Length = 4 × 1.5 + 4 × 2 + 2 ×
30◦
360◦
× 2π × 3 + 30◦
360◦
× 2π × 5
= 22.4
d) Find the length of the longest straight line that lies entirely within the volume: This will be between
the points A(ρ = 3, φ = 100◦, z = 3) and B(ρ = 5, φ = 130◦, z = 4.5). Performing point
transformations to cartesian coordinates, these become A(x = −0.52, y = 2.95, z = 3) and B(x =
−3.21, y = 3.83, z = 4.5). Taking A and B as vectors directed from the origin, the requested length
is
Length = |B − A| = |(−2.69, 0.88, 1.5)| = 3.21
1.24. At point P(−3, 4, 5), express the vector that extends from P to Q(2, 0,−1) in:
a) rectangular coordinates.
RPQ = Q − P = 5ax − 4ay − 6az
Then |RPQ| =
√
25 + 16 + 36 = 8.8
b) cylindrical coordinates. At P, ρ = 5, φ = tan−1(4/ − 3) = −53.1◦, and z = 5. Now,
RPQ · aρ = (5ax − 4ay − 6az) · aρ = 5 cos φ − 4 sin φ = 6.20
RPQ · aφ = (5ax − 4ay − 6az) · aφ = −5 sin φ − 4 cos φ = 1.60
Thus
RPQ = 6.20aρ + 1.60aφ − 6az
and |RPQ| =
√
6.202 + 1.602 + 62 = 8.8
c) spherical coordinates. At P, r =
√
9 + 16 + 25 =
√
50 = 7.07, θ = cos−1(5/7.07) = 45◦, and
φ = tan−1(4/ − 3) = −53.1◦.
RPQ · ar = (5ax − 4ay − 6az) · ar = 5 sin θ cos φ − 4 sin θ sin φ − 6 cos θ = 0.14
RPQ · aθ = (5ax − 4ay − 6az) · aθ = 5 cos θ cos φ − 4 cos θ sin φ − (−6) sin θ = 8.62
RPQ · aφ = (5ax − 4ay − 6az) · aφ = −5 sin φ − 4 cos φ = 1.60
10
1.24. (continued)
Thus
RPQ = 0.14ar + 8.62aθ + 1.60aφ
and |RPQ| =
√
0.142 + 8.622 + 1.602 = 8.8
d) Show that each of these vectors has the same magnitude. Each does, as shown above.
1.25. Given point P(r = 0.8, θ = 30◦
, φ = 45◦
), and
E = 1
r2
cos φ ar + sin φ
sin θ
aφ
a) Find E at P: E = 1.10aρ + 2.21aφ.
b) Find |E| at P: |E| =
√
1.102 + 2.212 = 2.47.
c) Find a unit vector in the direction of E at P:
aE = E
|E|
= 0.45ar + 0.89aφ
1.26. a) Determine an expression for ay in spherical coordinates at P(r = 4, θ = 0.2π, φ = 0.8π): Use
ay · ar = sin θ sin φ = 0.35, ay · aθ = cos θ sin φ = 0.48, and ay · aφ = cos φ = −0.81 to obtain
ay = 0.35ar + 0.48aθ − 0.81aφ
b) Express ar in cartesian components at P: Find x = r sin θ cos φ = −1.90, y = r sin θ sin φ = 1.38,
and z = r cos θ = −3.24. Then use ar · ax = sin θ cos φ = −0.48, ar · ay = sin θ sin φ = 0.35, and
ar · az = cos θ = 0.81 to obtain
ar = −0.48ax + 0.35ay + 0.81az
1.27. The surfaces r = 2 and 4, θ = 30◦ and 50◦, and φ = 20◦ and 60◦ identify a closed surface.
a) Find the enclosed volume: This will be
Vol =
60◦
20◦
50◦
30◦
4
2
r2 sin θdrdθdφ = 2.91
where degrees have been converted to radians.
b) Find the total area of the enclosing surface:
Area =
60◦
20◦
50◦
30◦
(42 + 22) sin θdθdφ +
4
2
60◦
20◦
r(sin 30◦ + sin 50◦
)drdφ
+ 2
50◦
30◦
4
2
rdrdθ = 12.61
c) Find the total length of the twelve edges of the surface:
Length = 4
4
2
dr + 2
50◦
30◦
(4 + 2)dθ +
60◦
20◦
(4 sin 50◦ + 4 sin 30◦ + 2 sin 50◦ + 2 sin 30◦
)dφ
= 17.49
11
1.27. (continued)
d) Find the length of the longest straight line that lies entirely within the surface: This will be from
A(r = 2, θ = 50◦
, φ = 20◦
) to B(r = 4, θ = 30◦
, φ = 60◦
) or
A(x = 2 sin 50◦ cos 20◦
, y = 2 sin 50◦ sin 20◦
, z = 2 cos 50◦
)
to
B(x = 4 sin 30◦ cos 60◦
, y = 4 sin 30◦ sin 60◦
, z = 4 cos 30◦
)
or finally A(1.44, 0.52, 1.29) to B(1.00, 1.73, 3.46). Thus B − A = (−0.44, 1.21, 2.18) and
Length = |B − A| = 2.53
1.28. a) Determine the cartesian components of the vector from A(r = 5, θ = 110◦
, φ = 200◦
) to B(r =
7, θ = 30◦
, φ = 70◦
): First transform the points to cartesian: xA = 5 sin 110◦ cos 200◦ = −4.42,
yA = 5 sin 110◦ sin 200◦ = −1.61, and zA = 5 cos 110◦ = −1.71; xB = 7 sin 30◦ cos 70◦ = 1.20,
yB = 7 sin 30◦ sin 70◦ = 3.29, and zB = 7 cos 30◦ = 6.06. Now
RAB = B − A = 5.62ax + 4.90ay + 7.77az
b) Find the spherical components of the vector at P(2,−3, 4) extending to Q(−3, 2, 5): First, RPQ =
Q − P = (−5, 5, 1). Then at P, r =
√
4 + 9 + 16 = 5.39, θ = cos−1(4/
√
29) = 42.0◦, and φ =
tan−1(−3/2) = −56.3◦. Now
RPQ · ar = −5 sin(42◦
) cos(−56.3◦
) + 5 sin(42◦
) sin(−56.3◦
) + 1 cos(42◦
) = −3.90
RPQ · aθ = −5 cos(42◦
) cos(−56.3◦
) + 5 cos(42◦
) sin(−56.3◦
) − 1 sin(42◦
) = −5.82
RPQ · aφ = −(−5) sin(−56.3◦
) + 5 cos(−56.3◦
) = −1.39
So finally,
RPQ = −3.90ar − 5.82aθ − 1.39aφ
c) If D = 5ar − 3aθ + 4aφ, find D · aρ at M(1, 2, 3): First convert aρ to cartesian coordinates at the
specified point. Use aρ = (aρ · ax )ax + (aρ · ay )ay. At A(1, 2, 3), ρ =
√
5, φ = tan−1(2) = 63.4◦,
r =
√
14, and θ = cos−1(3/
√
14) = 36.7◦. So aρ = cos(63.4◦
)ax + sin(63.4◦
)ay = 0.45ax + 0.89ay .
Then
(5ar − 3aθ + 4aφ) · (0.45ax + 0.89ay ) =
5(0.45) sin θ cos φ + 5(0.89) sin θ sin φ − 3(0.45) cos θ cos φ
− 3(0.89) cos θ sin φ + 4(0.45)(−sin φ) + 4(0.89) cos φ = 0.59
1.29. Express the unit vector ax in spherical components at the point:
a) r = 2, θ = 1 rad, φ = 0.8 rad: Use
ax = (ax · ar )ar + (ax · aθ )aθ + (ax · aφ)aφ =
sin(1) cos(0.8)ar + cos(1) cos(0.8)aθ + (−sin(0.8))aφ = 0.59ar + 0.38aθ − 0.72aφ
12
1.29 (continued) Express the unit vector ax in spherical components at the point:
b) x = 3, y = 2, z = −1: First, transform the point to spherical coordinates. Have r =
√
14,
θ = cos−1(−1/
√
14) = 105.5◦, and φ = tan−1(2/3) = 33.7◦. Then
ax = sin(105.5◦
) cos(33.7◦
)ar + cos(105.5◦
) cos(33.7◦
)aθ + (−sin(33.7◦
))aφ
= 0.80ar − 0.22aθ − 0.55aφ
c) ρ = 2.5, φ = 0.7 rad, z = 1.5: Again, convert the point to spherical coordinates. r =
√ ρ2 + z2 =
8.5, θ = cos−1(z/r) = cos−1(1.5/
√
8.5) = 59.0◦, and φ = 0.7 rad = 40.1◦. Now
ax = sin(59◦
) cos(40.1◦
)ar + cos(59◦
) cos(40.1◦
)aθ + (−sin(40.1◦
))aφ
= 0.66ar + 0.39aθ − 0.64aφ
1.30. Given A(r = 20, θ = 30◦
, φ = 45◦
) and B(r = 30, θ = 115◦
, φ = 160◦
), find:
a) |RAB|: First convert A and B to cartesian: Have xA = 20 sin(30◦
) cos(45◦
) = 7.07, yA =
20 sin(30◦
) sin(45◦
) = 7.07, and zA = 20 cos(30◦
) = 17.3. xB = 30 sin(115◦
) cos(160◦
) = −25.6,
yB = 30 sin(115◦
) sin(160◦
) = 9.3, and zB = 30 cos(115◦
) = −12.7. Now RAB = RB − RA =
(−32.6, 2.2,−30.0), and so |RAB| = 44.4.
b) |RAC|, given C(r = 20, θ = 90◦
, φ = 45◦
). Again, converting C to cartesian, obtain xC =
20 sin(90◦
) cos(45◦
) = 14.14, yC = 20 sin(90◦
) sin(45◦
) = 14.14, and zC = 20 cos(90◦
) = 0. So
RAC = RC − RA = (7.07, 7.07,−17.3), and |RAC| = 20.0.
c) the distance from A to C on a great circle path: Note that A and C share the same r and φ coordinates;
thus moving from A to C involves only a change in θ of 60◦. The requested arc length is then
distance = 20 ×
60
2π
360
= 20.9
13
CHAPTER 2
2.1. Four 10nC positive charges are located in the z = 0 plane at the corners of a square 8cm on a side.
A fifth 10nC positive charge is located at a point 8cm distant from the other charges. Calculate the
magnitude of the total force on this fifth charge for = 0:
Arrange the charges in the xy plane at locations (4,4), (4,-4), (-4,4), and (-4,-4). Then the fifth charge
will be on the z axis at location z = 4
√
2, which puts it at 8cm distance from the other four. By
symmetry, the force on the fifth charge will be z-directed, and will be four times the z component of
force produced by each of the four other charges.
F = 4 √
2
× q2
4π0d2
= 4 √
2
× (10−8)2
4π(8.85 × 10−12)(0.08)2
= 4.0 × 10−4 N
2.2. A charge Q1 = 0.1 μC is located at the origin, while Q2 = 0.2 μC is at A(0.8,−0.6, 0). Find the
locus of points in the z = 0 plane at which the x component of the force on a third positive charge is
zero.
To solve this problem, the z coordinate of the third charge is immaterial, so we can place it in the
xy plane at coordinates (x, y, 0). We take its magnitude to be Q3. The vector directed from the first
charge to the third is R13 = xax + yay ; the vector directed from the second charge to the third is
R23 = (x − 0.8)ax + (y + 0.6)ay . The force on the third charge is now
F3 = Q3
4π0
Q1R13
|R13|3
+ Q2R23
|R23|3
= Q3 × 10−6
4π0
0.1(xax + yay )
(x2 + y2)1.5
+ 0.2[(x − 0.8)ax + (y + 0.6)ay ]
[(x − 0.8)2 + (y + 0.6)2]1.5
We desire the x component to be zero. Thus,
0 =
0.1xax
(x2 + y2)1.5
+ 0.2(x − 0.8)ax
[(x − 0.8)2 + (y + 0.6)2]1.5
or
x[(x − 0.8)2 + (y + 0.6)2]1.5 = 2(0.8 − x)(x2 + y2)1.5
2.3. Point charges of 50nC each are located at A(1, 0, 0), B(−1, 0, 0), C(0, 1, 0), and D(0,−1, 0) in free
space. Find the total force on the charge at A.
The force will be:
F = (50 × 10−9)2
4π0
RCA
|RCA|3
+ RDA
|RDA|3
+ RBA
|RBA|3
where RCA = ax −ay , RDA = ax +ay , and RBA = 2ax . The magnitudes are |RCA| = |RDA| =
√
2,
and |RBA| = 2. Substituting these leads to
F = (50 × 10−9)2
4π0
1
2
√
2
+ 1
2
√
2
+ 2
8
ax = 21.5ax μN
where distances are in meters.
14
2.4. Let Q1 = 8 μC be located at P1(2, 5, 8) while Q2 = −5 μC is at P2(6, 15, 8). Let = 0.
a) Find F2, the force on Q2: This force will be
F2 = Q1Q2
4π0
R12
|R12|3
= (8 × 10−6)(−5 × 10−6)
4π0
(4ax + 10ay )
(116)1.5
= (−1.15ax − 2.88ay )mN
b) Find the coordinates of P3 if a charge Q3 experiences a total force F3 = 0 at P3: This force in
general will be:
F3 = Q3
4π0
Q1R13
|R13|3
+ Q2R23
|R23|3
where R13 = (x − 2)ax + (y − 5)ay and R23 = (x − 6)ax + (y − 15)ay . Note, however, that
all three charges must lie in a straight line, and the location of Q3 will be along the vector R12
extended past Q2. The slope of this vector is (15 − 5)/(6 − 2) = 2.5. Therefore, we look for P3
at coordinates (x, 2.5x, 8). With this restriction, the force becomes:
F3 = Q3
4π0
8[(x − 2)ax + 2.5(x − 2)ay ]
[(x − 2)2 + (2.5)2(x − 2)2]1.5
− 5[(x − 6)ax + 2.5(x − 6)ay ]
[(x − 6)2 + (2.5)2(x − 6)2]1.5
where we require the term in large brackets to be zero. This leads to
8(x − 2)[((2.5)2 + 1)(x − 6)2]1.5 − 5(x − 6)[((2.5)2 + 1)(x − 2)2]1.5 = 0
which reduces to
8(x − 6)2 − 5(x − 2)2 = 0
or
x = 6
√
8 − 2
√
5 √
8 −
√
5
= 21.1
The coordinates of P3 are thus P3(21.1, 52.8, 8)
2.5. Let a point charge Q125 nC be located at P1(4,−2, 7) and a charge Q2 = 60 nC be at P2(−3, 4,−2).
a) If = 0, find E at P3(1, 2, 3): This field will be
E = 10−9
4π0
25R13
|R13|3
+ 60R23
|R23|3
whereR13 = −3ax+4ay−4az andR23 = 4ax−2ay+5az. Also, |R13| =
√
41 and |R23| =
√
45.
So
E = 10−9
4π0
25 × (−3ax + 4ay − 4az)
(41)1.5
+ 60 × (4ax − 2ay + 5az)
(45)1.5
= 4.58ax − 0.15ay + 5.51az
b) At what point on the y axis is Ex = 0? P3 is now at (0, y, 0), so R13 = −4ax + (y + 2)ay − 7az
and R23 = 3ax +(y −4)ay +2az. Also, |R13| =
65 + (y + 2)2 and |R23| =
13 + (y − 4)2.
Now the x component of E at the new P3 will be:
Ex = 10−9
4π0
25 × (−4)
[65 + (y + 2)2]1.5
+ 60 × 3
[13 + (y − 4)2]1.5
To obtain Ex = 0, we require the expression in the large brackets to be zero. This expression
simplifies to the following quadratic:
0.48y2 + 13.92y + 73.10 = 0
which yields the two values: y = −6.89,−22.11
15
2.6. Point charges of 120 nC are located at A(0, 0, 1) and B(0, 0,−1) in free space.
a) Find E at P(0.5, 0, 0): This will be
EP = 120 × 10−9
4π0
RAP
|RAP |3
+ RBP
|RBP |3
where RAP = 0.5ax − az and RBP = 0.5ax + az. Also, |RAP| = |RBP| =
√
1.25. Thus:
EP = 120 × 10−9ax
4π(1.25)1.50
= 772 V/m
b) What single charge at the origin would provide the identical field strength? We require
Q0
4π0(0.5)2
= 772
from which we find Q0 = 21.5 nC.
2.7. A 2 μC point charge is located at A(4, 3, 5) in free space. Find Eρ, Eφ, and Ez at P(8, 12, 2). Have
EP = 2 × 10−6
4π0
RAP
|RAP |3
= 2 × 10−6
4π0
4ax + 9ay − 3az
(106)1.5
= 65.9ax + 148.3ay − 49.4az
Then, at point P, ρ =
√
82 + 122 = 14.4, φ = tan−1(12/8) = 56.3◦, and z = z. Now,
Eρ = Ep · aρ = 65.9(ax · aρ) + 148.3(ay · aρ) = 65.9 cos(56.3◦
) + 148.3 sin(56.3◦
) = 159.7
and
Eφ = Ep · aφ = 65.9(ax · aφ) + 148.3(ay · aφ) = −65.9 sin(56.3◦
) + 148.3 cos(56.3◦
) = 27.4
Finally, Ez = −49.4
2.8. Given point charges of −1 μC at P1(0, 0, 0.5) and P2(0, 0,−0.5), and a charge of 2 μC at the origin,
find E at P(0, 2, 1) in spherical components, assuming = 0.
The field will take the general form:
EP = 10−6
4π0
− R1
|R1|3
+ 2R2
|R2|3
− R3
|R3|3
whereR1, R2, R3 are the vectors to P from each of the charges in their original listed order. Specifically,
R1 = (0, 2, 0.5), R2 = (0, 2, 1), and R3 = (0, 2, 1.5). The magnitudes are |R1| = 2.06, |R2| = 2.24,
and |R3| = 2.50. Thus
EP = 10−6
4π0
−(0, 2, 0.5)
(2.06)3
+ 2(0, 2, 1)
(2.24)3
− (0, 2, 1.5)
(2.50)3
= 89.9ay + 179.8az
Now, at P, r =
√
5, θ = cos−1(1/
√
5) = 63.4◦, and φ = 90◦. So
Er = EP · ar = 89.9(ay · ar ) + 179.8(az · ar ) = 89.9 sin θ sin φ + 179.8 cos θ = 160.9
Eθ = EP · aθ = 89.9(ay · aθ ) + 179.8(az · aθ ) = 89.9 cos θ sin φ + 179.8(−sin θ) = −120.5
Eφ = EP · aφ = 89.9(ay · aφ) + 179.8(az · aφ) = 89.9 cos φ = 0
16
2.9. A 100 nC point charge is located at A(−1, 1, 3) in free space.
a) Find the locus of all points P(x, y, z) at which Ex = 500 V/m: The total field at P will be:
EP = 100 × 10−9
4π0
RAP
|RAP |3
where RAP = (x + 1)ax + (y − 1)ay + (z − 3)az, and where |RAP| = [(x + 1)2 + (y − 1)2 +
(z − 3)2]1/2. The x component of the field will be
Ex = 100 × 10−9
4π0
(x + 1)
[(x + 1)2 + (y − 1)2 + (z − 3)2]1.5
= 500 V/m
And so our condition becomes:
(x + 1) = 0.56 [(x + 1)2 + (y − 1)2 + (z − 3)2]1.5
b) Find y1 if P(−2, y1, 3) lies on that locus: At point P, the condition of part a becomes
3.19 =
1 + (y1 − 1)2
3
from which (y1 − 1)2 = 0.47, or y1 = 1.69 or 0.31
2.10. Charges of 20 and -20 nC are located at (3, 0, 0) and (−3, 0, 0), respectively. Let = 0.
Determine |E| at P(0, y, 0): The field will be
EP = 20 × 10−9
4π0
R1
|R1|3
− R2
|R2|3
where R1, the vector from the positive charge to point P is (−3, y, 0), and R2, the vector from
the negative charge to point P, is (3, y, 0). The magnitudes of these vectors are |R1| = |R2| =
9 + y2. Substituting these into the expression for EP produces
EP = 20 × 10−9
4π0
−6ax
(9 + y2)1.5
from which
|EP| = 1079
(9 + y2)1.5 V/m
2.11. A charge Q0 located at the origin in free space produces a field for which Ez = 1 kV/m at point
P(−2, 1,−1).
a) Find Q0: The field at P will be
EP = Q0
4π0
−2ax + ay − az
61.5
Since the z component is of value 1 kV/m, we find Q0 = −4π061.5 × 103 = −1.63 μC.
17
2.11. (continued)
b) Find E at M(1, 6, 5) in cartesian coordinates: This field will be:
EM =
−1.63 × 10−6
4π0
ax + 6ay + 5az
[1 + 36 + 25]1.5
or EM = −30.11ax − 180.63ay − 150.53az.
c) Find E at M(1, 6, 5) in cylindrical coordinates: At M, ρ =
√
1 + 36 = 6.08, φ = tan−1(6/1) =
80.54◦, and z = 5. Now
Eρ = EM · aρ = −30.11 cos φ − 180.63 sin φ = −183.12
Eφ = EM · aφ = −30.11(−sin φ) − 180.63 cos φ = 0 (as expected)
so that EM = −183.12aρ − 150.53az.
d) Find E at M(1, 6, 5) in spherical coordinates: At M, r =
√
1 + 36 + 25 = 7.87, φ = 80.54◦ (as
before), and θ = cos−1(5/7.87) = 50.58◦. Now, since the charge is at the origin, we expect to
obtain only a radial component of EM. This will be:
Er = EM · ar = −30.11 sin θ cos φ − 180.63 sin θ sin φ − 150.53 cos θ = −237.1
2.12. The volume charge density ρv = ρ0e
−|x|−|y|−|z| exists over all free space. Calculate the total charge
present: This will be 8 times the integral of ρv over the first octant, or
Q = 8
∞
0
∞
0
∞
0
ρ0e
−x−y−z dx dy dz = 8ρ0
2.13. A uniform volume charge density of 0.2 μC/m3 (note typo in book) is present throughout the spherical
shell extending from r = 3 cm to r = 5 cm. If ρv = 0 elsewhere:
a) find the total charge present throughout the shell: This will be
Q =
2π
0
π
0
.05
.03
0.2 r2 sinθ dr dθ dφ =
4π(0.2)
r3
3
.05
.03
= 8.21 × 10−5 μC = 82.1 pC
b) find r1 if half the total charge is located in the region 3 cm < r < r1: If the integral over r in part
a is taken to r1, we would obtain
4π(0.2)
r3
3
r1
.03
= 4.105 × 10−5
Thus
r1 =
3 × 4.105 × 10−5
0.2 × 4π
+ (.03)3
1/3
= 4.24 cm
18
2.14. Let
ρv = 5e
−0.1ρ (π − |φ|)
1
z2 + 10
μC/m3
in the region 0 ≤ ρ ≤ 10, −π < φ <π, all z, and ρv = 0 elsewhere.
a) Determine the total charge present: This will be the integral of ρv over the region where it exists;
specifically,
Q =
∞
−∞
π
−π
10
0
5e
−0.1ρ (π − |φ|)
1
z2 + 10
ρ dρ dφ dz
which becomes
Q = 5
e
−0.1ρ
(0.1)2 (−0.1 − 1)
10
0
∞
−∞
2
π
0
(π − φ)
1
z2 + 10
dφ dz
or
Q = 5 × 26.4
∞
−∞
π2 1
z2 + 10
dz
Finally,
Q = 5 × 26.4 × π2
1 √
10
tan−1
√z
10
∞
−∞
= 5(26.4)π3
√
10
= 1.29 × 103 μC = 1.29 mC
b) Calculate the charge within the region 0 ≤ ρ ≤ 4, −π/2 < φ < π/2, −10 < z < 10: With the
limits thus changed, the integral for the charge becomes:
Q
=
10
−10
2
π/2
0
4
0
5e
−0.1ρ (π − φ)
1
z2 + 10
ρ dρ dφ dz
Following the same evaulation procedure as in part a, we obtain Q
= 0.182 mC.
2.15. A spherical volume having a 2 μm radius contains a uniform volume charge density of 1015 C/m3.
a) What total charge is enclosed in the spherical volume?
This will be Q = (4/3)π(2 × 10−6)3 × 1015 = 3.35 × 10−2 C.
b) Now assume that a large region contains one of these little spheres at every corner of a cubical grid
3mm on a side, and that there is no charge between spheres. What is the average volume charge
density throughout this large region? Each cube will contain the equivalent of one little sphere.
Neglecting the little sphere volume, the average density becomes
ρv,avg = 3.35 × 10−2
(0.003)3
= 1.24 × 106 C/m3
2.16. The region in which 4 < r < 5, 0 < θ < 25◦, and 0.9π < φ < 1.1π contains the volume charge
density of ρv = 10(r − 4)(r − 5) sin θ sin(φ/2). Outside the region, ρv = 0. Find the charge within
the region: The integral that gives the charge will be
Q = 10
1.1π
.9π
25◦
0
5
4
(r − 4)(r − 5) sin θ sin(φ/2) r2 sinθ dr dθ dφ
19
2.16. (continued) Carrying out the integral, we obtain
Q = 10
r5
5
− 9
r4
4
+ 20
r3
3
5
4
1
2
θ − 1
4
sin(2θ)
25◦
0
−2 cos
θ
2
1.1π
.9π
= 10(−3.39)(.0266)(.626) = 0.57 C
2.17. A uniform line charge of 16 nC/m is located along the line defined by y = −2, z = 5. If = 0:
a) Find E at P(1, 2, 3): This will be
EP = ρl
2π0
RP
|RP |2
where RP = (1, 2, 3) − (1,−2, 5) = (0, 4,−2), and |RP |2 = 20. So
EP = 16 × 10−9
2π0
4ay − 2az
20
= 57.5ay − 28.8az V/m
b) Find E at that point in the z = 0 plane where the direction of E is given by (1/3)ay − (2/3)az:
With z = 0, the general field will be
Ez=0 = ρl
2π0
(y + 2)ay − 5az
(y + 2)2 + 25
We require |Ez| = −|2Ey |, so 2(y + 2) = 5. Thus y = 1/2, and the field becomes:
Ez=0 = ρl
2π0
2.5ay − 5az
(2.5)2 + 25
= 23ay − 46az
2.18. Uniform line charges of 0.4 μC/m and −0.4 μC/m are located in the x = 0 plane at y = −0.6 and
y = 0.6 m respectively. Let = 0.
a) Find E at P(x, 0, z): In general, we have
EP = ρl
2π0
R+P
|R+P |
− R−P
|R−P |
where R+P and R−P are, respectively, the vectors directed from the positive and negative line
charges to the point P, and these are normal to the z axis. We thus have R+P = (x, 0, z) −
(0,−.6, z) = (x, .6, 0), and R−P = (x, 0, z) − (0, .6, z) = (x,−.6, 0). So
EP = ρl
2π0
xax + 0.6ay
x2 + (0.6)2
− xax − 0.6ay
x2 + (0.6)2
= 0.4 × 10−6
2π0
1.2ay
x2 + 0.36
= 8.63ay
x2 + 0.36
kV/m
20
2.18. (continued)
b) Find E at Q(2, 3, 4): This field will in general be:
EQ = ρl
2π0
R+Q
|R+Q|
− R−Q
|R−Q|
where R+Q = (2, 3, 4)−(0,−.6, 4) = (2, 3.6, 0), and R−Q = (2, 3, 4)−(0, .6, 4) = (2, 2.4, 0).
Thus
EQ = ρl
2π0
2ax + 3.6ay
22 + (3.6)2
− 2ax + 2.4ay
22 + (2.4)2
= −625.8ax − 241.6ay V/m
2.19. A uniform line charge of 2 μC/m is located on the z axis. Find E in cartesian coordinates at P(1, 2, 3)
if the charge extends from
a) −∞ < z < ∞: With the infinite line, we know that the field will have only a radial component
in cylindrical coordinates (or x and y components in cartesian). The field from an infinite line on
the z axis is generally E = [ρl/(2π0ρ)]aρ. Therefore, at point P:
EP = ρl
2π0
RzP
|RzP |2
= (2 × 10−6)
2π0
ax + 2ay
5
= 7.2ax + 14.4ay kV/m
where RzP is the vector that extends from the line charge to point P, and is perpendicular to the z
axis; i.e., RzP = (1, 2, 3) − (0, 0, 3) = (1, 2, 0).
b) −4 ≤ z ≤ 4: Here we use the general relation
EP =
ρldz
4π0
r − r
|r − r |3
where r = ax + 2ay + 3az and r = zaz. So the integral becomes
EP = (2 × 10−6)
4π0
4
−4
ax + 2ay + (3 − z)az
[5 + (3 − z)2]1.5 dz
Using integral tables, we obtain:
EP = 3597
(ax + 2ay)(z − 3) + 5az
(z2 − 6z + 14)
4
−4
V/m = 4.9ax + 9.8ay + 4.9az kV/m
The student is invited to verify that when evaluating the above expression over the limits −∞ <
z < ∞, the z component vanishes and the x and y components become those found in part a.
2.20. Uniform line charges of 120 nC/m lie along the entire extent of the three coordinate axes. Assuming
free space conditions, find E at P(−3, 2,−1): Since all line charges are infinitely-long, we can write:
EP = ρl
2π0
RxP
|RxP |2
+ RyP
|RyP |2
+ RzP
|RzP |2
where RxP , RyP , and RzP are the normal vectors from each of the three axes that terminate on point
P. Specifically, RxP = (−3, 2,−1) − (−3, 0, 0) = (0, 2,−1), RyP = (−3, 2,−1) − (0, 2, 0) =
(−3, 0,−1), and RzP = (−3, 2,−1)−(0, 0,−1) = (−3, 2, 0). Substituting these into the expression
for EP gives
EP = ρl
2π0
2ay − az
5
+
−3ax − az
10
+
−3ax + 2ay
13
= −1.15ax + 1.20ay − 0.65az kV/m
21
2.21. Two identical uniform line charges with ρl = 75 nC/m are located in free space at x = 0, y = ±0.4 m.
What force per unit length does each line charge exert on the other? The charges are parallel to the z
axis and are separated by 0.8 m. Thus the field from the charge at y = −0.4 evaluated at the location
of the charge at y = +0.4 will be E = [ρl/(2π0(0.8))]ay . The force on a differential length of the
line at the positive y location is dF = dqE = ρldzE. Thus the force per unit length acting on the line
at postive y arising from the charge at negative y is
F =
1
0
ρ2
l dz
2π0(0.8)
ay = 1.26 × 10−4 ay N/m = 126 ay μN/m
The force on the line at negative y is of course the same, but with −ay .
2.22. A uniform surface charge density of 5 nC/m2 is present in the region x = 0, −2 < y < 2, and all z. If
= 0, find E at:
a) PA(3, 0, 0): We use the superposition integral:
E =
ρsda
4π0
r − r
|r − r |3
where r = 3ax and r = yay + zaz. The integral becomes:
EPA = ρs
4π0
∞
−∞
2
−2
3ax − yay − zaz
[9 + y2 + z2]1.5 dy dz
Since the integration limits are symmetric about the origin, and since the y and z components of
the integrand exhibit odd parity (change sign when crossing the origin, but otherwise symmetric),
these will integrate to zero, leaving only the x component. This is evident just from the symmetry
of the problem. Performing the z integration first on the x component, we obtain (using tables):
Ex,PA = 3ρs
4π0
2
−2
dy
(9 + y2)
z
9 + y2 + z2
∞
−∞
= 3ρs
2π0
2
−2
dy
(9 + y2)
= 3ρs
2π0
1
3
tan−1
y
3
2
−2
= 106 V/m
The student is encouraged to verify that if the y limits were −∞ to∞, the result would be that of
the infinite charged plane, or Ex = ρs/(20).
b) PB(0, 3, 0): In this case, r = 3ay , and symmetry indicates that only a y component will exist.
The integral becomes
Ey,PB = ρs
4π0
∞
−∞
2
−2
(3 − y) dy dz
[(z2 + 9) − 6y + y2]1.5
= ρs
2π0
2
−2
(3 − y) dy
(3 − y)2
= − ρs
2π0
ln(3 − y)
2
−2
= 145 V/m
22
2.23. Given the surface charge density, ρs = 2 μC/m2, in the regionρ < 0.2m, z = 0, and is zero elsewhere,
find E at:
a) PA(ρ = 0, z = 0.5): First, we recognize from symmetry that only a z component of E will be
present. Considering a general point z on the z axis, we have r = zaz. Then, with r = ρaρ, we
obtain r − r = zaz − ρaρ. The superposition integral for the z component of E will be:
Ez,PA
= ρs
4π0
2π
0
0.2
0
z ρ dρ dφ
(ρ2 + z2)1.5
= −2πρs
4π0
z
1
z2 + ρ2
0.2
0
= ρs
20
z
1 √
z2
− 1 √
z2 + 0.4
With z = 0.5 m, the above evaluates as Ez,PA
= 8.1 kV/m.
b) With z at −0.5 m, we evaluate the expression for Ez to obtain Ez,PB
= −8.1 kV/m.
2.24. Surface charge density is positioned in free space as follows: 20 nC/m2 at x = −3, −30 nC/m2 at
y = 4, and 40 nC/m2 at z = 2. Find the magnitude of E at the three points, (4, 3,−2), (−2, 5,−1),
and (0, 0, 0). Since all three sheets are infinite, the field magnitude associated with each one will be
ρs/(20), which is position-independent. For this reason, the net field magnitude will be the same
everywhere, whereas the field direction will depend on which side of a given sheet one is positioned.
We take the first point, for example, and find
EA = 20 × 10−9
20
ax + 30 × 10−9
20
ay − 40 × 10−9
20
az = 1130ax + 1695ay − 2260az V/m
The magnitude of EA is thus 3.04 kV/m. This will be the magnitude at the other two points as well.
2.25. Find E at the origin if the following charge distributions are present in free space: point charge, 12 nC
at P(2, 0, 6); uniform line charge density, 3nC/m at x = −2, y = 3; uniform surface charge density,
0.2 nC/m2 at x = 2. The sum of the fields at the origin from each charge in order is:
E =
(12 × 10−9)
4π0
(−2ax − 6az)
(4 + 36)1.5
+
(3 × 10−9)
2π0
(2ax − 3ay )
(4 + 9)
−
(0.2 × 10−9)ax
20
= −3.9ax − 12.4ay − 2.5az V/m
2.26. A uniform line charge density of 5 nC/m is at y = 0, z = 2 m in free space, while −5 nC/m is located
at y = 0, z = −2 m. A uniform surface charge density of 0.3 nC/m2 is at y = 0.2 m, and −0.3 nC/m2
is at y = −0.2 m. Find |E| at the origin: Since each pair consists of equal and opposite charges, the
effect at the origin is to double the field produce by one of each type. Taking the sum of the fields at
the origin from the surface and line charges, respectively, we find:
E(0, 0, 0) = −2 × 0.3 × 10−9
20
ay − 2 × 5 × 10−9
2π0(2)
az = −33.9ay − 89.9az
so that |E| = 96.1V/m.
23
2.27. Given the electric field E = (4x − 2y)ax − (2x + 4y)ay , find:
a) the equation of the streamline that passes through the point P(2, 3,−4): We write
dy
dx
= Ey
Ex
=
−(2x + 4y)
(4x − 2y)
Thus
2(x dy + y dx) = y dy − x dx
or
2 d(xy) = 1
2
d(y2) − 1
2
d(x2)
So
C1 + 2xy = 1
2
y2 − 1
2
x2
or
y2 − x2 = 4xy + C2
Evaluating at P(2, 3,−4), obtain:
9 − 4 = 24 + C2, or C2 = −19
Finally, at P, the requested equation is
y2 − x2 = 4xy − 19
b) a unit vector specifying the direction of E at Q(3,−2, 5): Have EQ = [4(3)+2(2)]ax −[2(3)−
4(2)]ay = 16ax + 2ay . Then |E| =
√
162 + 4 = 16.12 So
aQ = 16ax + 2ay
16.12
= 0.99ax + 0.12ay
2.28. Let E = 5x3 ax − 15x2y ay , and find:
a) the equation of the streamline that passes through P(4, 2, 1): Write
dy
dx
= Ey
Ex
=
−15x2y
5x3
=
−3y
x
So
dy
y
= −3
dx
x
⇒ ln y = −3 ln x + ln C
Thus
y = e
−3 ln xeln C = C
x3
At P, have 2 = C/(4)3 ⇒ C = 128. Finally, at P,
y = 128
x3
24
2.28. (continued)
b) a unit vector aE specifying the direction of E at Q(3,−2, 5): At Q, EQ = 135ax + 270ay , and
|EQ| = 301.9. Thus aE = 0.45ax + 0.89ay .
c) a unit vector aN = (l,m, 0) that is perpendicular to aE atQ: Since this vector is to have no z component,
wecan find it through aN = ±(aE×az). Performing this, wefind aN = ±(0.89ax − 0.45ay ).
2.29. If E = 20e
−5y
cos 5xax − sin 5xay
, find:
a) |E| at P(π/6, 0.1, 2): Substituting this point, we obtain EP = −10.6ax − 6.1ay , and so |EP| =
12.2.
b) a unit vector in the direction ofEP : The unit vector associated withE is just
cos 5xax − sin 5xay
,
which evaluated at P becomes aE = −0.87ax − 0.50ay .
c) the equation of the direction line passing through P: Use
dy
dx
=
−sin 5x
cos 5x
= −tan 5x ⇒ dy = −tan 5x dx
Thus y = 15
ln cos 5x + C. Evaluating at P, we find C = 0.13, and so
y = 1
5
ln cos 5x + 0.13
2.30. Given the electric field intensity E = 400yax + 400xay V/m, find:
a) the equation of the streamline passing through the point A(2, 1,−2): Write:
dy
dx
= Ey
Ex
= x
y
⇒ x dx = y dy
Thus x2 = y2 + C. Evaluating at A yields C = 3, so the equation becomes
x2
3
− y2
3
= 1
b) the equation of the surface on which |E| = 800 V/m: Have |E| = 400
x2 + y2 = 800. Thus
x2 + y2 = 4, or we have a circular-cylindrical surface, centered on the z axis, and of radius 2.
c) A sketch of the part a equation would yield a parabola, centered at the origin, whose axis is the
positive x axis, and for which the slopes of the asymptotes are ±1.
d) A sketch of the trace produced by the intersection of the surface of part b with the z = 0 plane
would yield a circle centered at the origin, of radius 2.
25
2.31. In cylindrical coordinates with E(ρ, φ) = Eρ(ρ, φ)aρ +Eφ(ρ, φ)aφ, the differential equation describing
the direction lines is Eρ/Eφ = dρ/(ρdφ) in any constant-z plane. Derive the equation of the line
passing through the point P(ρ = 4, φ = 10◦
, z = 2) in the field E = 2ρ2 cos 3φaρ + 2ρ2 sin 3φaφ:
Using the given information, we write
Eρ
Eφ
= dρ
ρdφ
= cot 3φ
Thus
dρ
ρ
= cot 3φ dφ ⇒ ln ρ = 1
3
ln sin 3φ + ln C
or ρ = C(sin 3φ)1/3. Evaluate this at P to obtain C = 7.14. Finally,
ρ3 = 364 sin 3φ
26
CHAPTER 3
3.1. An empty metal paint can is placed on a marble table, the lid is removed, and both parts are discharged
(honorably) by touching them to ground. An insulating nylon thread is glued to the center of the lid,
and a penny, a nickel, and a dime are glued to the thread so that they are not touching each other. The
penny is given a charge of +5 nC, and the nickel and dime are discharged. The assembly is lowered
into the can so that the coins hang clear of all walls, and the lid is secured. The outside of the can is
again touched momentarily to ground. The device is carefully disassembled with insulating gloves and
tools.
a) What charges are found on each of the five metallic pieces? All coins were insulated during the
entire procedure, so they will retain their original charges: Penny: +5 nC; nickel: 0; dime: 0. The
penny’s charge will have induced an equal and opposite negative charge (-5 nC) on the inside wall
of the can and lid. This left a charge layer of +5 nC on the outside surface which was neutralized
by the ground connection. Therefore, the can retained a net charge of −5 nC after disassembly.
b) If the penny had been given a charge of +5 nC, the dime a charge of −2 nC, and the nickel a
charge of −1 nC, what would the final charge arrangement have been? Again, since the coins are
insulated, they retain their original charges. The charge induced on the inside wall of the can and
lid is equal to negative the sum of the coin charges, or −2 nC. This is the charge that the can/lid
contraption retains after grounding and disassembly.
3.2. A point charge of 12 nC is located at the origin. four uniform line charges are located in the x = 0
plane as follows: 80 nC/m at y = −1 and −5 m, −50 nC/m at y = −2 and −4 m.
a) Find D at P(0,−3, 2): Note that this point lies in the center of a symmetric arrangement of line
charges, whose fields will all cancel at that point. Thus D arise from the point charge alone, and
will be
D = 12 × 10−9(−3ay + 2az)
4π(32 + 22)1.5
= −6.11 × 10−11ay + 4.07 × 10−11az C/m2
= −61.1ay + 40.7az pC/m2
b) How much electric flux crosses the plane y = −3 and in what direction? The plane intercepts all
flux that enters the −y half-space, or exactly half the total flux of 12 nC. The answer is thus 6 nC
and in the −ay direction.
c) How much electric flux leaves the surface of a sphere, 4m in radius, centered at C(0,−3, 0)? This
sphere encloses the point charge, so its flux of 12 nC is included. The line charge contributions
are most easily found by translating the whole assembly (sphere and line charges) such that the
sphere is centered at the origin, with line charges now at y = ±1 and ±2. The flux from the line
charges will equal the total line charge that lies within the sphere. The length of each of the inner
two line charges (at y = ±1) will be
h1 = 2r cos θ1 = 2(4) cos
sin−1
1
4
= 1.94m
That of each of the outer two line charges (at y = ±2) will be
h2 = 2r cos θ2 = 2(4) cos
sin−1
2
4
= 1.73m
27
3.2c. (continued) The total charge enclosed in the sphere (and the outward flux from it) is now
Ql + Qp = 2(1.94)(−50 × 10−9) + 2(1.73)(80 × 10−9) + 12 × 10−9 = 348 nC
3.3. The cylindrical surface ρ = 8 cm contains the surface charge density, ρs = 5e
−20|z| nC/m2.
a) What is the total amount of charge present? We integrate over the surface to find:
Q = 2
∞
0
2π
0
5e
−20z(.08)dφ dz nC = 20π(.08)
−1
20
e
−20z
∞
0
= 0.25 nC
b) How much flux leaves the surface ρ = 8 cm, 1 cm < z < 5cm, 30◦
< φ < 90◦? We just integrate
the charge density on that surface to find the flux that leaves it.
= Q
=
.05
.01
90◦
30◦
5e
−20z(.08) dφ dz nC =
90 − 30
360
2π(5)(.08)
−1
20
e
−20z
.05
.01
= 9.45 × 10−3 nC = 9.45 pC
3.4. The cylindrical surfaces ρ = 1, 2, and 3 cm carry uniform surface charge densities of 20, −8, and 5
nC/m2, respectively.
a) How much electric flux passes through the closed surface ρ = 5 cm, 0 < z < 1 m? Since the
densities are uniform, the flux will be
= 2π(aρs1 + bρs2 + cρs3)(1m) = 2π [(.01)(20) − (.02)(8) + (.03)(5)] × 10−9 = 1.2 nC
b) Find D at P(1 cm, 2 cm, 3 cm): This point lies at radius
√
5 cm, and is thus inside the outermost
charge layer. This layer, being of uniform density, will not contribute to D at P. We know that in
cylindrical coordinates, the layers at 1 and 2 cm will produce the flux density:
D = Dρaρ = aρs1 + bρs2
ρ
aρ
or
Dρ = (.01)(20) + (.02)(− √ 8)
.05
= 1.8 nC/m2
At P, φ = tan−1(2/1) = 63.4◦. Thus Dx = 1.8 cos φ = 0.8 and Dy = 1.8 sin φ = 1.6. Finally,
DP = (0.8ax + 1.6ay ) nC/m2
28
3.5. Let D = 4xyax + 2(x2 + z2)ay + 4yzaz C/m2 and evaluate surface integrals to find the total charge
enclosed in the rectangular parallelepiped 0 < x < 2, 0 < y < 3, 0 < z < 5 m: Of the 6 surfaces to
consider, only 2 will contribute to the net outward flux. Why? First consider the planes at y = 0 and 3.
The y component of D will penetrate those surfaces, but will be inward at y = 0 and outward at y = 3,
while having the same magnitude in both cases. These fluxes will thus cancel. At the x = 0 plane,
Dx = 0 and at the z = 0 plane, Dz = 0, so there will be no flux contributions from these surfaces.
This leaves the 2 remaining surfaces at x = 2 and z = 5. The net outward flux becomes:
=
5
0
3
0
D
x=2
· ax dy dz +
3
0
2
0
D
z=5
· az dx dy
= 5
3
0
4(2)y dy + 2
3
0
4(5)y dy = 360 C
3.6. Two uniform line charges, each 20 nC/m, are located at y = 1, z = ±1 m. Find the total flux leaving a
sphere of radius 2 m if it is centered at
a) A(3, 1, 0): The result will be the same if we move the sphere to the origin and the line charges to
(0, 0,±1). The length of the line charge within the sphere is given by l = 4 sin[cos−1(1/2)] =
3.46. With two line charges, symmetrically arranged, the total charge enclosed is given by Q =
2(3.46)(20 nC/m) = 139 nC
b) B(3, 2, 0): In this case the result will be the same if we move the sphere to the origin and keep
the charges where they were. The length of the line joining the origin to the midpoint of the line
charge (in the yz plane) is l1 =
√
2. The length of the line joining the origin to either endpoint
of the line charge is then just the sphere radius, or 2. The half-angle subtended at the origin by
the line charge is then ψ = cos−1(
√
2/2) = 45◦. The length of each line charge in the sphere
is then l2 = 2 × 2 sin ψ = 2
√
2. The total charge enclosed (with two line charges) is now
Q
= 2(2
√
2)(20 nC/m) = 113 nC
3.7. Volume charge density is located in free space as ρv = 2e
−1000r nC/m3 for 0 < r < 1 mm, and ρv = 0
elsewhere.
a) Find the total charge enclosed by the spherical surface r = 1 mm: To find the charge we integrate:
Q =
2π
0
π
0
.001
0
2e
−1000r r2 sinθ dr dθ dφ
Integration over the angles gives a factor of 4π. The radial integration we evaluate using tables;
we obtain
Q = 8π
−r2e
−1000r
1000
.001
0
+ 2
1000
e
−1000r
(1000)2 (−1000r − 1)
.001
0
= 4.0 × 10−9 nC
b) By using Gauss’s law, calculate the value of Dr on the surface r = 1 mm: The gaussian surface
is a spherical shell of radius 1 mm. The enclosed charge is the result of part a. We thus write
4πr2Dr = Q, or
Dr = Q
4πr2
= 4.0 × 10−9
4π(.001)2
= 3.2 × 10−4 nC/m2
29
3.8. Uniform line charges of 5 nC/m ar located in free space at x = 1, z = 1, and at y = 1, z = 0.
a) Obtain an expression for D in cartesian coordinates at P(0, 0, z). In general, we have
D(z) = ρs
2π
r1 − r
1
|r1 − r
1
|2
+
r2 − r
2
|r2 − r
2
|2
where r1 = r2 = zaz, r
1
= ay , and r
2
= ax + az. Thus
D(z) = ρs
2π
[zaz − ay ]
[1 + z2]
+ [(z − 1)az − ax ]
[1 + (z − 1)2]
= ρs
2π
−ax
[1 + (z − 1)2]
− ay
[1 + z2]
+
(z − 1)
[1 + (z − 1)2]
+ z
[1 + z2]
az
b) Plot |D| vs. z at P, −3 < z < 10: Using part a, we find the magnitude of D to be
|D| = ρs
2π
1
[1 + (z − 1)2]2
+ 1
[1 + z2]2
+
(z − 1)
[1 + (z − 1)2]
+ z
[1 + z2]
2
1/2
A plot of this over the specified range is shown in Prob3.8.pdf.
3.9. A uniform volume charge density of 80 μC/m3 is present throughout the region 8mm < r < 10mm.
Let ρv = 0 for 0 < r < 8mm.
a) Find the total charge inside the spherical surface r = 10 mm: This will be
Q =
2π
0
π
0
.010
.008
(80 × 10−6)r2 sinθ dr dθ dφ = 4π × (80 × 10−6)
r3
3
.010
.008
= 1.64 × 10−10 C = 164 pC
b) Find Dr at r = 10 mm: Using a spherical gaussian surface at r = 10, Gauss’ law is written as
4πr2Dr = Q = 164 × 10−12, or
Dr (10mm) = 164 × 10−12
4π(.01)2
= 1.30 × 10−7 C/m2 = 130 nC/m2
c) If there is no charge for r > 10 mm, find Dr at r = 20 mm: This will be the same computation
as in part b, except the gaussian surface now lies at 20 mm. Thus
Dr (20mm) = 164 × 10−12
4π(.02)2
= 3.25 × 10−8 C/m2 = 32.5 nC/m2
3.10. Let ρs = 8μC/m2 in the region where x = 0 and −4 < z < 4 m, and let ρs = 0 elsewhere. Find D at
P(x, 0, z), wherex > 0: The sheet charge can be thought of as an assembly of infinitely-long parallel
strips that lie parallel to the y axis in the yz plane, and where each is of thickness dz. The field from
each strip is that of an infinite line charge, and so we can construct the field at P from a single strip as:
dDP = ρs dz
2π
r − r
|r − r|2
30
3.10 (continued) where r = xax + zaz and r = z
az We distinguish between the fixed coordinate of P, z,
and the variable coordinate, z
, that determines the location of each charge strip. To find the net field at
P, we sum the contributions of each strip by integrating over z
:
DP =
4
−4
8 × 10−6 dz
(xax + (z − z
)az)
2π[x2 + (z − z)2]
We can re-arrange this to determine the integral forms:
DP = 8 × 10−6
2π
(xax + zaz)
4
−4
dz
(x2 + z2) − 2zz + (z)2
− az
4
−4
z
dz
(x2 + z2) − 2zz + (z)2
Using integral tables, we find
DP = 4 × 10−6
π
(xax + zaz)
1
x
tan−1
2z
− 2z
2x
−
1
2
ln(x2 + z2 − 2zz
+ (z
)2) + 2z
2
1
x
tan−1
2z
− 2z
2x
az
4
−4
which evaluates as
DP = 4 × 10−6
π
tan−1
z + 4
x
− tan−1
z − 4
x
ax + 1
2
ln
x2 + (z + 4)2
x2 + (z − 4)2
az
C/m2
The student is invited to verify that for very small x or for a very large sheet (allowing z
to approach
infinity), the above expression reduces to the expected form, DP = ρs/2. Note also that the expression
is valid for all x (positive or negative values).
3.11. In cylindrical coordinates, let ρv = 0 for ρ < 1 mm, ρv = 2 sin(2000πρ) nC/m3 for 1mm < ρ <
1.5mm, and ρv = 0 for ρ > 1.5mm. Find D everywhere: Since the charge varies only with radius,
and is in the form of a cylinder, symmetry tells us that the flux density will be radially-directed and will
be constant over a cylindrical surface of a fixed radius. Gauss’ law applied to such a surface of unit
length in z gives:
a) for ρ < 1 mm, Dρ = 0, since no charge is enclosed by a cylindrical surface whose radius lies
within this range.
b) for 1mm < ρ < 1.5mm, we have
2πρDρ = 2π
ρ
.001
2 × 10−9 sin(2000πρ
)ρ
dρ
= 4π × 10−9
1
(2000π)2 sin(2000πρ) − ρ
2000π
cos(2000πρ)
ρ
.001
or finally,
Dρ = 10−15
2π2ρ
sin(2000πρ) + 2π
1 − 103ρ cos(2000πρ)
C/m2 (1mm < ρ < 1.5mm)
31
3.11. (continued)
c) for ρ > 1.5mm, the gaussian cylinder now lies at radius ρ outside the charge distribution, so
the integral that evaluates the enclosed charge now includes the entire charge distribution. To
accomplish this, we change the upper limit of the integral of part b from ρ to 1.5 mm, finally
obtaining:
Dρ = 2.5 × 10−15
πρ
C/m2 (ρ > 1.5mm)
3.12. A nonuniform volume charge density, ρv = 120r C/m3, lies within the spherical surface r = 1m, and
ρv = 0 everywhere else.
a) Find Dr everywhere. For r < 1 m, we apply Gauss’ law to a spherical surface of radius r within
this range to find
4πr2Dr = 4π
r
0
120r
(r
)2 dr
= 120πr4
Thus Dr = (30r2) for r < 1 m. For r > 1 m, the gaussian surface lies outside the charge
distribution. The set up is the same, except the upper limit of the above integral is 1 instead of r.
This results in Dr = (30/r2) for r > 1 m.
b) What surface charge density, ρs2, should be on the surface r = 2 such that Dr,r=2− = 2Dr,r=2+?
At r = 2−, we have Dr,r=2− = 30/22 = 15/2, from part a. The flux density in the region r > 2
arising from a surface charge at r = 2 is found from Gauss’ law through
4πr2Drs = 4π(2)2ρs2 ⇒ Drs = 4ρs2
r2
The total flux density in the region r > 2 arising from the two distributions is
DrT = 30
r2
+ 4ρs2
r2
Our requirement that Dr,r=2− = 2Dr,r=2+ becomes
30
22
= 2
30
22
+ ρs2
⇒ ρs2 = −15
4
C/m2
c) Make a sketch of Dr vs. r for 0 < r < 5 m with both distributions present. With both charges,
Dr (r < 1) = 30r2, Dr (1 < r < 2) = 30/r2, and Dr (r > 2) = 15/r2. These are plotted on the
next page.
32
.
3.13. Spherical surfaces at r = 2, 4, and 6mcarry uniform surface charge densities of 20 nC/m2,−4 nC/m2,
and ρs0, respectively.
a) Find D at r = 1, 3 and 5 m: Noting that the charges are spherically-symmetric, we ascertain that
D will be radially-directed and will vary only with radius. Thus, we apply Gauss’ law to spherical
shells in the following regions: r < 2: Here, no charge is enclosed, and so Dr = 0.
2 < r < 4 : 4πr2Dr = 4π(2)2(20 × 10−9) ⇒ Dr = 80 × 10−9
r2 C/m2
So Dr (r = 3) = 8.9 × 10−9 C/m2.
4 < r < 6 : 4πr2Dr = 4π(2)2(20 × 10−9) + 4π(4)2(−4 × 10−9) ⇒ Dr = 16 × 10−9
r2
So Dr (r = 5) = 6.4 × 10−10 C/m2.
b) Determine ρs0 such that D = 0 at r = 7 m. Since fields will decrease as 1/r2, the question could
be re-phrased to ask for ρs0 such that D = 0 at all points where r > 6 m. In this region, the total
field will be
Dr (r > 6) = 16 × 10−9
r2
+ ρs0(6)2
r2
Requiring this to be zero, we find ρs0 = −(4/9) × 10−9 C/m2.
3.14. If ρv = 5 nC/m3 for 0 < ρ < 1 mm and no other charges are present:
a) find Dρ for ρ < 1 mm: Applying Gauss’ law to a cylindrical surface of unit length in z, and of
radiusρ < 1 mm, we find
2πρDρ = πρ2(5 × 10−9) ⇒ Dρ = 2.5 ρ × 10−9 C/m2
33
3.14b. find Dρ forρ > 1 mm: The Gaussian cylinder now lies outside the charge, so
2πρDρ = π(.001)2(5 × 10−9) ⇒ Dρ = 2.5 × 10−15
ρ
C/m2
c) What line charge ρL at ρ = 0 would give the same result for part b? The line charge field will be
Dr = ρL
2πρ
= 2.5 × 10−15
ρ
(part b)
Thus ρL = 5π × 10−15 C/m. In all answers, ρ is expressed in meters.
3.15. Volume charge density is located as follows: ρv = 0 forρ < 1mmand forρ > 2 mm, ρv = 4ρ μC/m3
for 1 < ρ < 2 mm.
a) Calculate the total charge in the region 0 < ρ < ρ1, 0 < z < L, where 1 < ρ1 < 2 mm: We find
Q =
L
0
2π
0
ρ1
.001
4ρ ρ dρ dφ dz = 8πL
3
[ρ3
1
− 10−9] μC
where ρ1 is in meters.
b) Use Gauss’ law to determine Dρ at ρ = ρ1: Gauss’ law states that 2πρ1LDρ = Q, where Q is
the result of part a. Thus
Dρ(ρ1) =
4(ρ3
1
− 10−9)
3ρ1
μC/m2
where ρ1 is in meters.
c) Evaluate Dρ at ρ = 0.8mm, 1.6mm, and 2.4mm: At ρ = 0.8mm, no charge is enclosed by a
cylindrical gaussian surface of that radius, so Dρ(0.8mm) = 0. At ρ = 1.6mm, we evaluate the
part b result at ρ1 = 1.6 to obtain:
Dρ(1.6mm) = 4[(.0016)3 − (.0010)3]
3(.0016)
= 3.6 × 10−6 μC/m2
At ρ = 2.4, we evaluate the charge integral of part a from .001 to .002, and Gauss’ law is written
as
2πρLDρ = 8πL
3
[(.002)2 − (.001)2] μC
from which Dρ(2.4mm) = 3.9 × 10−6 μC/m2.
3.16. Given the electric flux density, D = 2xy ax + x2 ay + 6z3 az C/m2:
a) use Gauss’ law to evaluate the total charge enclosed in the volume 0 < x, y, z < a: We call the
surfaces at x = a and x = 0 the front and back surfaces respectively, those at y = a and y = 0
the right and left surfaces, and those at z = a and z = 0 the top and bottom surfaces. To evaluate
the total charge, we integrate D · n over all six surfaces and sum the results:
= Q =
D · n da =
a
0
a
0
2ay dy dz
front
+
a
0
a
0
−2(0)y dy dz
back
+
a
0
a
0
−x2 dx dz
left
+
a
0
a
0
x2 dx dz
right
+
a
0
a
0
−6(0)3 dx dy
bottom
+
a
0
a
0
6a3 dx dy
top
34
3.16a. (continued) Noting that the back and bottom integrals are zero, and that the left and right integrals
cancel, we evaluate the remaining two (front and top) to obtain Q = 6a5 + a4.
b) use Eq. (8) to find an approximate value for the above charge. Evaluate the derivatives at
P(a/2, a/2, a/2): In this application, Eq. (8) states that Q
.=
(∇ · D
P )v. We find ∇ · D =
2x+18z2, which when evaluated at P becomes ∇ ·D
P
= a+4.5a2. ThusQ
.=
(a+4.5a2)a3 =
4.5a5 + a4
c) Show that the results of parts a and b agree in the limit as a → 0. In this limit, both expressions
reduce to Q = a4, and so they agree.
3.17. A cube is defined by 1 < x, y, z < 1.2. If D = 2x2yax + 3x2y2ay C/m2:
a) apply Gauss’ law to find the total flux leaving the closed surface of the cube. We call the surfaces
at x = 1.2 and x = 1 the front and back surfaces respectively, those at y = 1.2 and y = 1 the
right and left surfaces, and those at z = 1.2 and z = 1 the top and bottom surfaces. To evaluate
the total charge, we integrate D · n over all six surfaces and sum the results. We note that there
is no z component of D, so there will be no outward flux contributions from the top and bottom
surfaces. The fluxes through the remaining four are
= Q =
D · n da =
1.2
1
1.2
1
2(1.2)2y dy dz
front
+
1.2
1
1.2
1
−2(1)2y dy dz
back
+
1.2
1
1.2
1
−3x2(1)2 dx dz
left
+
1.2
1
1.2
1
3x2(1.2)2 dx dz
right
= 0.1028C
b) evaluate ∇ · D at the center of the cube: This is
∇ · D =
4xy + 6x2y
(1.1,1.1)
= 4(1.1)2 + 6(1.1)3 = 12.83
c) Estimate the total charge enclosed within the cube by using Eq. (8): This is
Q
=. ∇ · D
center
× v = 12.83 × (0.2)3 = 0.1026 Close!
3.18. Let a vector field by given by G = 5x4y4z4 ay . Evaluate both sides of Eq. (8) for this G field and the
volume defined by x = 3 and 3.1, y = 1 and 1.1, and z = 2 and 2.1. Evaluate the partial derivatives at
the center of the volume. First find
∇ · G = ∂Gy
∂y
= 20x4y3z4
The center of the cube is located at (3.05,1.05,2.05), and the volume is v = (0.1)3 = 0.001. Eq. (8)
then becomes
.=
20(3.05)4(1.05)3(2.05)4(0.001) = 35.4
35
3.19. A spherical surface of radius 3 mm is centered at P(4, 1, 5) in free space. Let D = xax C/m2. Use the
results of Sec. 3.4 to estimate the net electric flux leaving the spherical surface: We use
=. ∇ ·Dv,
where in this case ∇ · D = (∂/∂x)x = 1 C/m3. Thus
.=
4
3
π(.003)3(1) = 1.13 × 10−7 C = 113 nC
3.20. A cube of volume a3 has its faces parallel to the cartesian coordinate surfaces. It is centered at
P(3,−2, 4). Given the field D = 2x3ax C/m2:
a) calculate divD at P: In the present case, this will be
∇ · D = ∂Dx
∂x
= dDx
dx
= 54 C/m3
b) evaluate the fraction in the rightmost side of Eq. (13) for a = 1 m, 0.1 m, and 1 mm: With the
field having only an x component, flux will pentrate only the two surfaces at x = 3 ± a/2, each
of which has surface area a2. The cube volume is v = a3. The equation reads:
D · dS
v
= 1
a3
2
3 + a
2
3
a2 − 2
3 − a
2
3
a2
= 2
a
(3 + a
2
)3 − (3 − a
2
)3
evaluating the above formula at a = 1 m, .1 m, and 1 mm, yields respectively
54.50, 54.01, and 54.00 C/m3,
thus demonstrating the approach to the exact value as v gets smaller.
3.21. Calculate the divergence of D at the point specified if
a) D = (1/z2)
10xyz ax + 5x2z ay + (2z3 − 5x2y) az
at P(−2, 3, 5): We find
∇ · D =
10y
z
+ 0 + 2 + 10x2y
z3
(−2,3,5)
= 8.96
b) D = 5z2aρ + 10ρz az at P(3,−45◦
, 5): In cylindrical coordinates, we have
∇ · D = 1
ρ
∂
∂ρ
(ρDρ) + 1
ρ
∂Dφ
∂φ
+ ∂Dz
∂z
=
5z2
ρ
+ 10ρ
(3,−45◦
,5)
= 71.67
c) D = 2r sin θ sin φ ar +r cos θ sin φ aθ +r cos φ aφ at P(3, 45◦
,−45◦
): In spherical coordinates,
we have
∇ · D = 1
r2
∂
∂r
(r2Dr ) + 1
r sin θ
∂
∂θ
(sin θDθ ) + 1
r sin θ
∂Dφ
∂φ
=
6 sin θ sin φ + cos 2θ sin φ
sin θ
− sin φ
sin θ
(3,45◦
,−45◦
)
= −2
36
3.22. Let D = 8ρ sin φ aρ + 4ρ cos φ aφ C/m2.
a) Find div D: Using the divergence form
...