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**CHAPTER 1
**

1.1. Given the vectors **M **= −10**a***x *+ 4**a***y *− 8**a***z *and **N **= 8**a***x *+ 7**a***y *− 2**a***z*, find:
a) a unit vector in the direction of −**M **+ 2**N**.

−**M **+ 2**N **= 10**a***x *− 4**a***y *+ 8**a***z *+ 16**a***x *+ 14**a***y *− 4**a***z *= *(*26*, *10*, *4*)
*Thus

**a **= *(*26*, *10*, *4*)*|*(*26*, *10*, *4*)*| = *(*0*.*92*, *0*.*36*, *0*.*14*)
*

b) the magnitude of 5**a***x *+ **N **− 3**M**:
*(*5*, *0*, *0*)*+ *(*8*, *7*,*−2*)*− *(*−30*, *12*,*−24*) *= *(*43*,*−5*, *22*)*, and |*(*43*,*−5*, *22*)*| = 48*.*6.

c) |**M**||2**N**|*(***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*) *= 4**a***x *+ 3**a***y *+ 2**a***z
*

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 − 23*B)***a***x *− *(*2 − 23*B)***a***y *− *(*4 + 13*B)***a***z*| = 10
Expanding, obtain
36 − 8*B *+ 49*B*2 + 4 − 83*B *+ 49*B*2 + 16 + 83*B *+ 19*B*2 = 100
or *B*2 − 8*B *− 44 = 0. Thus *B *= 8±

√
64−176
2 = 11*.*75 (taking positive option) and so

**B **= 2
3
*(*11*.*75*)***a***x *− 2

3
*(*11*.*75*)***a***y *+ 1

3
*(*11*.*75*)***a***z *= 7*.*83**a***x *− 7*.*83**a***y *+ 3*.*92**a***z
*

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

**a***AB *= **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*.

**a**0*M *= *(***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 **= 24*xy***a***x *+ 12*(x*2 + 2*)***a***y *+ 18*z*2**a***z*. 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

**a***G *= *(*−48*, *72*, *162*)*|*(*−48*, *72*, *162*)*| = *(*−0*.*26*, *0*.*39*, *0*.*88*)
*

c) a unit vector directed from *Q *toward *P *:

**a***QP *= **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 = |*(*24*xy, *12*(x*2 + 2*), *18*z*2*)*|, or
10 = |*(*4*xy, *2*x*2 + 4*, *3*z*2*)*|, so the equation is

100 = 16*x*2*y*2 + 4*x*4 + 16*x*2 + 16 + 9*z*4

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*) *= *(*24*x, *12*x*2 + 24*, *18*)*, from which *Gx *= 24*x*, *Gy *= 12*x*2 + 24,
*Gz *= 18, and |**G**| = 6

√
4*x*4 + 32*x*2 + 25. Plots are shown below.

1.7. Given the vector field **E **= 4*zy*2 cos 2*x***a***x *+ 2*zy *sin 2*x***a***y *+ *y*2 sin 2*x***a***z *for the region |*x*|, |*y*|, and |*z*| less
than 2, find:

a) the surfaces on which *Ey *= 0. With *Ey *= 2*zy *sin 2*x *= 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 2*zy *sin 2*x *= *y*2 sin 2*x*, or on the plane 2*z *= *y*, with
|*x*| *< *2, |*y*| *< *2, |*z*| *< *1.

c) the region in which **E **= 0: We would have *Ex *= *Ey *= *Ez *= 0, or *zy*2 cos 2*x *= *zy *sin 2*x *=
*y*2 sin 2*x *= 0. This condition is met on the plane *y *= 0, with |*x*| *< *2, |*z*| *< *2.

1.8. Two vector fields are **F **= −10**a***x*+20*x(y*−1*)***a***y *and **G **= 2*x*2*y***a***x*−4**a***y*+*z***a***z*. 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
*(x*2 + *y*2*) (x***a***x *+ *y***a***y)
*

Find:
a) a unit vector in the direction of **G **at *P(*3*, *4*,*−2*)*: Have **G***p *= 25*/(*9 + 16*)*× *(*3*, *4*, *0*) *= 3**a***x *+ 4**a***y *,

and |**G***p*| = 5. Thus **a***G *= *(*0*.*6*, *0*.*8*, *0*)*.
b) the angle between **G **and **a***x *at *P *: The angle is found through **a***G *· **a***x *= 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 **· **a***ydzdx
*

∫ 4 0

∫ 2 0

25

*x*2 + *y*2 *(x***a***x *+ *y***a***y) *· **a***ydzdx *=
∫ 4

0

∫ 2 0

25

*x*2 + 49 × 7 *dzdx *=
∫ 4

0

350

*x*2 + 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 **R***AB *= *(*−3*, *1*, *7*) *and **R***AC *= *(*−1*,*−5*, *3*) *to form **R***AB *· **R***AC *= |**R***AB *||**R***AC *| cos *θA*. Obtain
3 + 5 + 21 = √59√35 cos *θA*. Solve to find *θA *= 65*.*3◦.

b) Use **R***BA *= *(*3*,*−1*,*−7*) *and **R***BC *= *(*2*,*−6*,*−4*) *to form **R***BA *· **R***BC *= |**R***BA*||**R***BC *| 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 **R***MN *: **R***MN *= *(*−0*.*2*, *0*.*1*, *0*.*3*)*− *(*0*.*1*,*−0*.*2*,*−0*.*1*) *= *(*−0*.*3*, *0*.*3*, *0*.*4*)*.
b) the dot product **R***MN *· **R***MP *: **R***MP *= *(*0*.*4*, *0*, *0*.*1*) *− *(*0*.*1*,*−0*.*2*,*−0*.*1*) *= *(*0*.*3*, *0*.*2*, *0*.*2*)*. **R***MN *·

**R***MP *= *(*−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 **R***MN *on **R***MP *:

**R***MN *· **a***RMP *= *(*−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 **R***MN *and **R***MP *:

*θM *= cos−1
(

**R***MN *· **R***MP
*|**R***MN *||**R***MP *|

) = 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 **R***AB *+ **R***BC *on **R***AD*: **R***AB *+ **R***BC *= **R***AC *= *(*8*, *1*, *4*) *− *(*10*, *12*,*−6*) *=

*(*−2*,*−11*, *10*) *Then **R***AD *= *(*−2*,*−5*, *8*) *− *(*10*, *12*,*−6*) *= *(*−12*,*−17*, *14*)*. So the projection will
be:

*(***R***AC *· **a***RAD)***a***RAD *=
[
*(*−2*,*−11*, *10*) *· *(*−12*,*−17*, *14*)*√

629

]
*(*−12*,*−17*, *14*)*√

629
= *(*−6*.*7*,*−9*.*5*, *7*.*8*)
*

b) the vector projection of **R***AB *+**R***BC *on **R***DC *: **R***DC *= *(*8*,*−1*, *4*)*− *(*−2*,*−5*, *8*) *= *(*10*, *6*,*−4*)*. The
projection is:

*(***R***AC *· **a***RDC)***a***RDC *=
[
*(*−2*,*−11*, *10*) *· *(*10*, *6*,*−4*)*√

152

]
*(*10*, *6*,*−4*)*√

152
= *(*−8*.*3*,*−5*.*0*, *3*.*3*)
*

c) the angle between **R***DA *and **R***DC *: Use **R***DA *= −**R***AD *= *(*12*, *17*,*−14*) *and **R***DC *= *(*10*, *6*,*−4*)*.
The angle is found through the dot product of the associated unit vectors, or:

*θD *= cos−1*(***a***RDA *· **a***RDC) *= 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**:

**F***pG *= **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**:

**G***pF *= **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

**R***BA *× **R***BC *= [*(*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:

**R***BA *× **R***BC
*|**R***BA *× **R***BC *| = *(*0*.*47*, *0*.*82*, *0*.*33*)
*

b) Find the area of the face *ABC*:

Area = 1
2
|**R***BA *× **R***BC *| = 0*.*43

5

1.15. Three vectors extending from the origin are given as **r**1 = *(*7*, *3*,*−2*)*, **r**2 = *(*−2*, *7*,*−3*)*, and **r**3 = *(*0*, *2*, *3*)*.
Find:

a) a unit vector perpendicular to both **r**1 and **r**2:

**a***p*12 = **r**1 × **r**2|**r**1 × **r**2| =
*(*5*, *25*, *55*)
*

60*.*6
= *(*0*.*08*, *0*.*41*, *0*.*91*)
*

b) a unit vector perpendicular to the vectors **r**1 − **r**2 and **r**2 − **r**3: **r**1 − **r**2 = *(*9*,*−4*, *1*) *and **r**2 − **r**3 =
*(*−2*, *5*,*−6*)*. So **r**1 − **r**2 × **r**2 − **r**3 = *(*19*, *52*, *32*)*. Then

**a***p *= *(*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 **r**1 and **r**2:

Area = 1
2
|**r**1 × **r**2| = 30*.*3

d) the area of the triangle defined by the heads of **r**1, **r**2, and **r**3:

Area = 1
2
|*(***r**2 − **r**1*)*× *(***r**2 − **r**3*)*| = 1

2
|*(*−9*, *4*,*−1*)*× *(*−2*, *5*,*−6*)*| = 32*.*0

1.16. Describe the surfaces defined by the equations:

a) **r **· **a***x *= 2, where **r **= *(x, y, z)*: This will be the plane *x *= 2.
b) |**r **× **a***x *| = 2: **r **× **a***x *= *(*0*, z,*−*y)*, and |**r **× **a***x *| =

√
*z*2 + *y*2 = 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, **R***AM *= *(*20*, *18*,*−10*) *and **R***AN *= *(*−10*, *8*, *15*)*, define a triangle.
a) Find a unit vector perpendicular to the triangle: Use

**a***p *= **R***AM *× **R***AN*|**R***AM *× **R***AN *| =
*(*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 **R***AN *:

**a***AN *= *(*−10*, *8*, *15*)*√
389

= *(*−0*.*507*, *0*.*406*, *0*.*761*)
*

Then

**a***pAN *= **a***p *× **a***AN *= *(*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*)(***a***AM *+ **a***AN)*, where

**a***AM *= *(*20*, *18*,*−10*)*|*(*20*, *18*,*−10*)*| = *(*0*.*697*, *0*.*627*,*−0*.*348*)
*

6

1.17c. (continued) Now

1

2
*(***a***AM *+ **a***AN) *= 1

2
[*(*0*.*697*, *0*.*627*,*−0*.*348*)*+ *(*−0*.*507*, *0*.*406*, *0*.*761*)*] = *(*0*.*095*, *0*.*516*, *0*.*207*)
*

Finally,

**a***bis *= *(*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 **R***AB *= *(*1*.*73*,*−1*, *1*) *− *(*1*.*71*, *4*.*70*,*−3*) *= *(*0*.*02*,*−5*.*70*, *4*) *and
therefore

**a***AB *= *(*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*: **a***AB *· **a***ρ *= 0*.*003 cos 70◦ − 0*.*82 sin 70◦ =
−0*.*77. **a***AB *· **a***φ *= −0*.*003 sin 70◦ − 0*.*82 cos 70◦ = −0*.*28. Thus

**a***AB *= −0*.*77**a***ρ *− 0*.*28**a***φ *+ 0*.*57**a***z
*

.

c) a unit vector in cylindrical coordinates at *B *directed toward *A*:
Use **a***BA *= *(*−0*, *003*, *0*.*82*,*−0*.*57*)*. Then **a***BA *·**a***ρ *= −0*.*003 cos*(*−30◦*)*+0*.*82 sin*(*−30◦*) *= −0*.*43, and
**a***BA *· **a***φ *= 0*.*003 sin*(*−30◦*)*+ 0*.*82 cos*(*−30◦*) *= 0*.*71. Finally,

**a***BA *= −0*.*43**a***ρ *+ 0*.*71**a***φ *− 0*.*57**a***z
*

1.19 a) Express the field **D **= *(x*2 + *y*2*)*−1*(x***a***x *+ *y***a***y) *in cylindrical components and cylindrical variables:
Have *x *= *ρ *cos*φ*, *y *= *ρ *sin *φ*, and *x*2 + *y*2 = *ρ*2. Therefore

**D **= 1
*ρ
(*cos*φ***a***x *+ sin *φ***a***y)
*

Then

*Dρ *= **D **· **a***ρ *= 1
*ρ
*

[
cos*φ(***a***x *· **a***ρ)*+ sin *φ(***a***y *· **a***ρ)
*

] = 1
*ρ
*

[
cos2 *φ *+ sin2 *φ
*

] = 1

*ρ
*

and

*Dφ *= **D **· **a***φ *= 1
*ρ
*

[
cos*φ(***a***x *· **a***φ)*+ sin *φ(***a***y *· **a***φ)
*

] = 1
*ρ
*

[cos*φ(*− sin *φ)*+ sin *φ *cos*φ*] = 0

Therefore

**D **= 1
*ρ
*

**a***ρ
*

7

1.19b. Evaluate **D **at the point where *ρ *= 2, *φ *= 0*.*2*π *, and *z *= 5, expressing the result in cylindrical and
cartesian coordinates: At the given point, and in cylindrical coordinates, **D **= 0*.*5**a***ρ *. To express this in
cartesian, we use

**D **= 0*.*5*(***a***ρ *· **a***x)***a***x *+ 0*.*5*(***a***ρ *· **a***y)***a***y *= 0*.*5 cos 36◦**a***x *+ 0*.*5 sin 36◦**a***y *= 0*.*41**a***x *+ 0*.*29**a***y
*

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 **R***AB *= *(*−4*.*77*,*−7*.*30*, *4*)*.

b) a unit vector at *B *directed toward *A*: Have **R***BA *= *(*4*.*77*, *7*.*30*,*−4*)*, and so

**a***BA *= *(*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 **r***B *= *(*−1*.*71*,*−4*.*70*, *2*)*, and so −**r***B *=
*(*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 **R***CD *= *(*−4*,*−6*, *9*) *and *Rρ *= **R***CD *· **a***ρ *= −4 cos*(*33*.*7*) *− 6 sin*(*33*.*7*) *= −6*.*66. Then
*Rφ *= **R***CD *· **a***φ *= 4 sin*(*33*.*7*)*− 6 cos*(*33*.*7*) *= −2*.*77. So **R***CD *= −6*.*66**a***ρ *− 2*.*77**a***φ *+ 9**a***z
*

b) a unit vector at *D *directed toward *C*:
**R***CD *= *(*4*, *6*,*−9*) *and *Rρ *= **R***DC *· **a***ρ *= 4 cos*(*−104*.*0*) *+ 6 sin*(*−104*.*0*) *= −6*.*79. Then *Rφ *=
**R***DC *· **a***φ *= 4[− sin*(*−104*.*0*)*] + 6 cos*(*−104*.*0*) *= 2*.*43. So **R***DC *= −6*.*79**a***ρ *+ 2*.*43**a***φ *− 9**a***z
*Thus **a***DC *= −0*.*59**a***ρ *+ 0*.*21**a***φ *− 0*.*78**a***z
*

c) a unit vector at *D *directed toward the origin: Start with **r***D *= *(*−1*,*−4*, *2*)*, and so the vector toward
the origin will be −**r***D *= *(*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*) *· **a***ρ *= 0*.*22 cos*(*−104*.*0*)*+ 0*.*87 sin*(*−104*.*0*) *= −0*.*90, and
*aφ *= *(*0*.*22*, *0*.*87*,*−0*.*44*) *· **a***φ *= 0*.*22[− sin*(*−104*.*0*)*] + 0*.*87 cos*(*−104*.*0*) *= 0, so that finally,
**a **= −0*.*90**a***ρ *− 0*.*44**a***z*.

1.22. A field is given in cylindrical coordinates as

**F **=
[

40

*ρ*2 + 1 + 3*(*cos*φ *+ sin *φ)
*]

**a***ρ *+ 3*(*cos*φ *− sin *φ)***a***φ *− 2**a***z
*

where the magnitude of **F **is found to be:

|**F**| =
√

**F **· **F **=
[

1600

*(ρ*2 + 1*)*2 +
240

*ρ*2 + 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

*(ρ*2 + 1*)*2 +
240

*ρ*2 + 1 + 22
]1*/*2

c) vs. *ρ *with *φ *= 45◦, in which

|**F***(φ *= 45◦*)*| = |*Fc*| =
[

1600

*(ρ*2 + 1*)*2 +
240

√ 2

*ρ*2 + 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 *dφ dz
*

+
∫ 4*.*5

3

∫ 130◦ 100◦

5 *dφ dz *+ 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◦
× 2*π *× 3 + 30

◦

360◦
× 2*π *× 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.

**R***PQ *= **Q **− **P **= 5**a***x *− 4**a***y *− 6**a***z
*Then |**R***PQ*| =

√
25 + 16 + 36 = 8*.*8

b) cylindrical coordinates. At *P *, *ρ *= 5, *φ *= tan−1*(*4*/*− 3*) *= −53*.*1◦, and *z *= 5. Now,
**R***PQ *· **a***ρ *= *(*5**a***x *− 4**a***y *− 6**a***z) *· **a***ρ *= 5 cos*φ *− 4 sin *φ *= 6*.*20

**R***PQ *· **a***φ *= *(*5**a***x *− 4**a***y *− 6**a***z) *· **a***φ *= −5 sin *φ *− 4 cos*φ *= 1*.*60
Thus

**R***PQ *= 6*.*20**a***ρ *+ 1*.*60**a***φ *− 6**a***z
*and |**R***PQ*| =

√
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◦.

**R***PQ *· **a***r *= *(*5**a***x *− 4**a***y *− 6**a***z) *· **a***r *= 5 sin *θ *cos*φ *− 4 sin *θ *sin *φ *− 6 cos *θ *= 0*.*14
**R***PQ *· **a***θ *= *(*5**a***x *− 4**a***y *− 6**a***z) *· **a***θ *= 5 cos *θ *cos*φ *− 4 cos *θ *sin *φ *− *(*−6*) *sin *θ *= 8*.*62

**R***PQ *· **a***φ *= *(*5**a***x *− 4**a***y *− 6**a***z) *· **a***φ *= −5 sin *φ *− 4 cos*φ *= 1*.*60

10

1.24. (continued)

Thus
**R***PQ *= 0*.*14**a***r *+ 8*.*62**a***θ *+ 1*.*60**a***φ
*

and |**R***PQ*| =
√

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
*r*2

(
cos*φ ***a***r *+ sin *φ
*

sin *θ
***a***φ
*

)

a) Find **E **at *P *: **E **= 1*.*10**a***ρ *+ 2*.*21**a***φ *.
b) Find |**E**| at *P *: |**E**| = √1*.*102 + 2*.*212 = 2*.*47.
c) Find a unit vector in the direction of **E **at *P *:

**a***E *= **E**|**E**| = 0*.*45**a***r *+ 0*.*89**a***φ
*

1.26. a) Determine an expression for **a***y *in spherical coordinates at *P(r *= 4*, θ *= 0*.*2*π, φ *= 0*.*8*π)*: Use
**a***y *· **a***r *= sin *θ *sin *φ *= 0*.*35, **a***y *· **a***θ *= cos *θ *sin *φ *= 0*.*48, and **a***y *· **a***φ *= cos*φ *= −0*.*81 to obtain

**a***y *= 0*.*35**a***r *+ 0*.*48**a***θ *− 0*.*81**a***φ
*

b) Express **a***r *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 **a***r *· **a***x *= sin *θ *cos*φ *= −0*.*48, **a***r *· **a***y *= sin *θ *sin *φ *= 0*.*35, and
**a***r *· **a***z *= cos *θ *= 0*.*81 to obtain

**a***r *= −0*.*48**a***x *+ 0*.*35**a***y *+ 0*.*81**a***z
*

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

*r*2 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

**R***AB *= **B **− **A **= 5*.*62**ax **+ 4*.*90**a***y *+ 7*.*77**a***z
*

b) Find the spherical components of the vector at *P(*2*,*−3*, *4*) *extending to *Q(*−3*, *2*, *5*)*: First, **R***PQ *=
**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

**R***PQ *· **a***r *= −5 sin*(*42◦*) *cos*(*−56*.*3◦*)*+ 5 sin*(*42◦*) *sin*(*−56*.*3◦*)*+ 1 cos*(*42◦*) *= −3*.*90

**R***PQ *· **a***θ *= −5 cos*(*42◦*) *cos*(*−56*.*3◦*)*+ 5 cos*(*42◦*) *sin*(*−56*.*3◦*)*− 1 sin*(*42◦*) *= −5*.*82
**R***PQ *· **a***φ *= −*(*−5*) *sin*(*−56*.*3◦*)*+ 5 cos*(*−56*.*3◦*) *= −1*.*39

So finally,
**R***PQ *= −3*.*90**a***r *− 5*.*82**a***θ *− 1*.*39**a***φ
*

c) If **D **= 5**a***r *− 3**a***θ *+ 4**a***φ *, find **D **· **a***ρ *at *M(*1*, *2*, *3*)*: First convert **a***ρ *to cartesian coordinates at the
specified point. Use **a***ρ *= *(***a***ρ *· **a***x)***a***x *+ *(***a***ρ *· **a***y)***a***y *. At *A(*1*, *2*, *3*)*, *ρ *=

√
5, *φ *= tan−1*(*2*) *= 63*.*4◦,

*r *= √14, and *θ *= cos−1*(*3*/*√14*) *= 36*.*7◦. So **a***ρ *= cos*(*63*.*4◦*)***a***x *+ sin*(*63*.*4◦*)***a***y *= 0*.*45**a***x *+ 0*.*89**a***y *.
Then

*(*5**a***r *− 3**a***θ *+ 4**a***φ) *· *(*0*.*45**a***x *+ 0*.*89**a***y) *=
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 **a***x *in spherical components at the point:
a) *r *= 2, *θ *= 1 rad, *φ *= 0*.*8 rad: Use

**a***x *= *(***a***x *· **a***r )***a***r *+ *(***a***x *· **a***θ )***a***θ *+ *(***a***x *· **a***φ)***a***φ *=
sin*(*1*) *cos*(*0*.*8*)***a***r *+ cos*(*1*) *cos*(*0*.*8*)***a***θ *+ *(*− sin*(*0*.*8*))***a***φ *= 0*.*59**a***r *+ 0*.*38**a***θ *− 0*.*72**a***φ
*

12

1.29 (continued) Express the unit vector **a***x *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

**a***x *= sin*(*105*.*5◦*) *cos*(*33*.*7◦*)***a***r *+ cos*(*105*.*5◦*) *cos*(*33*.*7◦*)***a***θ *+ *(*− sin*(*33*.*7◦*))***a***φ
*= 0*.*80**a***r *− 0*.*22**a***θ *− 0*.*55**a***φ
*

c) *ρ *= 2*.*5, *φ *= 0*.*7 rad, *z *= 1*.*5: Again, convert the point to spherical coordinates. *r *=
√
*ρ*2 + *z*2 =√

8*.*5, *θ *= cos−1*(z/r) *= cos−1*(*1*.*5*/*√8*.*5*) *= 59*.*0◦, and *φ *= 0*.*7 rad = 40*.*1◦. Now

**a***x *= sin*(*59◦*) *cos*(*40*.*1◦*)***a***r *+ cos*(*59◦*) *cos*(*40*.*1◦*)***a***θ *+ *(*− sin*(*40*.*1◦*))***a***φ
*= 0*.*66**a***r *+ 0*.*39**a***θ *− 0*.*64**a***φ
*

1.30. Given *A(r *= 20*, θ *= 30◦*, φ *= 45◦*) *and *B(r *= 30*, θ *= 115◦*, φ *= 160◦*)*, find:
a) |**R***AB *|: 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 **R***AB *= **R***B *− **R***A *=
*(*−32*.*6*, *2*.*2*,*−30*.*0*)*, and so |**R***AB *| = 44*.*4.

b) |**R***AC *|, 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
**R***AC *= **R***C *− **R***A *= *(*7*.*07*, *7*.*07*,*−17*.*3*)*, and |**R***AC *| = 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

(
2*π
*

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
× *q
*

2

4*π*0*d*2
= 4√

2
× *(*10

−8*)*2

4*π(*8*.*85 × 10−12*)(*0*.*08*)*2 = 4*.*0 × 10
−4 N

2.2. A charge *Q*1 = 0*.*1 *µ*C is located at the origin, while *Q*2 = 0*.*2 *µ*C 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 *Q*3. The vector directed from the first
charge to the third is **R**13 = *x***a***x *+ *y***a***y *; the vector directed from the second charge to the third is
**R**23 = *(x *− 0*.*8*)***a***x *+ *(y *+ 0*.*6*)***a***y *. The force on the third charge is now

**F**3 = *Q*3
4*π*0

[
*Q*1**R**13
|**R**13|3 +

*Q*2**R**23
|**R**23|3

]

= *Q*3 × 10
−6

4*π*0

[
0*.*1*(x***a***x *+ *y***a***y)
(x*2 + *y*2*)*1*.*5 +

0*.*2[*(x *− 0*.*8*)***a***x *+ *(y *+ 0*.*6*)***a***y*]
[*(x *− 0*.*8*)*2 + *(y *+ 0*.*6*)*2]1*.*5

]

We desire the *x *component to be zero. Thus,

0 = [

0*.*1*x***a***x
(x*2 + *y*2*)*1*.*5 +

0*.*2*(x *− 0*.*8*)***a***x
*[*(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)(x*2 + *y*2*)*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

4*π*0

[
**R***CA
*

|**R***CA*|3 +
**R***DA
*

|**R***DA*|3 +
**R***BA
*

|**R***BA*|3
]

where **R***CA *= **a***x *− **a***y *, **R***DA *= **a***x *+ **a***y *, and **R***BA *= 2**a***x *. The magnitudes are |**R***CA*| = |**R***DA*| =
√

2,
and |**R***BA*| = 2. Substituting these leads to

**F **= *(*50 × 10
−9*)*2

4*π*0

[ 1

2 √

2 + 1

2 √

2 + 2

8

]
**a***x *= 21*.*5**a***x µ*N

where distances are in meters.

14

2.4. Let *Q*1 = 8 *µ*C be located at *P*1*(*2*, *5*, *8*) *while *Q*2 = −5 *µ*C is at *P*2*(*6*, *15*, *8*)*. Let * *= 0.
a) Find **F**2, the force on *Q*2: This force will be

**F**2 = *Q*1*Q*2
4*π*0

**R**12
|**R**12|3 =

*(*8 × 10−6*)(*−5 × 10−6*)
*4*π*0

*(*4**a***x *+ 10**a***y)
(*116*)*1*.*5

= *(*−1*.*15**a***x *− 2*.*88**a***y)*mN

b) Find the coordinates of *P*3 if a charge *Q*3 experiences a total force **F**3 = 0 at *P*3: This force in
general will be:

**F**3 = *Q*3
4*π*0

[
*Q*1**R**13
|**R**13|3 +

*Q*2**R**23
|**R**23|3

]
where **R**13 = *(x *− 2*)***a***x *+ *(y *− 5*)***a***y *and **R**23 = *(x *− 6*)***a***x *+ *(y *− 15*)***a***y *. Note, however, that
all three charges must lie in a straight line, and the location of *Q*3 will be along the vector **R**12
extended past *Q*2. The slope of this vector is *(*15 − 5*)/(*6 − 2*) *= 2*.*5. Therefore, we look for *P*3
at coordinates *(x, *2*.*5*x, *8*)*. With this restriction, the force becomes:

**F**3 = *Q*3
4*π*0

[
8[*(x *− 2*)***a***x *+ 2*.*5*(x *− 2*)***a***y*]

[*(x *− 2*)*2 + *(*2*.*5*)*2*(x *− 2*)*2]1*.*5 −
5[*(x *− 6*)***a***x *+ 2*.*5*(x *− 6*)***a***y*]

[*(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 *P*3 are thus *P*3*(*21*.*1*, *52*.*8*, *8*)
*

2.5. Let a point charge *Q*125 nC be located at *P*1*(*4*,*−2*, *7*) *and a charge *Q*2 = 60 nC be at *P*2*(*−3*, *4*,*−2*)*.
a) If * *= 0, find **E **at *P*3*(*1*, *2*, *3*)*: This field will be

**E **= 10
−9

4*π*0

[
25**R**13
|**R**13|3 +

60**R**23
|**R**23|3

]

where **R**13 = −3**a***x*+4**a***y*−4**a***z *and **R**23 = 4**a***x*−2**a***y*+5**a***z*. Also, |**R**13| =
√

41 and |**R**23| =
√

45. So

**E **= 10
−9

4*π*0

[
25 × *(*−3**a***x *+ 4**a***y *− 4**a***z)
*

*(*41*)*1*.*5
+ 60 × *(*4**a***x *− 2**a***y *+ 5**a***z)
*

*(*45*)*1*.*5

]
= 4*.*58**a***x *− 0*.*15**a***y *+ 5*.*51**a***z
*

b) At what point on the *y *axis is *Ex *= 0? *P*3 is now at *(*0*, y, *0*)*, so **R**13 = −4**a***x *+ *(y *+ 2*)***a***y *− 7**a***z
*and **R**23 = 3**a***x *+ *(y *− 4*)***a***y *+ 2**a***z*. Also, |**R**13| =

√
65 + *(y *+ 2*)*2 and |**R**23| =

√
13 + *(y *− 4*)*2.

Now the *x *component of **E **at the new *P*3 will be:

*Ex *= 10
−9

4*π*0

[
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*.*48*y*2 + 13*.*92*y *+ 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

**E***P *= 120 × 10
−9

4*π*0

[
**R***AP
*

|**R***AP *|3 +
**R***BP
*

|**R***BP *|3
]

where **R***AP *= 0*.*5**a***x *− **a***z *and **R***BP *= 0*.*5**a***x *+ **a***z*. Also, |**R***AP *| = |**R***BP *| =
√

1*.*25. Thus:

**E***P *= 120 × 10
−9**a***x
*

4*π(*1*.*25*)*1*.*50
= 772 V*/*m

b) What single charge at the origin would provide the identical field strength? We require

*Q*0

4*π*0*(*0*.*5*)*2
= 772

from which we find *Q*0 = 21*.*5 nC.

2.7. A 2 *µ*C point charge is located at *A(*4*, *3*, *5*) *in free space. Find *Eρ *, *Eφ *, and *Ez *at *P(*8*, *12*, *2*)*. Have

**E***P *= 2 × 10
−6

4*π*0

**R***AP
*|**R***AP *|3 =

2 × 10−6
4*π*0

[
4**a***x *+ 9**a***y *− 3**a***z
*

*(*106*)*1*.*5

]
= 65*.*9**a***x *+ 148*.*3**a***y *− 49*.*4**a***z
*

Then, at point *P *, *ρ *= √82 + 122 = 14*.*4, *φ *= tan−1*(*12*/*8*) *= 56*.*3◦, and *z *= *z*. Now,
*Eρ *= **E***p *· **a***ρ *= 65*.*9*(***a***x *· **a***ρ)*+ 148*.*3*(***a***y *· **a***ρ) *= 65*.*9 cos*(*56*.*3◦*)*+ 148*.*3 sin*(*56*.*3◦*) *= 159*.*7

and

*Eφ *= **E***p *· **a***φ *= 65*.*9*(***a***x *· **a***φ)*+ 148*.*3*(***a***y *· **a***φ) *= −65*.*9 sin*(*56*.*3◦*)*+ 148*.*3 cos*(*56*.*3◦*) *= 27*.*4
Finally, *Ez *= −49*.*4

2.8. Given point charges of −1 *µ*C at *P*1*(*0*, *0*, *0*.*5*) *and *P*2*(*0*, *0*,*−0*.*5*)*, and a charge of 2 *µ*C at the origin,
find **E **at *P(*0*, *2*, *1*) *in spherical components, assuming * *= 0.
The field will take the general form:

**E***P *= 10
−6

4*π*0

[
− **R**1|**R**1|3 +

2**R**2
|**R**2|3 −

**R**3
|**R**3|3

]

where **R**1, **R**2, **R**3 are the vectors to*P *from each of the charges in their original listed order. Specifically,
**R**1 = *(*0*, *2*, *0*.*5*)*, **R**2 = *(*0*, *2*, *1*)*, and **R**3 = *(*0*, *2*, *1*.*5*)*. The magnitudes are |**R**1| = 2*.*06, |**R**2| = 2*.*24,
and |**R**3| = 2*.*50. Thus

**E***P *= 10
−6

4*π*0

[−*(*0*, *2*, *0*.*5*)
(*2*.*06*)*3

+ 2*(*0*, *2*, *1*)
(*2*.*24*)*3

− *(*0*, *2*, *1*.*5*)
(*2*.*50*)*3

]
= 89*.*9**a***y *+ 179*.*8**a***z
*

Now, at *P *, *r *= √5, *θ *= cos−1*(*1*/*√5*) *= 63*.*4◦, and *φ *= 90◦. So
*Er *= **E***P *· **a***r *= 89*.*9*(***a***y *· **a***r )*+ 179*.*8*(***a***z *· **a***r ) *= 89*.*9 sin *θ *sin *φ *+ 179*.*8 cos *θ *= 160*.*9

*Eθ *= **E***P *· **a***θ *= 89*.*9*(***a***y *· **a***θ )*+ 179*.*8*(***a***z *· **a***θ ) *= 89*.*9 cos *θ *sin *φ *+ 179*.*8*(*− sin *θ) *= −120*.*5
*Eφ *= **E***P *· **a***φ *= 89*.*9*(***a***y *· **a***φ)*+ 179*.*8*(***a***z *· **a***φ) *= 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:

**E***P *= 100 × 10
−9

4*π*0

**R***AP
*|**R***AP *|3

where **R***AP *= *(x *+ 1*)***a***x *+ *(y *− 1*)***a***y *+ *(z *− 3*)***a***z*, and where |**R***AP *| = [*(x *+ 1*)*2 + *(y *− 1*)*2 +
*(z*− 3*)*2]1*/*2. The *x *component of the field will be

*Ex *= 100 × 10
−9

4*π*0

[
*(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 *y*1 if *P(*−2*, y*1*, *3*) *lies on that locus: At point *P *, the condition of part *a *becomes

3*.*19 =
[
1 + *(y*1 − 1*)*2

]3
from which *(y*1 − 1*)*2 = 0*.*47, or *y*1 = 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

**E***P *= 20 × 10
−9

4*π*0

[
**R**1

|**R**1|3 −
**R**2

|**R**2|3
]

where **R**1, the vector from the positive charge to point *P *is *(*−3*, y, *0*)*, and **R**2, the vector from
the negative charge to point *P *, is *(*3*, y, *0*)*. The magnitudes of these vectors are |**R**1| = |**R**2| =√

9 + *y*2. Substituting these into the expression for **E***P *produces

**E***P *= 20 × 10
−9

4*π*0

[ −6**a***x
(*9 + *y*2*)*1*.*5

]

from which

|**E***P *| = 1079
*(*9 + *y*2*)*1*.*5 V*/*m

2.11. A charge *Q*0 located at the origin in free space produces a field for which *Ez *= 1 kV/m at point
*P(*−2*, *1*,*−1*)*.

a) Find *Q*0: The field at *P *will be

**E***P *= *Q*0
4*π*0

[−2**a***x *+ **a***y *− **a***z
*61*.*5

]

Since the *z *component is of value 1 kV/m, we find *Q*0 = −4*π*061*.*5 × 103 = −1*.*63 *µ*C.

17

2.11. (continued)

b) Find **E **at *M(*1*, *6*, *5*) *in cartesian coordinates: This field will be:

**E***M *= −1*.*63 × 10
−6

4*π*0

[
**a***x *+ 6**a***y *+ 5**a***z
*[1 + 36 + 25]1*.*5

]

or **E***M *= −30*.*11**a***x *− 180*.*63**a***y *− 150*.*53**a***z*.
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 *= **E***M *· **a***ρ *= −30*.*11 cos*φ *− 180*.*63 sin *φ *= −183*.*12

*Eφ *= **E***M *· **a***φ *= −30*.*11*(*− sin *φ)*− 180*.*63 cos*φ *= 0 *(*as expected*)
*so that **E***M *= −183*.*12**a***ρ *− 150*.*53**a***z*.

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 **E***M *. This will be:

*Er *= **E***M *· **a***r *= −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 *= *ρ*0*e*−|*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

*ρ*0*e
*−*x*−*y*−*z dx dy dz *= 8*ρ*0

2.13. A uniform volume charge density of 0.2 *µ*C*/*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 *=
∫ 2*π
*

0

∫ *π
*0

∫ *.*05
*.*03

0*.*2 *r*2 sin *θ dr dθ dφ *=
[

4*π(*0*.*2*)
r*3

3

]*.*05
*.*03

= 8*.*21 × 10−5 *µ*C = 82*.*1 pC

b) find *r*1 if half the total charge is located in the region 3 cm *< r < r*1: If the integral over *r *in part
*a *is taken to *r*1, we would obtain[

4*π(*0*.*2*)
r*3

3

]*r*1
*.*03

= 4*.*105 × 10−5

Thus

*r*1 =
[

3 × 4*.*105 × 10−5
0*.*2 × 4*π *+ *(.*03*)
*

3

]1*/*3
= 4*.*24 cm

18

2.14. Let

*ρv *= 5*e*−0*.*1*ρ (π *− |*φ*|*) *1
*z*2 + 10 *µ*C*/*m

3

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

5*e*−0*.*1*ρ (π *− |*φ*|*) *1
*z*2 + 10*ρ dρ dφ dz
*

which becomes

*Q *= 5
[
*e*−0*.*1*ρ
*

*(*0*.*1*)*2
*(*−0*.*1 − 1*)
*

]10 0

∫ ∞ −∞

2
∫ *π
*

0
*(π *− *φ) *1

*z*2 + 10*dφ dz
*

or

*Q *= 5 × 26*.*4
∫ ∞
−∞

*π*2
1

*z*2 + 10 *dz
*

Finally,

*Q *= 5 × 26*.*4 × *π*2
[

1√ 10

tan−1 (

*z*√
10

)]∞ −∞

= 5*(*26*.*4*)π
*3

√ 10

= 1*.*29 × 103 *µ*C = 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

5*e*−0*.*1*ρ (π *− *φ) *1
*z*2 + 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 *µ*m 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*.*9*π < φ < *1*.*1*π *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*.*1*π
.*9*π
*

∫ 25◦ 0

∫ 5 4

*(r *− 4*)(r *− 5*) *sin *θ *sin*(φ/*2*) r*2 sin *θ dr dθ dφ
*

19

2.16. (continued) Carrying out the integral, we obtain

*Q *= 10
[
*r*5

5
− 9 *r
*

4

4
+ 20 *r
*

3

3

]5 4

[ 1

2
*θ *− 1

4
sin*(*2*θ)
*

]25◦ 0

[ −2 cos

(
*θ
*

2

)]1*.*1*π
.*9*π
*

= 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

**E***P *= *ρl
*2*π*0

**R***P
*|**R***P *|2

where **R***P *= *(*1*, *2*, *3*)*− *(*1*,*−2*, *5*) *= *(*0*, *4*,*−2*)*, and |**R***P *|2 = 20. So

**E***P *= 16 × 10
−9

2*π*0

[
4**a***y *− 2**a***z
*

20

]
= 57*.*5**a***y *− 28*.*8**a***z *V*/*m

b) Find **E **at that point in the *z *= 0 plane where the direction of **E **is given by *(*1*/*3*)***a***y *− *(*2*/*3*)***a***z*:
With *z *= 0, the general field will be

**E***z*=0 = *ρl
*2*π*0

[
*(y *+ 2*)***a***y *− 5**a***z
(y *+ 2*)*2 + 25

]

We require |*Ez*| = −|2*Ey *|, so 2*(y *+ 2*) *= 5. Thus *y *= 1*/*2, and the field becomes:

**E***z*=0 = *ρl
*2*π*0

[
2*.*5**a***y *− 5**a***z
(*2*.*5*)*2 + 25

]
= 23**a***y *− 46**a***z
*

2.18. Uniform line charges of 0*.*4 *µ*C/m and −0*.*4 *µ*C/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

**E***P *= *ρl
*2*π*0

[
**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

**E***P *= *ρl
*2*π*0

[
*x***a***x *+ 0*.*6**a***y
x*2 + *(*0*.*6*)*2 −

*x***a***x *− 0*.*6**a***y
x*2 + *(*0*.*6*)*2

]
= 0*.*4 × 10

−6

2*π*0

[
1*.*2**a***y
*

*x*2 + 0*.*36
]
= 8*.*63**a***y
*

*x*2 + 0*.*36 kV*/*m

20

2.18. (continued)
b) Find **E **at *Q(*2*, *3*, *4*)*: This field will in general be:

**E***Q *= *ρl
*2*π*0

[
**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

**E***Q *= *ρl
*2*π*0

[
2**a***x *+ 3*.*6**a***y
*22 + *(*3*.*6*)*2 −

2**a***x *+ 2*.*4**a***y
*22 + *(*2*.*4*)*2

]
= −625*.*8**a***x *− 241*.*6**a***y *V*/*m

2.19. A uniform line charge of 2 *µ*C/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/(*2*π*0*ρ)*]**a***ρ *. Therefore, at point *P *:

**E***P *= *ρl
*2*π*0

**R***zP
*|**R***zP *|2 =

*(*2 × 10−6*)
*2*π*0

**a***x *+ 2**a***y
*5

= 7*.*2**a***x *+ 14*.*4**a***y *kV*/*m

where **R***zP *is the vector that extends from the line charge to point *P *, and is perpendicular to the *z
*axis; i.e., **R***zP *= *(*1*, *2*, *3*)*− *(*0*, *0*, *3*) *= *(*1*, *2*, *0*)*.

b) −4 ≤ *z *≤ 4: Here we use the general relation

**E***P *=
∫

*ρldz
*

4*π*0

**r **− **r**′
|**r **− **r**′|3

where **r **= **a***x *+ 2**a***y *+ 3**a***z *and **r**′ = *z***a***z*. So the integral becomes

**E***P *= *(*2 × 10
−6*)
*

4*π*0

∫ 4 −4

**a***x *+ 2**a***y *+ *(*3 − *z)***a***z
*[5 + *(*3 − *z)*2]1*.*5 *dz
*

Using integral tables, we obtain:

**E***P *= 3597
[
*(***a***x *+ 2**a***y)(z*− 3*)*+ 5**a***z
*

*(z*2 − 6*z*+ 14*)
*]4
−4

V*/*m = 4*.*9**a***x *+ 9*.*8**a***y *+ 4*.*9**a***z *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:

**E***P *= *ρl
*2*π*0

[
**R***xP
*

|**R***xP *|2 +
**R***yP
*

|**R***yP *|2 +
**R***zP
*

|**R***zP *|2
]

where **R***xP *, **R***yP *, and **R***zP *are the normal vectors from each of the three axes that terminate on point
*P *. Specifically, **R***xP *= *(*−3*, *2*,*−1*) *− *(*−3*, *0*, *0*) *= *(*0*, *2*,*−1*)*, **R***yP *= *(*−3*, *2*,*−1*) *− *(*0*, *2*, *0*) *=
*(*−3*, *0*,*−1*)*, and **R***zP *= *(*−3*, *2*,*−1*)*− *(*0*, *0*,*−1*) *= *(*−3*, *2*, *0*)*. Substituting these into the expression
for **E***P *gives

**E***P *= *ρl
*2*π*0

[
2**a***y *− **a***z
*

5
+ −3**a***x *− **a***z
*

10
+ −3**a***x *+ 2**a***y
*

13

]
= −1*.*15**a***x *+ 1*.*20**a***y *− 0*.*65**a***z *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/(*2*π*0*(*0*.*8*))*]**a***y *. The force on a differential length of the
line at the positive *y *location is *d***F **= *dq***E **= *ρldz***E**. Thus the force per unit length acting on the line
at postive *y *arising from the charge at negative *y *is

**F **=
∫ 1

0

*ρ*2*l dz
*

2*π*0*(*0*.*8*)
***a***y *= 1*.*26 × 10−4 **a***y *N*/*m = 126 **a***y µ*N*/*m

The force on the line at negative *y *is of course the same, but with −**a***y *.

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
*

4*π*0

**r **− **r**′
|**r **− **r**′|3

where **r **= 3**a***x *and **r**′ = *y***a***y *+ *z***a***z*. The integral becomes:

**E***PA *= *ρs
*4*π*0

∫ ∞ −∞

∫ 2 −2

3**a***x *− *y***a***y *− *z***a***z
*[9 + *y*2 + *z*2]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 *= 3*ρs
*4*π*0

∫ 2 −2

*dy
*

*(*9 + *y*2*)
*

[
*z*√

9 + *y*2 + *z*2

]∞ −∞

= 3*ρs
*2*π*0

∫ 2 −2

*dy
*

*(*9 + *y*2*)
*

= 3*ρs
*2*π*0

( 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 **= 3**a***y *, and symmetry indicates that only a *y *component will exist.
The integral becomes

*Ey,PB *= *ρs
*4*π*0

∫ ∞ −∞

∫ 2 −2

*(*3 − *y) dy dz
*[*(z*2 + 9*)*− 6*y *+ *y*2]1*.*5 =

*ρs
*

2*π*0

∫ 2 −2

*(*3 − *y) dy
(*3 − *y)*2

= − *ρs
*2*π*0

ln*(*3 − *y)
*∣∣∣2−2 = 145 V*/*m

22

2.23. Given the surface charge density, *ρs *= 2*µ*C*/*m2, in the region *ρ < *0*.*2 m, *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 **= *z***a***z*. Then, with **r**′ = *ρ***a***ρ *, we
obtain **r **− **r**′ = *z***a***z *− *ρ***a***ρ *. The superposition integral for the *z *component of **E **will be:

*Ez,PA *=
*ρs
*

4*π*0

∫ 2*π
*0

∫ 0*.*2
0

*z ρ dρ dφ
*

*(ρ*2 + *z*2*)*1*.*5 = −
2*πρs
*4*π*0

*z
*

[ 1√

*z*2 + *ρ*2

]0*.*2
0

= *ρs
*20

*z
*

[
1√
*z*2

− 1√
*z*2 + 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

**E***A *= 20 × 10
−9

20
**a***x *+ 30 × 10

−9

20
**a***y *− 40 × 10

−9

20
**a***z *= 1130**a***x *+ 1695**a***y *− 2260**a***z *V*/*m

The magnitude of **E***A *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*)
*

4*π*0

*(*−2**a***x *− 6**a***z)
(*4 + 36*)*1*.*5

]
+
[
*(*3 × 10−9*)
*

2*π*0

*(*2**a***x *− 3**a***y)
(*4 + 9*)
*

]
−
[
*(*0*.*2 × 10−9*)***a***x
*

20

]
= −3*.*9**a***x *− 12*.*4**a***y *− 2*.*5**a***z *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
**a***y *− 2 × 5 × 10

−9

2*π*0*(*2*)
***a***z *= −33*.*9**a***y *− 89*.*9**a***z
*

so that |**E**| = 96*.*1 V*/*m.

23

2.27. Given the electric field **E **= *(*4*x *− 2*y)***a***x *− *(*2*x *+ 4*y)***a***y *, find:
a) the equation of the streamline that passes through the point *P(*2*, *3*,*−4*)*: We write

*dy
*

*dx
*= *Ey
*

*Ex
*= −*(*2*x *+ 4*y)
*

*(*4*x *− 2*y)
*Thus

2*(x dy *+ *y dx) *= *y dy *− *x dx
*or

2 *d(xy) *= 1
2
*d(y*2*)*− 1

2
*d(x*2*)
*

So

*C*1 + 2*xy *= 1
2
*y*2 − 1

2
*x*2

or
*y*2 − *x*2 = 4*xy *+ *C*2

Evaluating at *P(*2*, *3*,*−4*)*, obtain:

9 − 4 = 24 + *C*2*, *or *C*2 = −19

Finally, at *P *, the requested equation is

*y*2 − *x*2 = 4*xy *− 19

b) a unit vector specifying the direction of **E **at *Q(*3*,*−2*, *5*)*: Have **E***Q *= [4*(*3*)*+ 2*(*2*)*]**a***x *− [2*(*3*)*−
4*(*2*)*]**a***y *= 16**a***x *+ 2**a***y *. Then |**E**| =

√
162 + 4 = 16*.*12 So

**a***Q *= 16**a***x *+ 2**a***y
*16*.*12

= 0*.*99**a***x *+ 0*.*12**a***y
*

2.28. Let **E **= 5*x*3 **a***x *− 15*x*2*y ***a***y *, and find:
a) the equation of the streamline that passes through *P(*4*, *2*, *1*)*: Write

*dy
*

*dx
*= *Ey
*

*Ex
*= −15*x
*

2*y
*

5*x*3
= −3*y
*

*x
*

So
*dy
*

*y
*= −3 *dx
*

*x
*⇒ ln *y *= −3 ln *x *+ ln*C
*

Thus

*y *= *e*−3 ln *xe*ln*C *= *C
x*3

At *P *, have 2 = *C/(*4*)*3 ⇒ *C *= 128. Finally, at *P *,

*y *= 128
*x*3

24

2.28. (continued)
b) a unit vector **a***E *specifying the direction of **E **at *Q(*3*,*−2*, *5*)*: At *Q*, **E***Q *= 135**a***x *+ 270**a***y *, and

|**E***Q*| = 301*.*9. Thus **a***E *= 0*.*45**a***x *+ 0*.*89**a***y *.
c) a unit vector **a***N *= *(l, m, *0*) *that is perpendicular to **a***E *at*Q*: Since this vector is to have no *z *compo-

nent, we can find it through **a***N *= ±*(***a***E*×**a***z)*. Performing this, we find **a***N *= ±*(*0*.*89**a***x *− 0*.*45**a***y)*.

2.29. If **E **= 20*e*−5*y *(cos 5*x***a***x *− sin 5*x***a***y*), find:
a) |**E**| at *P(π/*6*, *0*.*1*, *2*)*: Substituting this point, we obtain **E***P *= −10*.*6**a***x *− 6*.*1**a***y *, and so |**E***P *| =

12*.*2.

b) a unit vector in the direction of **E***P *: The unit vector associated with **E **is just
(
cos 5*x***a***x *− sin 5*x***a***y
*

) ,

which evaluated at *P *becomes **a***E *= −0*.*87**a***x *− 0*.*50**a***y *.
c) the equation of the direction line passing through *P *: Use

*dy
*

*dx
*= − sin 5*x
*

cos 5*x
*= − tan 5*x *⇒ *dy *= − tan 5*x dx
*

Thus *y *= 15 ln cos 5*x *+ *C*. Evaluating at *P *, we find *C *= 0*.*13, and so

*y *= 1
5

ln cos 5*x *+ 0*.*13

2.30. Given the electric field intensity **E **= 400*y***a***x *+ 400*x***a***y *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 *x*2 = *y*2 + *C*. Evaluating at *A *yields *C *= 3, so the equation becomes

*x*2

3
− *y
*

2

3 = 1

b) the equation of the surface on which |**E**| = 800 V/m: Have |**E**| = 400
√
*x*2 + *y*2 = 800. Thus

*x*2 + *y*2 = 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