engineering electromagnetic fields and waves 2nd edition.pdf

_________________________________________ CHAPTER 1

Vector Analysis

and Electromagnetic Fields

in Free Space

The introduction of vector analysis as an important branch of mathematics dates back to the midnineteenth century. Since then, it has developed into an essential tool for the physical scientist and engineer. The object of the treatment of vector analysis as given in the first two chapters is to serve the needs of the remainder of this book. In this chapter, attention is confined to the scalar and vector products as well as to certain integrals involving vectors. This provides a groundwork for the Lorentz force effects defining the electric and magnetic fields and for the Maxwell integral relationships among these fields and their chargc and current sources. The coordinate systems em-ployed are confined to the common rectangular, circular cylindrical, and spherical systems. To unifY their treatment, the generalized coordinate system is used. This time-saving approach permits developing the general rules for vcctor manipulations, to enable writing the desired vector operation in a given coordinate system by inspection. This avoids the rederivation of the desired operation for each new coordinate system employed.

Next arc postulated the Maxwell integral relations for the electric and magnetic fields produced by charge and current sources in free space. Applying the vector rules developed earlier, their solutions corresponding to simple classes of symmetric static charge and current distributions are considered. The chapter concludes with a discus-sion of transformations among the three common coordinate systems.

1·1 SCALAR AND VECTOR FIELDS

A field is taken to mean a mathematical function of space and time. Fields can be classified as scalar or vector fields. A scalar field is a function having, at each instant in

lJ

i'

2

VECTOR ANALYSIS AND ELECTROMAGNETIC FIELDS IN FREE SPACE

F

time, an assignable magnitude at every point of a region in space. Thus, the tem-perature field

t)

inside the block of material of Figure 1-1 (a) is a scalar field. To each point there exists a corresponding temperature T(x,]!,

z,

t)

at any instant

t

in time. The velocity of a fluid moving inside the pipe shown in Figure 1-1 (b) illustrates a vector field. A variable direction, as well as magnitude, of the fluid velocity occurs in the pipe where the cross-sectional area is changing. Other examples of scalar fields are mass, density, pressure, and gravitational potential. A force field, a velocity field, and an acceleration field are examples of vector fields.

The mathematical symbol for a scalar quantity is taken to be any letter: for example,

A, T,

Il,

f.

The symbol for a vector quantity is any letter set in boldface

roman type, ff)!' A, H, a, g. Vector quantities are represented graphically by

6

(x)

Heat source

(a)

FIGURE 1-1. Examples of

material. (b) Fluid velocity field ill,ide

(z)

6

200·

Temperature field at x

=

4 em

m-Id.

ny (b) ity lar ity for lce by

I

3--

1~!

~

----...

Unit

~\

~~\~I("'"

B=C vector a

y

FIGURE 1-2. Graphic representations of a vector, equal vectors, a uni t vector, and the representation of magnitude or length of a

vector.

means of arrows, or directed line segments, as shown in Figure 1-2. The magnitude or length of a vector A is written \A\ or simply

A,

a positive real scalar. The

negative

of a vector is tbat vector taken in an opposing direction, with its arrowhead on the opposite end. A

unit vector

is any vector having a magnitude of unity. The symbol a is used to denote a unit vector, with a subscript employed to specify a special direction. For example, ax means a unit vector having the positive-x direction. Two vectors are said to be

equal

if they have the same direction and the same magnitude. (They need not be collinear, but only parallel to each other.)

1·2 VECTOR SUMS

The vector sum of A and B is defined in relation to the graphic sketch of the vectors, as in Figure 1-3. A physical illustration of the vector sum occurs in combining dis-placements in space. Thus, if a particle were displaced consecutively by the vector distance A and then by

B,

its final position would be denoted by the vector sum A

+

B

=

C shown in Figure 1-3 (a). Reversing the order of these displacements pro-vides the same vector sum C, so that

A+B=B+A

( 1-1)

the commutative law of the addition of vectors. If several vectors are to be added, an associative law

(A

+

B)

+

D

=

A

+

(B

+

D)

follows £I'om the definition of vector sum and from Figure 1-3(b).

B (a) I I I I :A + B

=

C I I I I (b)

FIGURE 1-3. (a) The graphic definition of the sum of two vectors. (b) The associa-tive law of addition.

1·3 PRODUCT OF A VECTOR AND A SCALAR If a scalar

same direction

The f()lIowing laws hold

u and if B denotes a vector quantity, their produc a magnitude u times the magnitude of B, and having tht a positive scalar, or the opposite direction if u is negative

IiII' the products of vectors and scalars.

uB

Bu

Commutative law ( 1-3)

(uv)A Associative law ( 1-4)

(u o)A = uA

+

vA Distributive law (1-5)

u(A +B) uA

+

uB

Distributive law ( 1-6)

1·4 COORDINATE SYSTEMS

The solution of physical problems often requires that the framework of a coordinate system be introduced, particularly

if

explicit solutions are being sought. The system most familiar to engineers and scientists is the cartesian, or rectangular coordinate sys-tem, although two other ii'ames of reference often used are the circular cylindrical and the spherical coordinate systems, The symbols employed for the independent coordinate variables of these orthogonal systems are listed as follows.

1. Rectangular coordinates: (x,y, z)

2. Circular cylindrical coordinates: (p,

cj>,

z) 3. Spherical coordinates: (r, 8, cj»

In

Figure 1-4(a), the point

P

in space, relative to the origin 0, is depicted in terms of the coordinate variables of the three common orthogonal coordinate systems: as P(x,y, z) in the rectangular system, as P(p,

cj>,

z) in the circular cylindrical (or just "cylindrical") system, and as P(r, 8, cj» in the spherical coordinate system.

In

the cylindrical and spherical systems, it is seen that the rectangular coordinate axes, labeled (x), and ,are retained to establish proper angular references. You should observr that. the coordinate variable

cj>

(the azimuth angle) is common to both

(x) : (zi I I P(x, y, z) z

--y Rectangular

FIGURE 1-4. Notational convcnlions

(a) Location of a point P in space, (Ii) The

Circular cylindrical (a)

Spherical

in the three nnnmoll coordinate systems. p"im P Ie) The resolution of a vector A into its orthogonal COmpOllt'nts.

-

,

,

~ ~ ~~ ~!a ~

s·

~ &~ ~ ~ ("l) ("l) ,

S '""

i:

(.n ~ (x) -x=Constant (z) z = Constant - - ( y ) z= Constant (plane) (x) p=Constant (circular cylinder) (b) (z) 0 d> = Constant (plane) (y) r= Constant (sphere) I I I I

,

:(z) :(z) :(z) I I '

_--®~ZA:---..,

:

~a;~;

i

arA~P:

r::::"- ... A ... "",,-- I I ,,/ .. " I , - - " " a A I

, __ _. P " ' , n! I 4 ' ' q, <I> I

t I ' I Af - r; I :::,-_... I

, I ' ' a I , , " I

axAx~:-::=L---

ayAy

i : /

q,A<I>1 A " ::'>w'

O/,

~

z _-'-_

aA-~'biO

p p z

aeAo' "-t':

: r~O

_- --- 0 ----_ Jy) _- - - ---~y) -'--~;' -(x) _--- - _---- <I> - _---- : <1>/ _- y x (x) p (x)

r--'

Rectangular FIGURE 1-4 (continued) Circular cylindrical (c) Spherical

-~

(z) (y) M-"'''I 0.. :;:."'" .: ("l) :::r- (") • ("l) ... d> = Constant . (plane) , \ .-,1 (yl

...

c:

§

?: I:

:2

:>-...

to' u ~ u ~ to'

::

"

~

the cylindrical and tbe spherical systems, with the x-axis taken as the </> = 0 reference,

</> generated in the positive sense from (x) toward (y). (By the "right-hand rule,"

if the thum b of the right hand points in the positive z-dircction, the fingers will indicate the sense.) The radial distance in the cylindrical system is p, measured perpendicularly from the to the desired point P; in the spherical system, the radial distance is 1, measured from the origin 0 to the point P, with

°

denoting the

desired declination angle measured positively from the reference z-axis to 1, as shown

1-4( a). The th ree coordinate systems shown are so-called "right-handed" properly definable after first discussing the unit vectors at P.

A.

Unit Vectors and Coordinate Surfaces

To enable expressing any vector A at the point P in a desired eoordinate system, three orthogonal unit vectors, denoted by a and suitably subscripted, are defined at

P

in the positive-increasing sense of each of the coordinate variables of that system. Thus, as noted in Figure 1-4(b), ax, a y, az are the mutually perpendicular unit vectors

of the rectangular coordinate system, shown at P(x,y, z) as dimensionless arrows of unit length originating at P and directed in the positive X,], and;;; senses respectively. Note that the disposition of these unit vectors at the point

P

corresponds to a right-handed coordinate system, so-called because a rotation from the unit vector ax through thc smaller angle toward ay and denoted by the fingers of the right hand, corresponds to

the thumb pointing in the direction of az . Similarly, in the cylindrical coordinate

system of that figure, the unit vectors at P(p, </>, z) are ap ' aq,' az as shown, pointing

in the positive p, </>, and;;; senses; at P(r, 0,

<j»

in the spherical system, the unit vectors an ao, aq, are shown in the positive directions of the corresponding coordinates there. These are also right-handed coordinate systems, since on rotating the fingers of the right hand from the first-mentioned unit vector to the second, the thumb points in the direction of the last unit vector of each triplet.

Notice from Figure 1-4(b) that the only constant unit vectors in these coordinate systems are ax, ay , and az . The unit vectors ap and aq, in the circular cylindrical system,

II)r example, will change (in direction, not magnitude) as the angle </> rotates

P

to a new location. Thus, in certain differentiation or integration processes involving unit vectors, most unit vectors should not be treated as constants (see Example I-I in Section 1-6). I n Figure I , it is instructive to notice how the point P, in any of the co-ordinale systems, can be looked on as the intersection of three coordinate suifaces. A coordinate surf;tcC necessarily planar) is defmed as that surface formed by simply Ihe desired coordinate variable equal to a constant. Thus, the point P(x,], z) in the is the intersection of the three coordinate surfaces x = constant, y =

constallt, constant (in this case planes), thosc constants depending on the desired location fe)r P. two such coordinate surfaces intersect orthogonally to define a line;whiIe the perpt'IHlicular intersection of the line with the third surface pinpoints P.) The unit vectors at z) are thus perpendicular to their corresponding coordinate surfaces .g., ax is perpendicular to the surface x = constant). Because the coordinate surfaces are mutually perpendicular, so are the unit vectors.

Similar observations at in the cylindrical coordinate system are appli-cable.

P

is the intersection

or

the three orthogonal coordinate surfaces p = constant (a right circular cylindrical </> constant (a semi-infinite plane), and ,(; constant (a plane), to each of which thee corresponding unit vectors are perpendicular,

thus making a_{p '} aq" a_z welL comments apply to the unit

lce, le, " ;ate red the the 'wn ~d" ~m, at :m. ors nit ote ded the to ate ng Drs reo he .he He m, ew ,rs,

5).

:0-A )Iy

z)

==

ed a ~.) lte Ite li-nt If, lit I) ,

wherein the coordinate suriltces defining the intersection P in this instance become r = constant (a spherical surface), ()

=

constant (a conical j and 4>

= constant

(a semi-infinite plane).

B. Representations in Terms of

Vector Components

A use[ill application of the product of a vector and a scalar as described in Section

1-3

occurs in the representation, at any poin t

P

in space, of the vector

A

in terms of its coordinalf components. In the rectangular system of Figure 1-4(c) is shown the typical vector

A

at the point P(x,y, z) in space. The perpendicular projections of

A

along the unit vectors ax, a y and a_z yield the three vector components of

A

in rec-tangular coordinates, seen from the geometry to be the vectors

axAx, ayAy,

_{and azAz} in that figure. Their vector sum,

axAx

+

ayAy

+

azAz

=

A,

thus provides the desired

representatioIl of

A

in the rectangular coordinate system. Similar manipulations into circular cylindrical and spherical coordinate components yield the other two corre-sponding diagrams depicted in Figure 1-4(c), whence the representations of A in terms

of its components: 1

A

=

axAx

+

ayAy

+

azA

z Rectangular

A

=

apAp

+

a.pA.p

+

azA

z Circular cylindrical

A

= arAr

+

aoAo

+

a.pA.p

Spherical

(1-7)

Because of the mutual perpendicularity of the components of any of these representa-tions, it is clear that the geometrical figure denoted by each dashed-line representation

of Figure is a parallelepiped (or box), with A appearing as a principal diagonal within each. The magnitude (or length) of each

A

in

(1

thus becomes

A

=

[A~

+

A;

+

A;)

1/2 Rectangular

A

=

[A;

+

A~

+

A;

11/2 Circular cylindrical

A

=

[A;

+

A~

+

A~]1/2 Spherical

C. Representation in Terms of Generalized

Orthogonal Coordinates

(1-8)

Noting the several similarities in the charaeterizations of the unit vectors and the vector A in the three common coordinate systems just described, and to permit unifying and shortening many discussions later on relative to scalar and vector fields, the system or generalized orthogonal coordinates is introduced. In this system, u_{I ,} u2 , U3 denote the

generalized coordinate variables, as suggested by Figure i-5(a). The generalized ap-proach to developing properties of fields in terms of (UI' 112, 113) has the advantage of

making it unnecessary to rederive certain desired expressions each time a new coordi-nate system is encountered.

Just as I(x the three common coordinate systems already described relative to Fignre 1-4, the point

P(uj,

112) 113) in generalized coordinates, as seen in Fignre 1-5(a),

lThus, the components of A in the rectangular coordinate system are the vectors axA" ayAy, and azAz'

Another usage is to rekr to only the scalar multipliers (lengths) AX' and A_z as the components of A,

8

VECTOR ANALYSIS AND ELECTROMAGNETIC FIELDS IN FREE SPACE Increasing u 1 I I (z) , y== x= Constant Constant az Rectangular (b) /llncreaSing u:l I I

Generalized orthogonal coordinates (a) Circular cylindrical (e) Increasing 112 Spherical (d)

FIGURE 1-5. The coordinate surfaces defining the typical point P and the unit vectors at P.

is the intersection of three perpendicular coordinate surfaces, Ul

=

constant, U2

=

constant, U3 = constant. The intersections of pairs of such surfaces moreover define coordinate lines. The unit vectors, denoted aI, a2, a3, are mutually perpendicular, tan-gent to the coordinate lines, and intersect the coordinate surfaces perpendicularly. The one-to-one correspondence of the:;e generalized coordinate variables Ill' U2, U 3 to their coordinate surfaces, and the generalized unit vectors aI, a2 , a3 to the equivalent vec-tors of the three common coordinate systems, can be better appreciated on making a direct visual comparison of the generalized sketch of Figure 1-5(a) with (b), (e), and (d) of that figure.

If the vector A were components alAI, and expression for A would he

A I ts magnitude is

The scalars AI, A2l and A

specialized to the three COllllllOII and (1-8).

the point P(uI' U2, in Figure 1-5(a), with the

in the directions of the unit vectors shown, the ( 1-9) construction for (1-9).

( 1-10)

'"''I'IJI''''"'''''

lIf A. Kxarnples of these expressions already been given in (1-7)

z=

fine an-fhe 1eir lec-19 a (d) the the [-9) -9). ·10) Ions 1-7)

1-5 DIFFERENTIAL ELEMENTS OF SPACE

9

1·5 DIFFERENTIAL ELEMENTS OF SPACE

In the processes of integration in space to be considered shortly, the differential ele-ments of volume, surface, and line are frequently needed. A differential element of volume dv is generated in the vicinity of a point P(Ub U2, U3) in space by means of the

displacements dtb dt2 , and dt3 on the coordinate surfaces, through the differential changes dUll duz, and dU3 in the coordinate variables. This situation is represented geometrical! y in Figure 1-6 (a). Thus, a volume-elemen t dv is represented in generalized orthogonal coordinates by means of the product of the differential length-elements as follows

(1-11 ) The relation or the length-elements to differential changes in the coordinate variables Ul' U2' and U3 is provided by the relations

(x)

113 + clu3

=

Constant

113

=

Constant Generalized (curvilinear) coordinates

(al (z)

,

I I / I _ (z) Rectangular (6) (z)

--"'::

-(;j- -Spheric;,i (<I)

FIGURE 1-6. The generation of a volume-element dv = dt1dt2dl'} at

orthogonal coordinate systems.

I

10

VECTOR ANALVSIS AND ELECTROMAGNETIC FIELDS IN FREE SPACE

so that (I-II) is written

(1-13~

The coefllcients hi'

dt of (1-12) their geometry of dv in each

! and h3 are called metric codfieients, needed to give the element~

dimension of length (meter). From a consideration of tht of Figure 1-6(b), (e), and (d), it is evident that tht and metric coefficients are applicable to the three commor: following

systems.

dx dt

_z

=

dy dt3 = dz

hi

=

h3

=

I Rectangular (1-14) dp dt

_z

=

pdp dt3

=

dz

hi p, h3

=

I Circular cylindrical ( I IS)

= rdO dt3

=

r sin Odp

hi I, h2 r, h3

=

r sin

0

Spherical ( 1-16) The substitution of these results into (1-13) therefore provides the volume-element dL

in each system as follows.

do dx

dv II

dv

Rectangular Circular cylindrical

sin OdrdOdp Spherical (1-17)

S in space may be left in its scalar f(nm ds, although for some purposes it a vector characterization, ds, if desired. Suppose ds coincides with a cOIlrdillatt' surface Ul = constant, as shown in Figure I-7(a).

ds = at

FIGURE 1-7. Typical as a vector element through 011 the coordina te surf~~,r{'

ds on the coordinate I (z) I

i

r = Constant (0) Iht, characterization of ds (a) A surface element ds

-13) Lents fthe • the mon -14) -15) -16) it dv -17) ugh [lose '(a). 1-6 POSITION VECTOR

11

Expressed as a scalar element, ds

=

_{dt2 dt3}

=

_{h2h3 du z dU3} for that example. An illustra-tion in spherical coordinates is shown in Figure 1-7 (b); on the r

=

constant coordinate surface, ds

=

r2 sin

0

dO d¢. A vector quality is given dol' through multiplying it with either the positive or the negative of the unit vector normal to ds. Thus, in Figure 1-7 (b),

the vector surface-element ds

=

ar ds is illustrated; ds

=

ar ds is the other possible

choice on the coordinate surface r

=

constant exemplified. These concepts are partic-ularly useful in the flux-integration techniques discussed in Section 1-9.

Differential line-elements are frequently of interest in applications to vector integration. This subject is introduced in terms of the position vector r of spatial points treated in the next section.

*1·6 POSITION VECTOR2

In field theory, reference may be made to a point P(Ub Ul, U3) in space by use of the position vector, denoted by the symbol r. The position vector of the point P in Figure 1-4, for example, is the vector r drawn from the origin 0 to the point

P.

Thus in rectangular coordinates, r is written

(1-18) and in circular cylindrical coordinates

( 1-19) while in spherical coordinates

r = arr ( 1-20)

A further application of the position vector r occurs in the symbolic designation of points in space. Instead of using the symbol P(Ul' U

_z,

U3) or P(x,y, z), you may employ the abbreviated notation P(r). By the same token, a scalar fielel F(ub _{U2, U3 ,} t)

can be more compactly represented by the equivalent symbol F(r, I), if desired. The differential element of length separating the points P(r) and P(r

+

dr) in space is denoted by the vector differential displacement dr. The differential change dr does not in general occur in the same direction as the position vector r; this is exemplified in Figure 1-8 (a). (The vector symbol de is sometimes used interchangeably with dr, particularly in line-integration applications.) The difierential displacement dr (or de) is written in terms of its generalized orthogonal components as follows.

dr

==

de

=

al dtl

+

a2dt2

+

a3dt3 alh l dUl

+

a2h2 dU2

+

a3h3 dU3

(1-21 )

(1-22)

It is illustrated graphically in Figure I-8(b) by means of the usual rectangular paral-lelepiped construction for a vector in terms of its components. Furthermore, the magnitude dt of the vector dt is given by the diagonal of the rectangular parallelepiped; thus,

(1-23) 2Throughout the text, sections marked with an asterisk (*) may be omitted to conserve time if desired.

12

VECTOR ANALYSIS AND ELECTROMAGNETIC FIELDS IN FREE SPACE Pathf \

,

(a)

o

-~-- (y) (b)

FIGURE 1-8. The position vector r used in defining points of space and its differential dr. (aJ The position vector r and a difrerenlial position change dr along an arbitrary path. (b) Showing the components of dr in generalized orthogonal coordinates.

For example, in spherical coordinates hl

=

1, h2

=

r, and h3 r sin 8, so that (1-2: and (1-23) are written

with

( 1-2. The simplest expression for a differential vector displacement dt occurs in tl

rectangular coordinate system, for which, from (1-14), with hi = 112 = h3 = 1 and wi at = ax, a2 = a y and _{a 3 = a}z , the general form (1-22) becomes

(1-2 while its magnitude dt is written, from the generalized (1-23), as

dt

( 1-2

Similarly, in the circular ~ylindrical coordinate system, the substitution of (1-1 into (1-22) and (I and with at ap ' a2 = a4> and a 3 = az , the vector displa, ment dt and its magIlitude hecoltw

ell

( 1-:

2 _d</J)2

₊

_{( 1-:}

The position vector r has usdi.ll applications in the dynamics of particles sud electrons and ions, fiJI' A of Figure 1-8 reveals that if the vector t

placement dr of a particle occurs in the time interval dt, then the ratio dr/dt dell( the vector velocity of the at Per). This particle velocity v is defined by

1-22) 1-24) :1-25) in the 1 with (1-26) ( 1-27) (1-15) splace-(1-28) ( 1-29) such as tor dis-::lenotes by the 1-6 POSITION VECTOR

13

derivative of the position vector r(t)

v dr dt . r(t

+

lit) - r(t) hm --'---:--'----'-At-+O lit ( 1-30)

A second such derivative of r(t) provides the vector acceleration <L

=

dv/dt of the

particle.

Because the vector displacement dr of the particle is tangent to its path t as shown in Figure 1-8, the velocity v

=

dr/dt will also be tangent at every point on

t.

This property of tangency does not hold for acceleration, however, except in purely straight-line motion. The velocity at the point P(r) can be expressed systematically in terms of its generalized orthogonal coordinate velocity components by means of

( 1-31)

For example, in a rectangular coordinate system, the notations Vb 1}2, and V3 mean

tf", v_{Y '} and V z respectively.

In all orthogonal coordinate systems except the rectangular system, some of or

all the unit vectors may change direction as their location

P

moves in space. A graphical approach to obtaining the spatial derivatives of the unit vectors in an explicit coordi-nate system is described in the following example.

EXAMPLE 1·1. Find the following partial derivatives of the unit vector ar : (a) Oar/Br;

(b) Bar/BO; (c) oa,/ocp.

(a) The partial derivative oar/Br equals zero, since the unit vector ar does not vary in

direction with r (nor does it vary in magnitnde, by the definition of a unit vector).

(b) The partial derivative oa,/oa can be found graphically from the accompanying figure. If _{a r is allowed only the differential change} dar in the

a

sense, then dar has

(a) (b)

EXAMPLE 1-1. (a) Differential dar generated by rotating ar 8-wise. (b) Differential dar

I

14

VECTOR ANALYSIS AND ELECTROMAGNETIC FIELDS IN FREE SPACE

the direction of the unit vector aoo The length of dar is given precisely by the

dO, irom the dc1inition of angle divided by radius, and the radius is unit make day become

whence the desired result is

dar] r

~

constant = ae dB

$ = constant

dar]

dO r ~ constant ¢=constant

The partial derivative 8aJikp is found sim.ilarly from (b) of the figmeo All only the changoe

d4>

in the position ofay generates the diHcrential vector dan

r

a direction specified by the unit vector a.p and a magnitude given by dq, sin (;

makes day (for r

=

constant, () = constant) become a,,> sin 8 dq, as shown, '"

By means of graphic techniques simila,o to those used in Example 1-1, 011 show for spherical coordinates that all the spatial partial derivatives of the unit v in that system are zero except for

Ja,o ° [ )

J¢ = aq, 3m tJ

ay sin ()

while in the circular cylindrical system, all are zero except for

1·7 SCALAR AND VECTOR PRODUCTS OF VECTORS

Besides the simple product of a vector with a scalar quantity discussed in Secli( two other kinds of products involving only vector quantities are now discussel lirst of these, called the scalar product (or dot product), is defined as followso

A· B

==

AB cos ()

in which () signifies the angle between the vectors A and B. Noting from (1-3 A· B may be written either (A cos 8)B or A(B cos 0) makes it evident that th, product A . B denotes the product of the scalar projection of either vector 0' other, times the magnitude of the other vector. The definition of A . B makes th,

the angle unity), to . Allowing Ian having sin O. This Il, whence , one can lit vectors ( 1-32) ( 1-33) eclioll 1-3, ussed. The (1-34) (1-34) that t the scalar )r onto the s the scalar

1-7 SCALAR AND VECTOR PRODUCTS OF VECTORS

15

useful, tor example, in computing the work done by a constant force acting a distance expressed as a vector. A generalization of this idea extended to the

expression for work is taken up in the next section.

Definition (1-34) permits the conclusion that if A and B are perpendicular, cos () zero, making their scalar product zero. Again, if A and B happen to lie in the same then A • B denotes the product of their lengths. These observations lead to results involving the scalar products of the orthogonal unit vectors _{a l , a2,} and

8 3 coordinate systems illustrated in Figure 1-5. For example, a l • a2 a2 • a3

=

8:\ • a l = 0, while at • at = a_{z .} a2 = a3 • a3 = l.

From the definition (1-34), and since B· A means BA cos 0, the commutative fhr the dot product follows.

A·B=B·A (1-35)

distributive law for the dot product of the sum of two vectors with a third vector

A . (B

+

C)

= A . B

+

A . C ( 1-36)

also be proved.

IXAMPLE 1·2. Vector analysis can be used to shorten a number of proo[~ of g-eometry. Sup-pose one is to show that the diagonals of a rhombus arc perpendicular. Represent its sides and diagonals by means of the veetors shown in the diagram. The diagonals are A

+

B C and A B D. Form the dot product of C and D.

(A

+

B) . (A - B) = A· A - B . B = A2 - B2

which must equal zero because A B for a rhombus. Thercf()[(~ C and D are perpendicular.

1 f rhe vectors A and B are expressed in terms of their generalized orthogonal c:omponents in the manner of (1-9), their scalar product can be written

expanding this expression by means of the distributive law (1-36) and applying results obtained earlier fix the dot products of the unit vectors, one obtains

(1-37a)

if

F

II.

16

VECTOR ANALYSIS AND ELECTROMAGNETIC FIELDS IN FREE SPACE

For example, the expansion of the dot product of two vectors in rectangular COOl nates is

( 1-3' and in circular cylindrical coordinates

(1-3

EXAMPLE 1-3. (a) At the point P(3, 5, 6), shown in (a) of the figure, are given the two veet D

= -

50a_x

+

60ay

+

100a_z and E

=

12a_{x -} 24ay- Find the vector magnitudes and dot product D . E. Use these to determine the projection D cos

e

of D onto E, and angk

e

between the vectors. (b) In (b) of the figure, at point P(5, 60°, 9) are given two vectors F = IOap

+

Ba", 4az and G = - 20ap

+

BOaz in cylindrical coord ina

Find the vector magnitudes and F . G as well as the angle 0 between the vectors.

(2) (xl ~-5 (al (xl EXAMPLE I G D= 50a., +603.,+ 1003, '~ (yl F

coordi-1-37b) vectors and the md the ven the :Iinates.

1-7 SCALAR AND VECTOR PRODUCTS OF VECTORS

17

(a) By usc of (I the vector magnitudes are

while the dot product is found from expansion (1-37b)

50(12)

+

60( -24) 2040 The latter, by (1-34), also means DE cos 0, whence the projection D cos 8 becomes

J) cos (J D·E

E

-2040

26.833 76.03

This nCl-iative result shows that the projection D cos 0 alonl-i E is in the negative-E sense (meaninl-i that 0 exceeds 90°). The value of 0 is found from the definition (1-34), yieldinl-i

.. 1 D . E .. 1 2040 , . , 0

0= cos ~-= cos -~-.. - - - - -= 126.82 DE 126.886(26.883)

(b) The mal-iniludcs and dot product, from (1-7) and coordinates, arc in circular cylindrical F [F~

+

F~

+

1';]112 [102

₊

₈2

₊

₄2 ] = 13.416 G = [202

+

802

F

I2 = 86.462 F' G = 10(-20) - 4(80» = -520

The anl-ik () between F and G is found from definition (1-34), obtaining

.. 1 F . G .. 1 520 , 0

0= cos = cos --~"'-'-- - = 117.93 FG 13.416(82.462)

From this result you may determine that the projection of F ncgativc-G sellSe.

G is in the

The second kind

or

product of one vector with another is called the vector product

cross product), defined as l()Uows

A x B

=

a"AB sin 0 ( 1-38)

in which

e

is the angle measured between A and B, and a" is a unit vector taken to be perpendicular to both A and B and having a direction determined {i-om the right-hand rule provided that the rotation is taken {i'om A to B through the angle O. The vector product A x B is illustrated graphically in Figure 1-9. One may show from the diagram that

18

VECTOR ANALYSIS AND ELECTROMAGNETIC FIELDS IN FREE SPACE AxB A~--~ () B A

--

_",

Positiv€~""­ (J sense from A to B

FIGURE 1-9. Illustrating the cross product.

AxB ,f.

which means that the vector product does not obey a commutative law. In forming t1: cross product, the ordering of the vectors, therefore, is an important consideratiOl

If A and B are parallel vectors, sin

e

is zero to make their cross product zen If A and B happen to be perpendicular vectors, then A X B is a vector having a lengt

AB and a direction perpendicular to both A and B, with the ambiguity in the directio resolved by means of the right-hand rule. These observations applied to the crm products of the orthogonal unit vectors of Figure 1-5, for example, lead to the sped" results: al X a l = az X az = a3 X a3 = 0; a l X az = _{a 3, az X a 3}

=

al, and a3

>

a l

=

az. However, note that ai X a3 az.

A distributive law can be shown to hold for the cross product

A X (B

+

C) = A X B

+ A

X C (1-40 Because of the noncommutativity of the cross product as expressed by (1-39), the orde of the factors in (1-40) is important.

If the vectors A and B are given in terms of their orthogonal components il the manner of (1-9), then their vector product is written

The use of the distributive law (1-40) and the special results obtained for the cros products of the orthogonal unit vectors provides the following expansion.

which can alternatively be put into the compact determinentaI form

a l a z a 3 A X B = Ai _{A z A3}

Bl B2 B3

Ig the Hion. zero. ~ngth 'ction cross )ecial a3 X 1-40) )rder Its in cross -41 )

1-7 SCALAR AND VECTOR PRODUCTS OF VECTORS

19

Pivot P

EXAMPLE 1-4

EXAMPLE 1·4. The definition of the cross product can be used to express the moment of a force F about a point P in space. Suppose R is a vector connecting the point P with the point of application Qofthe force vector F, as shown in the diagram. Then the vector moment M has the magnitude M = RF sin (} =

IR

X Fl. The turning direction of the moment, as well as its magnitude, are thus expressed by the vector product

M RxF (1-42)

EXAMPLE 1·5. A force F = !Oay N is applied at a point Q(O, 3, 2) in space. Find the moment ofF about the point P(2, 0, 0).

The vector distance R between P and Q)s

The vector moment at P is found by means of (l-42) and the determinant (1-41).

ax a y a z

M=RxF= -2 3 2 = -20ax-20azN-m

o

10 0

M, shown at P in the sketch, is a vector perpendicular to the plane formed by F and R.

EXAMPLE 1·6. Given the two vectors F and G in (b) of the figure in Example 1-3, determine their vector cross product F x G, as well as the magnitude of the latter. Find the unit veetor an in the direction of the vector F X G. Verify that an is perpendicular to F and to G. 1 1 (z) 21

d-,.

/1

---_9(0,3,2) I I F = lOay I I 1 / I R / I P(2 0 0) I ---0 --_ 1 , ,

-

"-_(-)---M

--3'--_

x - -_(y) EXAMPLE 1-5

20

VECTOR ANALYSIS ANI) ELECTROMAGNETIC FIELDS IN FREE SPACE

From (I I) in circular cylindrical coordinates, F x G becomes ap a", az

FxG IO 8 -4

20 0 80

+

a",[ -4( -20) - 10(80)]

+

a_z[IO(O) 8( 20)] 160a_z

The F G is IF X GI

=

[6402

₊

₇₂₀2

₊

₁₆₀2_{] 1/2}

₌

_{976.5, while the uni} vector an in the directioll of the vector F x G is given by

FxG

a

=

n

'iF

x

Gi

0.655ap 0.737a",

+

0.1638az

The dot an' F [wcnmes, from (l-37b), the zero result 10 0.737(8)

+

0.1638( -4) = 0

verifying frorn til!' definition (I that an and F are perpendicular vectors. You ma) similarly show that an and G are perpendicular.

1·8 VECTOR INTEGRATION

Vector integration, f()f the purposes of field theory, encompasses integrals in space along lines, over surfaces, or throughout volume regions, as well as integrals in the time domain and the domain. The subject of the present discussion concerns only integrations in space.

Tne

vector notation embodies compactness as an important feature, so it is always worthwhile to examine the integrand ofa vector integral carefully. The integrand may be either a scalar or a vector Thus, the integrals

possess scalar hand, the

[ A' Bdt

~I' Line integral

J,

(C X D) • ds Surface integral

J:

F'· Gdll Volume integTal

amI produce scalar results on integration. On the other

G Line integral

Hx

Surfilce in tegral

J

X

K

Volume

and t1H'IT/ill'C vector results. In the last three examples,

acroullt the different directions assumed by the on the surhce ,,)', or in the volume V defined.

, unit may )ace the erns vays nay her es, he ~d. Patht dt (Scalar displacement) 1-8 VECTOR INTEGRATION

21

Typical di (Vector displacement) ' " P2 "" P2

~~-;;J

.l---R

Pl (b)

~:XAMPLE \-7. (a) Integration of the scalar dt over a path t. (b) Integration the vector dt over the path t.

EXAMPLE 1·7. The difterent results provided by scalar and vector integrands is exemplified by simple integrals of scalar and vector displacements dt or dt along some prescribed path in space. The integral

summed over the path t shown in (a) of the figure, provides its true scalar length d. On the other hand, the integral of the vector displacement dt on the same path

R=

r

dt

Jt

produces quite a diftercnt answer, a vector result R determined only by the endpoints Pl and P2 of that path rather than by the form of the path between the endpoints.

This vector R is illustrated in (b) of the accompanying figure. So the line integral of dt

about a closed path is zero, whereas if dt is the integrand, the perimeter of the closed pa th is the result.

An integral flllding extensive utility in work or energy calculations is the scalar line integral

L

F .

dt

==

L

F dt cos

e

(1-43) This integral sums the scalar product F . dt over the path

t,

as suggested by Figure 1-10. Only the projection ofF along

de

at each point on the path contributes to the integral result. The line integral (1-43) can be expressed in terms of the generalized orthogonal components of F and of

de

in the following way, making use of (l-9), (1-21), and

1-37 a)

(1-44)

In the rectangular coordinate system, in which hi = h2 = h3

=

1, (1-44) is written (1-45)

j' (

22

VECTOR ANALYSIS AND ELECTROMAGNETIC FIELDS IN FREE SPACE

-

--

11----""'-F

(a) (b) (c)

FIGURE J-lO. A palh and the field F in space. (0) Division of t into vector elements dt. (c) product F· dt (to be summed over the path) shown at the typical point P on the path.

assuming (Xi,_Vl'

.::tl

and of the path

t.

are the coordinates of the endpoints P1 and P

EXAMPLE 1·8. Evaluate the line integral (1-43) between the points PI(O, 0,1) and P2(2, 4,1

ovcr a path t defined the intersection of the two surfaces y = x2 and z = 1, if F is thl v(,ctor fidd

The path

t

is illustrated ill the Inserting = lOx,

dz

=

0 from the definitiD!!

.')x2y, and f~

it {()llows that

(1

into (1-45) and since x2 = y all(

£

F . dt

fx2~O

lOx dx -

f

_y

4=O

5y2 dy

+

0

20 106.7 = 86.7

the desired resnlt.

(2,0,

4, I) is the

(I)

, and

1-9 ELECTRIC CHARGES, CURRENTS, AND THEIR DENSITIES

23

This answer can also be obtained by expressing the dificrential displacement dx

along the path in terms of From the definition of l, dy = 2x dx and dz O. Thus

r

F . dt: =

r

2 lOx dx

Jt

Jo

j

'4

o 5y2 dy

4y

= -36.7

IXAMPlE 1·9, A line integral such as (1-4·3) in gcncral has a value depending on the shape of the path connecting the endpoints PI and P_{2 .} Evaluate the integral of Example 1-3 for the same function F and the same endpoints PI(O, 0,1) and P2(2, 4,1), but deform

t: into the straight-line path given by the intersection of the surElCes y = 2x and z = I.

Integral (1-43) now becomes

dy

+

0 60

obviously dilll'rent from the result obtained over the parabolic path in the last example.

F is f()r tbis reason called a nonconservalive field. A vector field fell' which the line integral (1-43) is independent of the shape of the path connecting a fixed pair of emlpoints is said to be conservative. More is said later of such fields in connection with static electric charge distributions in Chapter 4.

1·9 ELECTRIC CHARGES, CURRENTS, AND THEIR DENSITIES

The physical and the chemical properties of matter are known to be governed by the eitcctric and magnetic forces that act among the particles comprising all material

sub-!ltalH~es, whether inorganic or living cells. The fundamental electric panicles of matter

of two varieties, commonly called positive and negative electric charges. Many experiments have provided the following conclusions concerning electric charges.

1. The algebraic sum oCthe positive and negative electric charges in a closed system never changes; that is, the total electric charge of a defined aggregate of matter is consewed.

2. Electric charge exists only in positive or negative integral multiples of the mag-nitude of the elect mnic charge, e = 1.60 X 10 - 19 C; this implies that electric charge is quantized.

From the viewpoint of classical electromagnetic theory, an electric charge aggre-gate will be treated as though it were capable of being indefinitely divisible, such that a volume electric-charge density, denoted by the symbol Pv is defined as follows3

Aq , 3

P = - -

elm

v Ali ( 1-46a)

This limit of this ratio is taken such that the volume-element in space does not be-come so small that it contains so few charged particles that the relatively smooth property of the density quantity p" is lost, although Ali is kept small enough thal thl' integration

or

the quantities containing Av becomes a meaningful process.

I-II (a) illustrates the meaning of these quantities relative to a volume eiemellt

3It is dear thaI Ihe symbol p, for volume ('haq;;" density should not be confused with the lIn.l11/" , the radial variahle of the circular cylindrical coordinales (p, 4>, 'c).

:!i

I

24

VECTOR

(a)

FIGURE I-IL

point in a

ANn ELECTROMAGNETIC FIELDS IN FREE SPACE

(b)

de

ex.·

..

'~ dq = p{ dt on dt (c)

used in ddining volume, surface, and line charge densities in space. Qualltit;t·, defining Ps' (el Quantities defining pt.

Aq

residing within any element Av may vary from pOil function of space as

Pv(ur, U2' U 3 , t) or p,,(r,

it is evident from (1-46a) that charge density possibly of time. Thus Pv is a .field, written in ger In some physical the charge

I1q

is identified with an element of su

or line instead of a volume. The limiting ratio (1-46a) should then be defined as foil

Aq

2 pS=AC/m L.l.S ( 1-'

Aq

Pc

=

111' C/m

(1-The quantltles associated with these definitions of volume, surface, and line ch. densities are illustrated in 1-11.

In some systems densities may be

aggregates, two species of positive and negative ch: simultaneously. A net charge density p" (volume, sud such an instance defined

or line density) is

p"

p,;

+

pv

C/m3 ( 1

in which

P:

and denote limiting ratios defined due to the positive negative charges + ami

Aq

respectively in Av. occurrence of both pos: metallic ions and mobile electrons in a conductor is an example to which (1-47) rna

applied. The ill this being of eqnal magnitudes but opposite s

41n some physical ent simultaneously characterized by

if a total of

discharge, electrons and several kinds of ions maybe Their net density at any point in the region may th,

(I

point to lsity is a general )f surface follows. (1-46b) (1-46e) ~ charge e charge surface, ( 1-47) rive and positive maybe te signs ly be pres-y then be (1-47a)

1-9 ELECTRIC CHARGES, CURRENTS, AND THEIR DENSITIES

25

P;;

= -

p;;), cancel, providing the net density

Pv

0 in such a compensated charge system.

. The total amount of charge contained by a volume, surface, or line region is

obtained from the integral of the appropriate density function (1-46a), (1-46b), or 1-46c). Thus in some volume region, each element dv contains the charge dq

=

Pvdv,

making the total charge in 1) the integral

q =

Iv

dq =

Iv

Pv du C

Similar integral expressions may be constructed to yield the total charge on a given surface or a line in space.

EXAMPLE 1·10. (a) The radially dependent volume charge density Pv = 50r2 C/m3 _exists

within a sphere of radius r 5 cnL Find tlfe total charge if contained by that sphere.

(b) The same sphere of is now covered with the angularly dependent surface charge density Ps 2 x 1O~ 3 0 C/m2 Find the total charge on the spherical surface.

(a) Making usc of( 1-47) and dv of (1-17) obtains

q

=

Sv

p" dv

SSS

(50r2)r2 sin 0 dr dO dcjJ

= 50

s:n

d(p

S:

sin 0 dO

S:·os

r4 dr

,5 JO.os

= 50(2n)2 -- = 3.927 x 10 -5 = 39.27 j1C.

5

°

Attention is called to the "product separability" of the integrand in this example, enabling the expression of the triple integrand as the product of three separate integrals in r, 0, and cjJ.

(b) Using q

=

Is

Ps lis in this case, along with the scalar surhrce clement ds

=

r2 sin 0 dO dcjJ

on this sphere

or

radius r = OJ)5 m, as suggested by ds shown in Figure 1-7(b),

yields on the complete sphere

If

=

f

p"d.1

=

ff(2

X 10-3

cos2

0)r2 sin OdO r/cjJ]

~

S r-O.OS

2 x 1O~ 3(0.05)2

r.

2n dcjJ

In

(OS2 0 sin 0 dO = 5 x

10-. 0

Jo

[

_

cos~Jn

3 0

= 20.9 /lC

A vector field F(Ul' U2, U,' t) at some given instant t, can be represented graphi-by use of a myriad of vectors of appropriate lengths and directions at many

in a region of space. A vector field plotted in this way is shown in Figure 1-12 (a).

is, however, a cumbersome way to graph a vector field; usually a much more representation is by use of a/lux plot, a method replacing the vectors with

lines (called jlux lines) drawn in accordance with the i()llowing rules.

1. The directions of the flux lines agree with the directions of the field vectors. The transverse densities of the flux lines are the same as the magnitudes of the fidd vectors.

The flux plot of the vector field of Figure 1-12 (a), sketched in accordance with these is noted in (b) of that figure. If a surhlce S is, moreover, drawn in the region

26

VECTOR ELECTROMAGNETIC FIELDS IN FREE SPACE

(a) (b) (c)

FIGURE 1-12. A veCWr field F, its flux and the flux through typical surfaces (a) A vector field F, denoted by "farrows. The flux map of the vector field F, showing an open surface S through a net flux passes. (e) A closed surface S, showing zero net flux emergent from it.

of space embracing that flux, then the net lines of flux

r/J

passing through S can be a measure of some physical quantity (such as charge,current, or power flow), depending on the physical meaning ofF. The differential amount of flux dr/J passing through any surface-element ds in space is defined by the scalar dr/J = F ds cos

e

= F • ds, a positive or negative l-esuit, depending on the angle between F and ds. The net (positive or negative) flux of F through S is therefore the integral of dr/J over S

Is

F' ds (1-48)

in which ds is taken to emerge from that side of S assumed positive, as shown in Figure 1-12 (b). If S is a dosed surface, the net flux through it is given by

(1-49) as noted in Figure \-12 The la Her will integrate to zero (an indication that just as many flux Jines leave S' as enter it) unless the interior volume of S contains sources or sinks offlux lines. This view will be amplified later in the discussion of the divergence of a vector field.

The current flow through a surb.ce embodies a good illustration of the flux COIl-cept. Supr:iose that there are electric charges of density Pv(Ul, U2 , U3, t) in a region, and imagine that the cllarges have velocities averaging to the function v(ul' U2, U3, t)

within the elements dv with which the densities Pv are identified. A current density func-tion

J

may then be defined at any point P in the region by

or C/sec/m2 ( 1-50a)

This function is a measure, in the vicinity of any point P in space, of the instantaneous rate of flow of charge per unit cross-sectional area. If two species of charge density

be a :ling any ltive e or 48) -ure 49) ust ces Ice )0-)n, t) IC-a) us ty

l-9 ELECTRIC CHARGES, CURRENTS, AND THEIR DENSITIES

27

of opposite kinds, designated by P;; and Pv , exist simultaneously in a region of space, then their total current density

J

at each point is written

( 1-50b) In general, for n species with densities Pi and velocities Vi (e.g., electrons plus a mixture of ions)

( 1-50c) The differential current flux di flowing through a surface element ds at which the current density

J

exists, is di

J .

ds amperes, to make the net current i (current

through S

i

=

S~

J .

ds C/sec or A (I-51 )

IXAMPLE 1·11. An electron bcam of circular cross-section 1 mm in diameter in a cathode ray tube (CRT) has a measured current of I itA, and a known average electron speed of 106 _{m/sec. Calculate the average current density, charge density, and rate of mass transport}

in the beam.

Assuming a constant current density

J

=

azJz

in the cross-section (I-51), yields the following current through any cross-section.

in which A denotes the cross-sectional area of the beam. Thus the average current density is i 10-6 ₄ 2

Jz=

A = n(1O-3)2 = n Aim ----~ 4

The charge density in the beam, from (1-50a) in which

J

az4/n imd v - - azI06,

becomes

Jz

ds = azds 1 mm i = l/1A Cross-section A ---l>-'(z) I':XAMPLE I-II

I

il

28

VECTOR ANALYSIS AND ELECTROMAGNETIC FIELDS IN FREE SPACE

The rate of mass transport in the heam is the current times the electronic mass-to-charge ratio; this yields 5.7 x 10-18 _{kg/sec, assuming an electron mass of9.! x 10}

31 kg.

1·10 ELECTRIC AND MAGNETIC FIELDS IN TERMS OF THEIR FORCES

Electric and magnetic fields are fundamentally fields of force that ongmate from electric charges. Whether a force field may be termed electric, magnetic, or electromagnetic hinges on the motional state of the electric charges relative to the point at which the field observations arc. made. Electric charges at rest relative to an observation point give rise to an electrostatic (time-independent) field there. The relative motion of the charges provides an additional force field called magnetic. That added field is

magneto-static if the charges are moving at constant velocities relative to the observation point. Accelerated motiolls, on the other hand, produce both time-varying electric and magnetic fields termed electromagnetic fields.

The connection of the electric and magnetic fields to their charge and current sources is provided by an elegant set of relations known as Maxwell's equations, attributed historically to the work of many scientists and mathematicians well before Maxwell's time,5 but to which he made significant contributiohs. They are introduced in the next section. Suppose that electric and magnetic fields have been established in some region of space. The symbol for the electric field intensity (or just electric intensity) is the vector E; its units are force per unit charge (newtons per coulomb). The magnetic field is represented by means of the vector B called magnetic flux density; it has the unit weber per square meter. If the fields E and B exist at a point P in space, their presence may be detected physically by means of a charge q placed at that point. The force F acting on that charge is given by the Lorentz force law

in which

F = q(E

+

v

x

B)

=FE+FBN

q is the charge (coulomb) at the point P

v is the velocity (meter per second) of the charge q

E is the electric intensity (newton per coulomb) at P

B is thc magnetic flux density (weber per square meter or tesla) at P

FE

=

qE, the electric field force acting on q F B = qv

x

B, the magnetic field force acting on q

(1-52a) (1-52b)

In Figure 1-13, these quantities are illustrated typically in space. The force

FE

has the same direction as the applied field E, whereas the magnetic field force F B is at right angles to both the applied field B and the velocity v of the charged particle.

The Lorentz force expression (1-52) may be used for discussing the ballistics of charged particles traveling in a region of space on which the electric and magnetic fields E and B are imposed. The deflection or the focusing of an electron beam in a cathode ray tube are common examples.

I-II MAXWELL'S INTEGRAL RELATIONS FOR FREE SPACE

29

B flux

----

" . , -fa) (b) I I FE

,I

I I I

,~

_{: B} I I I

t

FB (c)

:FIGURE 1-13. Lorentz forces acting on a moving charge q in the presence of (a) only an E field, (b) only a B Geld, and (e) both electromagnetic Gelds,

EXAMPLE 1·12. An electron at a given instant has the velocity v (3) I05

ay +

105az m/see at some position in empty space. At that point, the electric and magnetic fields are known to be E

=

400a_z V/m and B

=

O.005ay WbJm

2

, Find the total force acting on the electron.

The total force is found from the Lorentz reaction (1-52a)

F = q[E

+

v X B) = _{-1.6(10-19)[az400}

+

(ay3' 105

+

az4' 105) X ayO.005)

= (a_x32 - a_z6.4)IO-17 N

Although this is quite a small force, the very small mass of the electron charge provides a tremendous acceleration to the partiele, namely

a

F

1m

= (a_x3.51 - azO. 7) 1014 m/sec2_.

1·11 MAXWELL'S INTEGRAL RELATIONS FOR FREE SPACE

The relationships among the electric and magnetic force fields and their associated charge and current distributions in space are provided by Maxwell's equations, postu-lated here in integral form for the fields E and B in free space.

~s

(EoE) . ds

Iv

pvdv _C

J.

B· ds

= 0 Wb

:Vs

J,

E. dt

=

-~

r

B . ds V

'Ji

dt

Js

( 1-53) (1-54) ( 1-55) ( 1-56)

30

VECTOR ANALYSIS AND ELECTROMAGNETIC FIELDS IN FREE SPACE

in which

E

=

E(Ul' U 2 , U3 , t) is the electric intensity field B B(ul' U2, U3, t) is the magnetic flux density field

Iv

p" do

=

q(t) is the net charge inside any dosed surface S

itt) is the net current flowing through any open surface S bounded by the closed line t

Eo is the permittivity offi-ee space (~10 9/36n F/m)

110 is the permeability of free space (= 4n x 10 7 Him) The Maxwell equatious6

(I-53) through (1-56) must be simultaneously sati~fied

by the field solutions E and B for all possible closed paths t and surfaces S in the

region of space occupied by these fields. This strict requirement might appear to limit severely the number of practical problems that can be solved by means of these in-tegrals. Indeed, their application to the discovery offield solutions E(u l , U2 , U 3 , t) and B(ul' 11 2 , 113 , t) is restricted, in the present treatment, to problems in which the charge or current distributions have particular symmetries that serve t~ simplify the solutions. The equivalent differential forms of Maxwell's equations, developed in the next chapter, have a somewhat wider range of application in problem solving at the introductory level.

The reader is to be assured that only a low-level introduction to methods for obtaining electric and magnetic field solutions of Maxwell's integral relations (1-53)

through (I-56) is attempted here. For the purposes of this introductory treatment, the Maxwell relations are simplified by considering only the field solutions of a few simple, symmetrical geometries of static charge or current distributions. In Examples 1-13 through 1-17 that follow, these simplifications are shown to enable, in one or two steps, solving for the electric or magnetic field of a given charge or current dis-tribution. The symmetry of the distribution will be seen to be the key to providing quick solutions for the desired field. Symmetries about a point, a line, or a plane are considered.

A. Gauss's Law for Electric Fields in Free Space Maxwell's integral law (1-53)

[I-53

J

is also known as Gauss's law for electric fields in free space. The meanings of the quantities are illustrated in Figure 1-14. Thus, suppose that there is in free space an electric field E(Ub U2, U3, t) (denoted by the E-field flux line distribution in that figure), plus some related electric charge distribution of density Pv(Ul> U2, U3, t) as shown. Con-struct in this region a closed surface S, with S having any desired shape and enclosing

6 Although given the collective name Maxwell's equations, historically they were in a gradual process of evolution over many years before Maxwell's time. For an enjoyable and first-rate account of the details, you are encouraged to read the historical surveys at the beginning of each chapter in R. S. Elliott, Electromagnetics. New York: McGraw-Hill. 1966.