field theory Vector Algebra

Vector Calculus ENEL2FT Field Theory 1

ENEL2FTFIELD THEORY

REFERENCES

1. M.N. Sadiku: Elements of Electromagnetics, Oxford University Press, 1995, ISBN 0-19-510368-8.

2. N.N. Rao: Elements of Engineering Electromagnetics, Prectice-Hall, 1991, ISBN:0-13-251604-7.

3. P. Lorrain, D. Corson: Electromagnetic Fields and Waves, W.H. Freeman & Co, 1970, ISBN: 0-7167-0330-0.

4. David T. Thomas: Engineering Electromagnetics, Pergamon Press, ISBN: 08-016778-0.


VECTOR CALCULUS VECTOR ALGEBRA Electromagnetics deals with the study of electric and magnetic

fields. Therefore one needs to understand the concepts of a field. Electric and Magnetic Fields are vector quantities and their

behaviour is governed by a set of laws known as Maxwell’s equations.

The mathematical formulation of Maxwell’s equations and their subsequent application require the understanding of the basic rules pertinent to mathematical manipulations involving vector quantities.

We first introduce simple rules of vector algebra without the manipulation of the coordinate system; thereafter, we introduce the Cartesian, cylindrical, and spherical coordinate systems.

After introducing the vector algebraic rules, we introduce the concepts of scalar and vector fields, static as well as time-varying.


VECTOR CALCULUS VECTOR ALGEBRA In the study of elementary physics, we come across several

quantities such as mass, temperature, velocity, acceleration, force, and charge.

Some of these quantities have associated with them not only a magnitude, but also a direction in space whereas others are characterized by magnitude only.

The former class of quantities are known as vectors, and the latter class are known as scalars.

Mass, temperature, and charge are scalars, whereas velocity, acceleration, and force are vectors.

Other examples are voltage and current for scalars , and electric and magnetic fields for vectors.


VECTOR CALCULUS VECTOR ALGEBRA Graphically, a vector, A, is represented by a straight line with an arrowhead

pointing in the direction of A and having a length proportional to the magnitude of A.

If the top of the page represents North, then vectors A and B are directed eastward, with the magnitude of B being twice that of A.

Vector C is directed towards the northeast and has a magnitude three times that of A. Vector D is directed towards the southwest, and has a magnitude equal to that of C. Thus C and D are equal in magnitude but opposite in phase.

A

B

CD


VECTOR CALCULUS VECTOR ALGEBRA - THE UNIT VECTOR Since a vector may have, in general, an arbitrary orientation in three-

dimensional space, we need to define a set of three reference directions at each and every point in space in terms of which we can describe vectors drawn at that point.

Thus if we define three mutually orthogonal reference directions as shown below, and direct unit vectors along the three directions as shown, where a unit vector has magnitude of unity.

i1

i2

i3

Set of three orthogonal unit vectors in a right-handed coordinate system.


VECTOR CALCULUS VECTOR ALGEBRA A vector of magnitude different from unity along any reference

directions can be represented in terms of the unit vector along that direction.

Thus 4i1 represents a vector of magnitude 4 units in the direction of i1, 6i2 represents a vector of magnitude 6 units in the direction of i2, and -2i3 represents a vector of magnitude 2 units in the direction opposite to that of i3.

Thus we define vector A as the sum of 4i1+6i2. That is:

The magnitude of vector A is given by:

21ˆ6ˆ4 iiA

211.764 22 A


VECTOR CALCULUS VECTOR ALGEBRA If vector B is defined as:

then the magnitude of B is:

In general, a vector A is said to have components A1, A2, and A3 along the directions 1, 2, and 3 is written as:

Now consider three vectors A,B, and C given by:

321 2ˆ6ˆ4 iiiB

4833.72642ˆ6ˆ4 222321 iiiB

332211ˆˆ iAiAiAA

332211

332211

332211

ˆˆ

ˆˆ

ˆˆ

iCiCiCC

iBiBiBB

iAiAiAA


VECTOR CALCULUS VECTOR ALGEBRA - Vector Addition, Subtraction, Multiplication Then the sum of vectors A and B, (A+B), is given by:

Vector subtraction is a special case of vector addition; thus:

The multiplication of a vector, A, by a scalar m, is the same as repeated addition of the vector:

33322111

332211332211

ˆˆˆ

ˆˆˆˆ

2 iBAiBAiBA

iBiBiBiAiAiABA

33322111

332211332211

ˆˆˆ

ˆˆˆˆ

2 iCBiCBiCB

iCiCiCiBiBiBCB

332211

332211

ˆˆ

ˆˆ

imAimAimA

iAiAiAmAm


VECTOR CALCULUS VECTOR ALGEBRA The magnitude of vector A is given by:

The unit vector along A , iA, has a magnitude equal to unity, but its direction is the same as that of A. Thus:

Two vectors A and B are equal if and only if the corresponding components of A and B are equal. That is:

23

22

21332211

ˆˆ AAAiAiAiAA

33

22

11 i

A

Ai

A

Ai

A

A

A

AiA

33;22;11

;332211332211ˆˆˆˆ

BABABA

iBiBiBiAiAiABA


VECTOR CALCULUS VECTOR ALGEBRA - SCALAR OR DOT PRODUCT The scalar or dot product of two vectors A and B is a scalar quantity defined

as:

Here is the angle between A and B. For mutually orthogonal unit vectors i1, i2, and i3, we have:

Thus we have the dot product between A and B as:

BA

BABABA

.coscos. 1

1.;0.;0.

0.;1.;0.

0.;0.;1.

33233

32222

32

1

1

1111

iiiiii

iiiiii

iiiiii

33211

33.3322.3311.33

33.2222.2211.2233.1122.1111.11

332211332211

2

ˆˆˆˆˆˆ

ˆˆˆˆˆˆˆˆˆˆˆˆ

ˆˆ.ˆˆ.

BABABA

iBiAiBiAiBiA

iBiAiBiAiBiAiBiAiBiAiBiA

iBiBiBiAiAiABA


VECTOR CALCULUS VECTOR ALGEBRA - VECTOR OR CROSS PRODUCT The vector or cross product of two vectors, A and B, is a vector quantity

whose magnitude is equal to the product of the magnitudes of A and B and the sine of the smaller angle between A and B whose direction is normal to the plane containing A and B.

For mutually orthogonal unit vectors i1, i2, and i3, we have:

Note that the cross-product is not commutative, and also the distributive property holds for the cross product:

NiBABxA ˆsin

0;;

;0;

;;0

3312323

1322232

2332

1

1

1111

xiiixiiixii

ixiixiiixii

ixiiixiixii

CxABxACBxA

BxAiABAxB N

sin


VECTOR CALCULUS VECTOR ALGEBRA - VECTOR OR CROSS PRODUCT Using the above properties, we obtain:

This can be expressed in determinant form in the manner:

The cross product is useful in obtaining the unit vector normal to the plane containing the two vectors A and B:

122131132332

123213132312231321

333322331133

332222221122331122111111

332211332211

ˆˆˆˆˆˆ

ˆˆˆˆˆˆ

ˆˆˆˆˆˆˆˆˆˆˆˆ

ˆˆˆˆ

BABABABABABA

iBAiBAiBAiBAiBAiBA

iBiAiBiAiBiA

iBiAiBiAiBiAiBiAiBiAiBiA

iBiBiBxiAiAiABxA

xxx

xxxxxx

321

321

321ˆˆˆ

BBB

AAA

iii

BxA

BxA

BxAiN


VECTOR CALCULUS VECTOR ALGEBRA - TRIPLE PRODUCTS The scalar triple product involves three vectors in a dot product operation

and a cross product operation, such as, A.BxC. It is not necessary to include parentheses since this quantity can be

evaluated in only one manner - by evaluating BxC first, and then dotting the resulting vector with A.

We therefore have,

Since the value of the determinant on the right side remains unchanged if the rows are interchanged in a cylindrical manner, we have

321

321

321

321

321

321

332211

.

ˆˆˆ

.ˆˆˆ.

CCC

BBB

AAA

CxBA

CCC

BBB

iii

iAiAiACxBA

BxACAxCBCxBA

...


VECTOR CALCULUS VECTOR ALGEBRA - TRIPLE PRODUCTS The triple cross product involves three vectors in two cross product

operations. Caution must however be exercised in evaluating a triple cross

product since the order of evaluation is important; that is:

As an example, let us equate the three vectors to unit vectors as follows:

Therefore in a vector triple product, the parentheses are so important and must be included.

CxBxACxBxA

0ˆ0ˆˆˆ

ˆˆˆˆˆˆ

ˆ;ˆ;ˆ

2211

231211

211

ixixixiCxBxA

iixiixixiCxBxA

iCiBiA


VECTOR CALCULUS CARTESIAN COORDINATE SYSTEM So far, we have expressed a vector at a point in space in terms of its

component vectors along a set of three mutually orthogonal directions defined by three mutually orthogonal unit vectors at that point.

However, in order to relate vectors at one point in space to vectors at another point in space, we must define the set of three reference directions at each and every point in space. Thus we need a coordinate system.

Although there are several different coordinate systems, we are normally concerned with only three of these, namely, the Cartesian, cylindrical, and spherical coordinate systems.

The Cartesian coordinate system, also known as the rectangular coordinate system, is the simplest of the three since it permits the geometry to be simple, yet sufficient to study many of the elements of engineering electromagnetics.


VECTOR CALCULUS CARTESIAN COORDINATE SYSTEM The Cartesian coordinate system is defined by a set of three

mutually orthogonal vectors, x,y, and z, as shown below.

The coordinate axes are denoted as the x-, y-, and z-axes. The directions in which values of x, y, and z increase along the

respective coordinate axes are indicated by the arrowheads. Note that the positive x-, y-, and z-directions are chosen such that

they form a right-handed system.

i1=x

i2=y

i3=z


VECTOR CALCULUS CARTESIAN COORDINATE SYSTEM Therefore we have:

For a right-handed coordinate system, we have:

Consider two points, P1(x1,y1,z1), and P2 (x2,y2,z2) in the rectangular coordinate system. The position vector, r1, drawn from the origin to pint P1 and position vector r2 drawn from the origin to P2 are given by:

The resultant vector, R12, is given by:

ziyixi ˆˆ;ˆˆ;ˆˆ321

yxxzxzxyzyxx ˆˆˆ;ˆˆˆ;ˆˆˆ

zzyyxxr

zzyyxxr

ˆˆˆ

ˆˆˆ

222

111

2

1

zzzyyyxxxrrR ˆˆˆ 1212121212


VECTOR CALCULUS CARTESIAN COORDINATE SYSTEM

We can obtain the unit vector along the line drawn from P1 to P2 to be:

As an example, if P1 is (1,-2,0) and P2 is (4,2,5), then:

x

y

z

P1(x1,y1,z1)

P2(x2,y2,z2)

r1 r2

R12

2/12

122

122

12

121212

12

1212

ˆˆˆˆzzyyxx

zzzyyyxxx

R

Ri


VECTOR CALCULUS CARTESIAN COORDINATE SYSTEM

In our study of electromagnetic fields, we have to work with line, surface, and volume integrals.

These involve differential lengths, surfaces, and volumes obtained by incrementing the coordinates by infinitesimal amounts.

Since in the Cartesian coordinate system the three coordinates represent lengths, the differential length elements obtained by incrementing one coordinate at a time, keeping the other two constant, are, for the x-, y-, and z-coordinates respectively:

zyxi

zyxR

ˆ5ˆ4ˆ325

1ˆ

ˆ5ˆ4ˆ3

12

12

zdzandydyxdx ˆ,ˆ,ˆ


VECTOR CALCULUS DIFFERENTIAL LENGTH VECTOR

The differential length vector, dl, is the vector drawn from a point P(x,y,z) to a neighboring point Q(x+dx,y+dy,z+dz) obtained by incrementing the coordinates of P by infinitesimal amounts.

Thus it is the vector sum of the three differential elements as follows:

x

y

z

P(x,y,z)

Q(x+dx,y+dy,z+dz)

r1 r2

dl

zdzydyxdxld ˆˆˆ


VECTOR CALCULUS DIFFERENTIAL LENGTH VECTOR The differential lengths, dx, dy, and dz are, however, not independent

of each other since in the evaluation of line integrals, the integration is performed along a specified path on which the points P and Q lie.

As an example, consider the curve

Let us obtain the expression for the differential length vector dl along the curve at the point (1,1,1) and having the projection dz on the z axis. Then:

Note that x=1, y=1, z=1.

2zyx

dzzyxzdzydzxdzld

zdzydyxdxld

zdzdydx

ˆˆ2ˆ2ˆˆ2ˆ2

ˆˆˆ

2


VECTOR CALCULUS DIFFERENTIAL LENGTH VECTOR Differential length vectors are useful for finding the unit vector normal

to a surface at a point on that surface. This is done by considering two differential length vectors at the point

under consideration and tangential to the two curves on the surface then using the cross-product operation, which gives a vector that is normal to the crossed vectors.

Thus the unit vector normal to the surface is given by:

dl1

dl2

Surface

Curve 1

Curve 2

21

21ˆlxdld

lxdldin


VECTOR CALCULUS DIFFERENTIAL SURFACE VECTOR Two differential length vectors, dl1 and dl2 originating at a point define

a differential surface whose area dS is that of the parallelogram having dl1 and dl2 as two of its adjacent sides, as shown below:

From simple geometry and the definition of cross-product of two vectors, it can be seen that:

dS

dl1

dl2

in

2121 sin lxdlddldldS


VECTOR CALCULUS DIFFERENTIAL SURFACE VECTOR In the evaluation of surface integrals, it is convenient to define a

differential surface vector dS whose magnitude is the area dS and whose direction is normal to the differential surface.

Thus recognizing that the normal vector can be directed to either side of the surface, we have:

If we apply these equations to differential surface vectors in Cartesian coordinates, we obtain:

nidSlxdldSd ˆ21

zdxdyyxdyxdxtconszplaneFor

ydzdxxxdxzdztconsyplaneFor

xdydzzxdzydytconsxplaneFor

ˆˆˆ:tan

ˆˆˆ:tan

ˆˆˆ:tan


VECTOR CALCULUS DIFFERENTIAL VOLUME Three different length vectors, dl1, dl2, and dl3 originating at a point

define a differential volume dv which is that of the parallelepiped having dl1, dl2, and dl3 as three of its contiguous edges, as shown below.

It can be seen that:

dl1

dl2

dl3

dv

321

213

21

213

21321

.

..

ˆ.

)).((

lxdldlddv

lxdldldlxdld

lxdldldlxdldildlxdld

ipedparallelepofheightipedparallelepofareabasedv

n


VECTOR CALCULUS CYLINDRICAL COORDINATE SYSTEM Just like the Cartesian coordinate system is defined by a set of three

mutually orthogonal surfaces, the cylindrical coordinate system also involves a set of three mutually orthogonal surfaces.

For the cylindrical coordinate system, the three one of the planes is z=constant

x

y

z

P(r,,z)

r


VECTOR CALCULUS CYLINDRICAL COORDINATE SYSTEM One of these planes is the same as the z=constant plane in the

Cartesian coordinate system. The second plane contains the z-axis and makes an angle with a

reference plane, chosen to be the x-z plane of the Cartesian coordinate system. This plane is called the =constant plane.

The cylindrical coordinate system has the z-axis as its axis. But since the radial distance r from the z-axis to points on the cylindrical surface is constant, this surface is defined by r=constant.

Thus the three orthogonal surfaces defining the cylindrical coordinate system are: r=constant; =constant; and z=constant.

Only two of the coordinates (r and z) are distances; the third coordinate () is an angle.

We note that the entire space is spanned by varying r from 0 to ; z from - to +; and from 0 to 2.


VECTOR CALCULUS CYLINDRICAL COORDINATE SYSTEM

To obtain the expressions for the differential lengths, surfaces, and volumes in the cylindrical coordinate system, we now consider two points, P(r,,z) and Q(r+dr, +d, and z+dz) where Q is obtained by incrementing infinitesimally each coordinate from its value at P.

The three orthogonal surfaces intersecting at P, and the three orthogonal surfaces intersecting at Q, define a box which can be considered to be rectangular since dr,d, and dz are infinitesimal.

dz

rdr

dx

y

z

P

Q


VECTOR CALCULUS CYLINDRICAL COORDINATE SYSTEM The three differential length elements forming the contiguous sides of

the box are: The differential length vector dl from P to Q is thus given by:

The differential surface vectors defined by the pairs of the differential length elements are:

Finally, the differential volume dv is the volume of the box:

zdzrdrdr ˆ,ˆ,ˆ

zdzrdrdrld ˆˆˆ

zrdrdxrdrdr

drdzrxdrzdz

rdzrdzxdzrd

ˆˆˆ

;ˆˆˆ

;ˆˆˆ

dzrdrddzrddrdv


VECTOR CALCULUS SPHERICAL COORDINATE SYSTEM For the spherical coordinate system, the three mutually orthogonal

surfaces are a sphere, a cone, and a plane.

The three orthogonal surfaces defining the spherical coordinates of a point are:

r

x

y

z

tcons

tcons

tconsr

tan

tan

tan


VECTOR CALCULUS SPHERICAL COORDINATE SYSTEM The differential length elements and the differential length vector dl are

given by:

The differential surface vectors defined by pairs of differential length elements are:

The differential volume formed by the three differential lengths is:

ˆsinˆˆ

ˆsin,ˆ,ˆ

drrdrdrld

drrdrdr

ˆˆˆ

ˆsinˆˆsin

ˆsinˆsinˆ 2

rdrdxrdrdr

drdrrxdrdr

rddrdxrrd

ddrdrdrrddrdv sinsin 2


VECTOR CALCULUS CONVERSIONS BETWEEN THE COORDINATE SYSTEMS In the study of electromagnetics, it is useful to be able to convert from

one coordinate system to another, particularly from the Cartesian to the cylindrical system and vice-versa, and from the spherical system to the Cartesian system and vice-versa.

If rc is r in cylindrical coordinate, and rs is the designation of r in spherical coordinates, then we have the following conversions:

x

y

z

yxzyxr

zzx

yyxr

rzryrx

zzryrx

s

c

sss

cc

122

1222

122

tantan

tan

cossinsin;cossin

;sin;cos


VECTOR CALCULUS CONVERSIONS BETWEEN THE COORDINATE SYSTEMS Next consider the conversion of vectors from one coordinate system to

another. To do this, we need to express each of the unit vectors of the first

coordinate system in terms of its components along the unit vectors in the second coordinate system.

From the definition of the dot product of two vectors, the component of a unit vector along another unit vector, that is, the cosine of the angle between the two unit vectors, is simply the dot product of the two unit vectors.

For the sets of unit vectors in the cylindrical and Cartesian coordinate systems, we have:

1ˆ.ˆ0ˆ.ˆ0ˆ.ˆ

0ˆ.ˆcosˆ.ˆsinˆ.ˆ0ˆ.ˆsinˆ.ˆcosˆ.ˆ

zzyzxz

zyx

zryrxr ccc


VECTOR CALCULUS CONVERSIONS BETWEEN THE COORDINATE SYSTEMS Similarly, for the set of unit vectors in the spherical and Cartesian

coordinate systems, we obtain the dot products as follows:

Therefore when given a vector in spherical or cylindrical coordinates, it is possible to convert it into Cartesian coordinates, and vice-versa.

This is particularly so when solving electromagnetic radiation problems.

0ˆ.ˆcosˆ.ˆsinˆ.ˆsinˆ.ˆsincosˆ.ˆcoscosˆ.ˆcosˆ.ˆsinsinˆ.ˆcossinˆ.ˆ

zyx

zyx

zryrxr sss


VECTOR CALCULUS VECTOR DERIVATIVES AND INTEGRALS For a scalar function, F(t), we have:

Now suppose that F(t) were one component of a vector function, say Ax. Since each component would be a new scalar function, it follows that:

Suppose, instead, we asked for the partial derivative of vector A with respect to x? This asks for the change in A as we move along the x direction. This becomes:

t

tFttF

dt

dFt

)()(lim0

dt

dAz

dt

dAy

dt

dAx

dt

Ad zyx ˆˆˆ

x

Az

x

Ay

x

Ax

x

A zyx

ˆˆˆ


VECTOR CALCULUS VECTOR DERIVATIVES AND INTEGRALS The definition of a partial derivative is identical to the definition of an

ordinary derivative:

The only difference is that the function F(x,y) has now two independent variables, x and y.

Many such functions of two or more independent variables exist. For example, the height of a point above sea level depends on the position on the earth and requires two variables, latitudes (x) and longitude (y), to describe that position.

The partial derivative with respect to y is also defined as:

x

yxFyxxF

x

Fx

),(),(lim0

y

sFyyxF

y

Fy

)(),(lim0


VECTOR CALCULUS DIRECTIONAL DERIVATIVES The partial derivatives of F(x,y) with respect to x and y are both special

cases of a more general derivative, the directional derivative. Consider the same function, F(x,y), but now instead of partial derivative

with respect to x or y, we compute the derivative in a direction s, as shown below:

We wish to determine the partial derivative with respect to s:

x

y

st

s

sFssF

s

Fs

)()(lim0


VECTOR CALCULUS DIRECTIONAL DERIVATIVES The variables s,t, are orthogonal and related to x and y by the

equations:

Also recall the chain rule of differentiation from ordinary calculus:

Looking at the coordinate transformations, we find that:

cossin

sincos

tsy

tsx

y

F

s

y

x

F

s

x

s

F

sincos

;sin;cos

y

F

x

F

s

Fs

y

s

x


VECTOR CALCULUS DIRECTIONAL DERIVATIVES - THE GRADIENT What would be the maximum directional derivative of F(x,y) at the point

(x,y)? This is determined by setting the derivative of the directional derivative with respect to s equal to zero.

This would be denoted by F, the gradient of F, given by:

Here the del operator, , is defined as:

Thus the gradient is a vector operator, with the del operator, , operating on a scalar, F(x,y,z).

z

Fz

y

Fy

x

FxF

ˆˆˆ

zz

yy

xx

ˆˆˆ


VECTOR CALCULUS THE DIVERGENCE AND CURL OF A VECTOR Like the dot product of two vectors, the divergence of a vector field is

a scalar function, which, in rectangular coordinates, is given by:

The vector derivative or curl of a vector is defined in rectangular coordinates as:

z

A

y

A

x

AA zyx

.

zyx

zyx

AAAzyx

zyx

Ax

AzAyAxxz

zy

yx

xAx

ˆˆˆ

ˆˆˆˆˆˆ


VECTOR CALCULUS SOME VECTOR IDENTITIES Some useful vector identities are given below: 1. The Laplacian is defined as:

2. The Curl of the Gradient of a scalar:

3. The divergence of the curl of a vector:

4. The curl of the curl of a vector:

2

2

2

2

2

22 .

z

F

y

F

x

FFF

0 Fx

0. Ax

AAAxx

2.

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