ece 307: electricity and magnetism fall 2012 - electrical & computer 307 chapter 1-3... · ece...

ECE 307: Electricity and Magnetism

Fall 2012

Instructor: J.D. Williams, Assistant Professor

Electrical and Computer Engineering

University of Alabama in Huntsville

406 Optics Building, Huntsville, Al 35899

Phone: (256) 824-2898, email: [email protected]

Course material posted on UAH Angel course management website

Textbook:

M.N.O. Sadiku, Elements of Electromagnetics 5th ed. Oxford University Press, 2009.

Optional Reading:

H.M. Shey, Div Grad Curl and all that: an informal text on vector calculus, 4th ed. Norton Press, 2005.

All figures taken from primary textbook unless otherwise cited.

mailto:[email protected]

Course Material Chapters 1-3. (Review Material)

Vectors Algebra, Coordinate Transformations, Vector Calculus

Chapter 4.

Coulomb’s Law, Electric Field Intensity, Charge Distribution, Electric Flux Density, Gauss’ Law, Electric Potential, Energy

Chapters 5-6.

Properties of Materials, Currents, Continuity Equation, Poisson’s Equation, Laplace’s Equation, Resistance, Capacitance, Image Theory (Opt.)

Chapters 7-8.

Biot-Savart Law, Ampere’s Law, Magnetic Flux Density, Maxwell’s Equations (Static), Magnetic Vector Potentials, Magnetic Forces, Magnetic Materials, Boundary Conditions

Chapter 9. (Partial)

Faraday’s Law, Maxwell’s Equations (Time Varying),

Optional: Time-Harmonic Fields, Plane Waves Poynting Vectors

• Grading: 80 -100% A, 70-80% B, 60-70% C.

• Homework (20%): Turned in weekly. Graded on attempted effort.

• Exams (50%): 2 per Semester. Exam scores historically increase over the semester – Exam 1: Chapters 4, 5 and 6

– Exam 2: Chapters 7, 8 and 9

• Final Comprehensive Exam (30%): Includes material from Chapters 4 through 9

8/17/2012 3

• It was James Clark Maxwell that put all of this together and reduced electromagnetic field

theory to 4 simple equations. It was only through this clarification that the discovery of

electromagnetic waves were discovered and the theory of light was developed.

• The equations Maxwell is credited with to completely describe any electromagnetic field

(either statically or dynamically) are written as:

Maxwell’s Time Dependent Equations

Differential Form Integral Form Remarks

Gauss’s Law

Nonexistence of the

Magnetic Monopole

Faraday’s Law

Ampere’s Circuit Law

t

DJH

t

BE

B

D v

0

0 SdBS

Sdt

DJldH

SL

SL

SdBt

ldE

dvSdD v

S


Chapters 1-3: Review of Mathematical Essentials

• Chapter 1: Vector Analysis

– Scalars and Vectors

– Unit Vector

– Vector Addition and Subtraction

– Position and Distance Vectors

– Vector Multiplication

– Components of a Vector

• Chapter 2: Coordinate Systems and

Transformations

– Cartesian Coordinates

– Circular Cylindrical Coordinates

– Spherical Coordinates

– Constant Coordinate Surfaces

Note: Students will not be tested specifically on Chapters 1-3. However, the information contained

within them will be used in almost every aspect of the course. Knowledge and skilled practice of

these concepts will be required to complete all homework, and examinations throughout the semester

• Chapter 3: Vector Calculus

– Differential Length, Area, and Volume

– Line, Surface, and Volume Integrals

– Del Operator

– Gradient of a Scalar

– Divergence of a Vector and the

Divergence Theorem

– Curl of a Vector and Stokes’s Theorem

– Laplacian of a Scalar

– Classification of Vector Fields

• Homework • Ch. 1: 5c, 6a, 8a, 8b, 10, 26

• Ch. 2: 7, 8, 10, 15, 17

• Ch. 3: 1b, 2b, 2c, 3b, 6, 12a, 12b, 23, 30



• Chapter 1: Vector Analysis – Scalars and Vectors

– Unit Vector

– Vector Addition and Subtraction

– Position and Distance Vectors

– Vector Multiplication

– Components of a Vector




Vectors and Scalars

• Scalar: quantity defined only by its magnitude

– speed: 4 m/s

– Charge: 3 Coulombs

– Capacitance: 5 farads

• Vector: quantity defined by both its magnitude and direction in

space

– Force:

• Field: Function that specifies a particular quantity everywhere

within a spatial domain.

– Electric field (vector field):

– Voltage (scalar field):

NzyxF )ˆ4ˆ5ˆ3(

Vr

kqV

CNrr

kqE

/ˆ

2

Unit Vector

• A vector A has both magnitude and direction

• The magnitude of A is a scalar written as IAI.

• A unit vector aA along A is defined as a vector

along the direction of A with magnitude of 1.

• As such, the vector A may be defined as

• And thus

2

3

2

2

2

1

332211

2

3

2

2

2

1

332211

ˆˆˆˆ

ˆˆˆ

ˆ

AAA

aAaAaA

A

Aa

AAAA

aAaAaAA

aAA

A

A

zyxa

zyxzyx

a

A

zyxA

A

A

ˆ2

2ˆ

5

22ˆ

10

23ˆ

ˆ10

25ˆ

10

24ˆ

10

23

25

ˆ5ˆ4ˆ3ˆ

2550543

ˆ5ˆ4ˆ3

222

Example:

Vector Addition and Subtraction

• Two vectors, A and B, can be added together to generate a third vector, C

• Make sure to abide by the following basic laws of algebra as applied to vectors

Addition Multiplication

BkAkBAk

CBACBA

ABBA

AklAlk

kAAk

Commutative

Associative

Distributive

333222111

333222111

332211

332211

ˆˆˆ

ˆˆˆ

ˆˆˆ

ˆˆˆ

aBAaBAaBACBA

aBAaBAaBACBA

aBaBaBB

aAaAaAA

Distance Vectors

• Vectors can also be used to define the distance between two points in a

coordinate system, or between a line and a plane within a coordinate system

– If given two points, P and Q, one can find the distance between them as the vector, r

312212112

222322212

111312111

222

111

ˆˆˆ

,,ˆˆˆ

,,ˆˆˆ

),,(

),,(

azzayyaxxrrr

zyxazayaxr

zyxazayaxr

zyxQ

zyxP

PQPQ

Q

P

Vector Multiplication: Dot Product

• Two vectors, A and B, can be multiplied together to generate a third vector, C

– Scalar Product

– Vector Product

– Scalar Triple Product

– Vector Triple Product

• Make sure to abide by the following basic laws of algebra as applied to dot products

DCBA

lCBA

CBA

kBA

22

AAAA

CABACBA

ABBA

Commutative

Associative

ABBABA

BABABAkBA

aBaBaBB

aAaAaAA

cos

ˆˆˆ

ˆˆˆ

332211

332211

332211

Note: orthogonal vector dot products multiply to a cosine value of zero

parallel vector dot products multiply to a cosine value of 1

Vector Multiplication: Cross Product

• Make sure to abide by the following basic laws of algebra as applied to cross products

0

AA

CABACBA

CBACBA

ABBA

Anti-Commutative

Not Associative

Distributive

nABaBABA

aBABAaBABAaBABABA

BBB

AAA

aaa

BA

aBaBaBB

aAaAaAA

ˆsin

ˆˆˆ

ˆˆˆ

ˆˆˆ

ˆˆˆ

312212133112332

321

321

321

332211

332211

Note: orthogonal vector dot products multiply to a sine value of 1

BACCABCBA

BACACBCBA

CCC

BBB

AAA

CBA

321

321

321Scalar Triple Product

Vector Triple Product

Components of a Vector

• Given two vectors, A and B, one can directly find the scalar component of A along B as

This scalar product is known as the projection of A along the aB direction.

• The vector component of A along B is simply scalar component multiplied by the unit vector

along B

• One can also find the angle between A and B using the cross and dot product of the two

BABBABB aAaAAA ˆcosˆcos

BBBB aaAaAA ˆˆˆ

ABBA

BAsin

ABBA

BAcos



• Chapter 2: Coordinate Systems and

Transformations – Cartesian Coordinates

– Circular Cylindrical Coordinates

– Spherical Coordinates

– Constant Coordinate Surfaces




Cartesian Coordinates

• Coordinate system represented by (x,y,z) that are three orthogonal vectors in strait lines

that intersect at a single point (the origin).

• The vector A in this coordinate system can be written as

z

y

x

zzyyxx aAaAaAA ˆˆˆ

Cylindrical Coordinates

• Coordinate system represented by (,,z) that are three

orthogonal vectors

• The vector A in this coordinate system can be written as

• Where the following equations can be used to convert

between cylindrical and cartesian coordinate systems

zz

x

y

yx

1

22

tan

z

20

0

zzaAaAaAA ˆˆˆ

zz

y

x

sin

cos

Matrix Transformations: Cart. and Cyl.

zz

y

x

z

y

x

z

A

A

A

A

A

A

A

A

A

A

A

A

100

0cossin

0sincos

100

0cossin

0sincos

Spherical Coordinates

• Coordinate system represented by (r,,) that are

three orthogonal vectors emanating from or revolving

around the origin

• The vector A in this coordinate system can be written

as

• Where the following equations can be used to convert

between spherical and Cartesian coordinate systems

x

y

z

yx

zyxr

1

221

222

tan

tan

20

0

0

r

aAaAaAA rrˆˆˆ

cos

sinsin

cossin

rz

ry

rx

Matrix Transformations: Cart. and Sph.

A

A

A

A

A

A

A

A

A

A

A

A

r

z

y

x

z

y

xr

0sincos

cossincossinsin

sincoscoscossin

0cossin

sinsincoscoscos

cossinsincossin






• Chapter 3: Vector Calculus – Differential Length, Area, and Volume

– Line, Surface, and Volume Integrals

– Del Operator

– Gradient of a Scalar

– Divergence of a Vector and the Divergence

Theorem

– Curl of a Vector and Stokes’s Theorem

– Laplacian of a Scalar

– Classification of Vector Fields

Differential Length, Area, and Volume


• Differential Displacement (dl)

• Differential Surface Area (dS)

• Volume Differential (dv)

dxdydzdv

zyx adzadyadxld ˆˆˆ

z

y

x

adxdySd

adxdzSd

adydzSd

ˆ

ˆ

ˆ

Note: Differential length and surface areas are vectors. Differential volume is a scalar






dzdddv

zadzadadld ˆˆˆ

zaddSd

adzdSd

adzdSd

ˆ

ˆ

ˆ

Note: term is used as multiplier to complete units associated with arc radians of d






dddrrdv sin2

adrardadrld rˆsinˆˆ

addrrSd

addrrSd

addrSd r

ˆ

ˆsin

ˆsin2


Line Integrals

• The line integral is the integral of the tangential component of A along the curve L

• Requires L be a smooth, continuous curve.

• The vector A, may be a vector field component

• Line integrals are said to be path independent if the solution of the tangential

component of A is independent to the path L taken within the field. The most

common example of path independent integrals used are work (energy) solutions

integrating the force over the path length, L. Path independence of A occurs if xA=0 (the curl of A is equal to zero)

ldA

dlAldA

b

aL

cos

• For a contour of length, L

• Gives the circulation of A

around the contour, L

Example of a Line Integral • Given the following equation for F = [xy,-y2]

• Find the line integral of F along the following paths between (0,0) and (2,1)

• Path 1: strait line

• Path 2: parabola

188

3

88

4

2

1

42

2

1

2

1

2

0

32

0

22

0

222

0

22

x

dxx

dxx

dxx

dxx

dxx

W

dxdy

xy

3/2192163242

1

164

2

1

4

1

2

0

642

0

532

0

42

2

xx

dxx

dxx

xdxx

xdxx

W

xdxdy

xy

dyyxydxrdFW

dyyxydxrdF

dydxrd

yxr

2

2

,

,

Path Dependent!!!

Show (on your own)

that the curl does not

equal 0

Surface Integrals

• The surface integral is the integral of the vector field, A , over the closed contour, S

provides the net outward flux of A through the surface, S

surfacethetonormala

SddS

SdA

dSaAdSASdA

n

S

n

b

aS

___ˆ

ˆcos

Note: The surface integral will become the basis for flux of the electric

field through a Gaussian surface in Chapter 4

Surface Integrals: Finding the Area

of the Surface • Surface integrals are double integrals that calculate the area of a closed surface

• Examples

– Circle

– Cylinder ( two circles with a tubular surface between them

rLdrLdzddzddS

rdS

dSdSdS

r

L

S

S

SSS

2

00

2

02

2

1

21

2

22

2

2

1

21

2

2

0

222

00 2

22

rr

dr

dddddSr

S

S1

S2

r

L

r

Surface Integrals (cont.) • Unit conversions allow one to simplify problems to ease calculations

• Example: Find the area cut from the upper half of a sphere

by the cylinder

This is the sum of the area of the sphere which projects onto the disk

in the (x,y) plane. Thus, we want to integrate the area of the disk. The

geometry presented shows the area that will be integrated. Integration are

The desired area can then be calculated as:

1222 zyx

2/

0

cos

1

2/

0

sin

0 2

2

2/

0

sin

0 2

1

00 22

221

2

1,

12

12

2

zr

ry

yy

xS

dzdr

rdrd

r

rdrdzwhere

r

rdrd

yx

dxdyAreadS

221

11

yxz

022 yyx

20 yyx 10 y

x

y

Area and Volume

vv

dzdddvV SS

dzddSS

Volume Integrals

• Used to calculate the volume, mass, centroid, charge, etc of a solid

• Volume of a sphere

• Volume of a cylinder

3

4

3

2sin

3sin

32

0

32

00

32 r

dr

ddr

ddrdrdvvv

v

v

v

v dvdv

))(___(

2

2

0

2

0

2

heightcircletheofareaVolume

Ldzddzdddvv

L

v

v is a volumetric density function

Del Operator

• Del is a vector differential operator. The del operator will be used in 4 differential operations throughout any course on field theory. The following equation is the del operator for different coordinate systems

– Gradient of a Scalar, V is written as vector

– The divergence of a vector, A, is written as scalar

– The curl of a vector, A, is written as vector

– The Laplacian of a scalar, V, is written as scalar

ar

ar

ar

az

aa

az

ay

ax

r

z

zyxzyx

ˆsin

1ˆ

1ˆ

ˆˆ1

ˆ

ˆˆˆ,,

A

A

V2

V

Gradient of a Scalar

• The gradient of a scalar field, V, is a vector that represents both the magnitude and the direction of the maximum space rate of increase of V.

• To help visualize this concept, take for example a topographical map. Lines on the map represent equal magnitudes of the scalar field. The gradient vector crosses map at the location where the lines packed into the most dense space and perpendicular (or normal) to them. The orientation (up or down) of the gradient vector is such that the field is increased in magnitude along that direction.

• Example

zyxzyx Vaz

Va

y

Va

x

VV ,,

ˆˆˆ

xyzxzyzV

axyzaxzayzaz

Va

y

Va

x

VV

xyzV

zyxzyx

4,2,2

ˆ4ˆ2ˆ2ˆˆˆ

2

22

22

2

Gradient of a Scalar (2)

• Fundamental properties of the gradient of a scalar field

– The magnitude of gradient equals the maximum rate of change in V per unit distance

– Gradient points in the direction of the maximum rate of change in V

– Gradient at any point is perpendicular to the constant V surface that passes through that point

– The projection of the gradient in the direction of the unit vector a, is

and is called the directional derivative of V along a. This is the rate of change of V in the direction of a.

– If A is the gradient of V, then V is said to be the scalar potential of A

• Easily Proven Mathematical Relations

aV ˆ

VUUVVU

UVUV

VnVV

U

UVVU

U

V

n

12

2

Divergence of a Vector

• The divergence of a vector, A, at any given point P is the outward flux per unit volume as volume shrinks about P.

• The following relations can be easily derived for the divergence of a field

Ar

Ar

Arrr

A

Az

AAA

Az

Ay

Ax

A

v

SdAAAdiv

aAaAaAA

r

z

zyx

s

v

sin

1sin

sin

11

11

lim

ˆˆˆ

2

2

0

332211

VAAVAV

BABA

Image from www.ux1.eiu.edu

http://www.ux1.eiu.edu/

Divergence Theorem

• The divergence theorem states that the total outward flux of a vector field, A, through the closed surface, S, is the same as the volume integral of the divergence of A.

• This theorem is easily shown from the equation for the divergence of a vector field.

k

k

k

k k

s

kss

sv

s

v

dvAvv

SdASdASdA

proof

SdAdvA

v

SdAAAdiv

aAaAaAA

k

k

:

lim

ˆˆˆ

0

332211

Divergence of vector A (red) out of a spherical surface (green)

http://www.math.brown.edu/~banchoff/multivarcalc2/multivarcalc3-5.html




Divergence Theorem

Examples:

3210

3

52

2

52

55

ˆˆ5ˆˆ2ˆˆ5

501011

ˆ5ˆ2ˆ5

2322

22

zz

zz

dzddzzd

addazadzdaadzdazSdA

zAz

AAA

azaazA

S

S

zzs

z

z

Suppose we wanted to find the flux of A through a cylinder of radius and height z

Curl of a Vector

• The curl of a vector, A is an axial vector whose magnitude is the maximum circulation of A per unit area as the area tends to zero and whose direction is the normal direction of the area when the area is oriented to make the circulation maximum.

• The following relations can be easily derived for the divergence of a field

321

321

321

max

0

332211

ˆˆˆ

ˆlim

ˆˆˆ

AAA

xxx

aaa

A

aS

ldAAAcurl

aAaAaAA

nL

S

AVAVAV

BAABABBABA

BABA

0

0

V

A

Insert: Field lines of A (in

black) defined through the

closed loop dl (green)

Curl of a Vector (2)

• Curl of a vector in each of the three primary coordinate systems discussed in this course

aA

r

rA

ra

A

r

rA

ra

AA

rA

ArrAAr

arara

rA

aAA

az

AAa

z

AA

AAAz

aaa

A

ay

A

x

Aa

z

A

x

Aa

z

A

y

A

AAA

zyx

aaa

A

rrr

r

r

zzz

z

z

zxy

yxz

x

yz

zyx

zyx

ˆ1

ˆsin

11ˆ

sin

sin

1

sin

ˆsinˆˆ

sin

1

ˆ1

ˆˆ1

ˆˆˆ

1

ˆˆˆ

ˆˆˆ

2

Cartesian

Cylindrical

Spherical

Curl of a Vector (Example)

aaaA

arr

arr

arr

A

aA

r

rA

ra

A

r

rA

ra

AA

rA

rA

rA

rA

ararar

A

r

r

rrr

r

r

ˆ10ˆ12ˆcot6

ˆ101

ˆ121

ˆcos6sin

1

ˆ1

ˆsin

11ˆ

sin

sin

1

6

5

10

ˆ6ˆ5ˆ10

2

2

Stokes Theorem

• Stokes theorem states that the circulation of a vector field A, around a closed path, L is equal to the surface integral of the curl of A over the open surface S bounded by L. This theorem has been proven to hold as long as A and the curl of A are continuous along the closed surface S of a closed path L

• This theorem is easily shown from the equation for the curl of a vector field.

k

k

nk

k

k

k k

L

kLL

sL

nL

S

SdAadSASS

ldAldAldA

proof

SdAldA

aS

ldAAAcurl

aAaAaAA

k

k

k

ˆ

:

ˆlim

ˆˆˆ

max

0

332211

For example, see line integral

Laplacian of a Scalar

• The Laplacian differential operator is a combination of gradient and

divergence operators.

• The Laplacian of a scalar field is the divergence of the gradient of V

– n a harmonic field Laplace’s Eqn.

– Field with internal divergence Poison’s Eqn.

– The following vector identity holds true for all Laplacian fields governed

by a vector, A

2

2

222

2

2

2

2

2

2

2

2

2

2

2

2

2

2

22

2

sin

1sin

sin

11

11

V

r

V

rr

Vr

rrV

z

VVVV

Vzyx

V

VVLaplacianV

AAA

2

02 V

02 V

• Vector fields are uniquely defined by their divergence and curl. However

neither the divergence or the curl alone of a vector field is sufficient to

completely describe the field. Both flux and rotation must be considered

concurrently.

• Thus we use the four following equations to mathematically describe and

predict the effects of a vector field

Classification of Vector Fields

0,0

0,0

0,0

0,0

AA

AA

AA

AA

(a)

(b)

(c)

(d)

Classification of Vector Fields

• The vector field, A, is said to be divergenceless ( or solenoidal) if

– Such fields have no source or sink of flux, thus all the vector field lines entering an enclosed surface, S, must also leave it.

– Examples include magnetic fields, conduction current density under steady state, and imcompressible fluids

– The following equations are commonly utilized to solve divergenceless field problems

• The vector field, A, is said to be potential (or irrotational) if

– Such fields are said to be conservative. Examples include gravity, and electrostatic fields.

– The following equations are commonly used to solve potential field problems

0 A

0

0

A

V

0 A

AF

dvASdA

A

vS

0

0

VA

SdAldASL

0

Classification of Vector Fields (2)

• Return again to our initial statement. A vector field is defined by both its

divergence and its curl. Thus, we can describe any field by evaluating the

following two equations.

where is a density component of the field, v represents volume, and S

represents surface area. We refer to the volume density component as

the source density, and the area component as the circulation density

• Hemholtz’s Theorem states that any field satisfying the conditions above whose

density components vanish at infinity can be completely described as the sum

of two vectors. One of the vectors is irrotational, and the other is solenoidal.

– In our case, the irrotational component is the electric field, and the solenoidal

component is the magnetic field. As such one may completely describe an

electromagnetic field using the following relations

vA

SvA

BVA

BvEqF

2

SA

(Optional Material)

Alternative Approach to Vector

Derivatives Using Scale Factors

(Optional Material)

Derivatives in Space • Scalar Field: a scalar function, S, of position coordinates in a finite dimensional

vector space.

• For most of our purposes, the finite vector dimensional vector space will be

limited to 3 dimensions (or physically real space).

• Defining a coordinate system {x1,x2,x3}within V, then allows for differentiation to

be written as:

Name of the

coordinate

system

Cartesian Cylindrical Spherical

x1 x r

x2 y

x3 z z

h1 1 1 1

h2 1 r

h3 1 1 rsin 45

2

3

2

3

2

2

2

2

2

1

2

1

2 dxhdxhdxhds

3

1i

ii

k

i

i

i

i dxhdxdx

dssd

or

Derivatives in Space: Example • Derive the scale factors for the spherical coordinate system

46

2

3

2

3

2

2

2

2

2

1

2

1

2 dxhdxhdxhds

222

2

222

2

222

2

d

dz

d

dy

d

dxh

d

dz

d

dy

d

dxh

dr

dz

dr

dy

dr

dxhr

cos

sinsin

cossin

rz

ry

rx

sin

1

3

2

1

rhh

rhh

hh r

Here we take only

the positive roots






dxdydz

dxdxdxhhh

dxdxdxdvijk

kjiijk

321321

zyx

zyzi

i

ii

adzadyadx

adzhadyhadxhadxhld

ˆˆˆ

ˆˆˆˆ321

3

1

32121 edxdxhh

dxdxSdk

jiijkk


47

zz

yy

xx

adxdySd

adxdzSd

adydzSd

ˆ

ˆ

ˆ






dzdd

dxdxdxhhhdv

321321

z

zi

i

ii

adzadad

adzhadhadhadxhld

ˆˆˆ

ˆˆˆˆ321

3

1

zz addedxdxhhSd

adzdedxdxhhSd

adzdedxdxhhSd

ˆˆ

ˆˆ

ˆˆ

12121

13131

13232

Note: term is used as multiplier to complete units associated with arc radians of d

48






ddrdr

ddrdhhhdv r

sin2

adrardadr

adhadhadrhadxhld

r

rri

i

ii

ˆsinˆˆ

ˆˆˆˆ3

1

ardrdadrdhhSd

adrdradrdhhSd

addraddhhSd

r

r

rrr

ˆˆ

ˆsinˆ

ˆsinˆ 2


49

Del Operator

• Del is a vector differential operator. The del operator will be used in 4 differential operations throughout any course on field theory. The following equation is the del operator for different coordinate systems

– Gradient of a Scalar, V is written as vector

– The divergence of a vector, A, is written as scalar

– The curl of a vector, A, is written as vector

– The Laplacian of a scalar, V, is written as scalar

ar

ar

ar

az

aa

az

ay

ax

r

z

zyxzyx

ˆsin

1ˆ

1ˆ

ˆˆ1

ˆ

ˆˆˆ,,

A

A

V2

V

50

k

i

i

ii

i

i ii

exh

exh

ˆ1

ˆ13

1

• The general form of the ith component

of the gradient vector follows from the

definition of the gradient

• Yielding the general equation

Derivatives in Space: The

Gradient • Definition: The gradient is a vector

field generated by the product of the

differential operator, , on a scalar

field, .

• As such, a scalar is simply the product

of two vectors.

• The gradient is the maximum rate of

change of the scalar field on the point

at which the derivative is taken.

• It follows that is the maximum rate

of change of

• Finally, is always perpendicular to

equipotential lines of the surface

51

iii

iii

dsi

xhds

xdxx

i

1lim

0 dsdsdgrad

dssdd cos

dsd

i

i ii

exh

grad ˆ13

1

Derivatives in Space:

Divergence

52

321

3

213

2

132

1321

21

1

1

Vhhx

Vhhx

Vhhxhhh

kjikjiforVhhxhhh

divVk

i

iii

ikji

321321 dxdxdxhhhVolume of the surface:

• The divergence of the scalar field over the surface is written as:

• General Equations for the curl in any orthogonal curvilinear coordinate system can be extended

from our calculations of the vector cross product:

Derivatives in Space: The Curl

53

i

k

i i

ii

i

ii

ii

ex

Ah

x

Ah

hh

ex

Ah

x

Ah

hhe

x

Ah

x

Ah

hhe

x

Ah

x

Ah

hh

AhAhAh

xxx

ahahah

hhhAAcurl

ˆ)()(1

ˆ)()(1

ˆ)()(1

ˆ)()(1

ˆˆˆ

1

2

11

1

22

21

3

2

11

1

22

21

2

1

33

3

11

13

1

3

22

2

33

32

332211

321

332211

321

Derivatives in Space: Laplacian • Definition: The Laplacian of a scalar field is defined as the divergence of the

gradient that scalar field, .

• Generalized form of the Laplacian of a vector field, V, in orthogonal curvilinear

coordinates

54

k

i ii

ii

i x

V

h

hh

xhhh

x

V

h

hh

xx

V

h

hh

xx

V

h

hh

xhhhV

VVVLaplacian

21

321

33

21

322

13

211

32

1321

2

2

1

1

VVV

2

ece 307: electricity and magnetism fall 2012 - electrical & computer 307 chapter 1-3... · ece...

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