Fundamentals of ElectromagneticsFundamentals of Electromagneticsfor Teaching and Learning:for Teaching and Learning:

A Two-Week Intensive Course for Faculty inA Two-Week Intensive Course for Faculty inElectrical-, Electronics-, Communication-, and Electrical-, Electronics-, Communication-, and

Computer- Related Engineering Departments in Computer- Related Engineering Departments in Engineering Colleges in IndiaEngineering Colleges in India

byby

Nannapaneni Narayana RaoNannapaneni Narayana RaoEdward C. Jordan Professor EmeritusEdward C. Jordan Professor Emeritus

of Electrical and Computer Engineeringof Electrical and Computer EngineeringUniversity of Illinois at Urbana-Champaign, USAUniversity of Illinois at Urbana-Champaign, USADistinguished Amrita Professor of EngineeringDistinguished Amrita Professor of Engineering

Amrita Vishwa Vidyapeetham, IndiaAmrita Vishwa Vidyapeetham, India

Program for Hyderabad Area and Andhra Pradesh FacultySponsored by IEEE Hyderabad Section, IETE Hyderabad

Center, and Vasavi College of EngineeringIETE Conference Hall, Osmania University Campus

Hyderabad, Andhra PradeshJune 3 – June 11, 2009

Workshop for Master Trainer Faculty Sponsored byIUCEE (Indo-US Coalition for Engineering Education)

Infosys Campus, Mysore, KarnatakaJune 22 – July 3, 2009

3-3

Module 3Maxwell’s Equationsin Differential Form

3.1 Faraday’s law and Ampere’s Circuital Law3.2 Gauss’ Laws and the Continuity Equation3.3 Curl and Divergence

3-4

Instructional Objectives16. Obtain the simplified forms of Faraday’s law and Ampere’s circuital law in differential forms for any special cases of electric and magnetic fields, respectively, or the particular differential equation that satisfies both laws for a special case of electric or magnetic field17. Determine if a given time-varying electric/magnetic field satisfies Maxwell’s curl equations, and if so find the corresponding magnetic/electric field, and any required condition, if the field is incompletely specified18. Find the magnetic field due to one-dimensional static current distribution using Maxwell’s curl equation for the magnetic field19. Find the electric field due to one-dimensional static charge distribution using Maxwell’s divergence equation for the electric field

3-5

Instructional Objectives (Continued)20. Establish the physical realizability of a static electric field by using Maxwell’s curl equation for the static case, and of a magnetic field by using the Maxwell’s divergence equation for the magnetic field21. Investigate qualitatively the curl and divergence of a vector field by using the curl meter and divergence meter concepts, respectively22. Apply Stokes’ and divergence theorems in carrying out vector calculus manipulations

3-6

3.1 Faraday’s Law andAmpère’s Circuital Law(EEE, Sec. 3.1; FEME, Secs. 3.1, 3.2)

3-7

Maxwell’s Equations in Differential Form

Why differential form?

Because for integral forms to be useful, an a priori knowledge of the behavior of the field to be computed is necessary.

The problem is similar to the following:

There is no unique solution to this.

If y(x) dx 2, what is y(x)?01

3-8

However, if, e.g., y(x) = Cx, then we can find y(x), since then

On the other hand, suppose we have the following problem:

Then y(x) = 2x + C

Thus the solution is unique to within a constant.

If dy

dx2, what is y?

121

00

2 or 2 or 42

4

xCx dx C C

y x x

3-9

FARADAY’S LAW

First consider the special case

and apply the integral form to the rectangular path shown, in the limit that the rectangle shrinks to a point.

(x, z)

x S C

z (x, z + z)

x

(x + x, z) (x + x, z + z)

zy

and( , ) ( ) E a H ax x y yE z t H z,t

3-10

,

00

Limy x z

xz

d

dtB x z

x z

yxBE

z t

SC

dd d

dt E l B S

,x x yz z z x z

dE x E x B x z

dt

00

Limx xz z z

xz

E E x

x z

3-11

General Case

E Ex (x, y, z,t)a x Ey (x, y, z,t)ay Ez (x, y,z, t)az

H Hx (x, y, z,t)a x Hy (x, y, z,t)ay Hz (x, y, z, t)az

x

y

z

, ,a x y z

, ,c x y y z z , ,d x y z z

z

x

, ,b x y y z , ,e x x y z z

, ,f x x y z , ,g x x y y z

y

3-12

E Ex (x, y, z,t)a x Ey (x, y, z,t)ay Ez (x, y,z, t)az

H Hx (x, y, z,t)a x Hy (x, y, z,t)ay Hz (x, y, z, t)az

Lateral space derivatives of the components of E

Time derivatives of the components of B

– –

– –

– –

yz x

yzx

y zx

EE B

y z t

BEE

z x t

E BE

x y t

The terms on the left sides are the net right-lateral differentials of pairs of components of E. For example, in the first equation, it is the net right-lateral differential of Ey and Ez normal to the x-direction. The figure below illustrates (a) the case of zero value, and (b) the case of nonzero value, for this quantity.

3-13

zE

yE

(a) (b)

yE yE

yE

zE zE zEx y

z

×

3-14

Combining into a single differential equation,

Differential formof Faraday’s Law

–

a a a

Bx y z

x y z

x y z t

E E E

–

×B

Et

a a ax y zx y z

Del Cross or Curl of = – B

t

E E

3-15

AMPÈRE’S CIRCUITAL LAW

Consider the general case first. Then noting that

we obtain from analogy,

E –t(B)

E • dl –d

dtB • dS

SC

H • dl J • dS ddt

D • dSSSC

H J t(D)

3-16

Thus

Special case:

Differential form of Ampère’s circuital law

E Ex (z,t)a x , H Hy (z,t)ay

–

y xx

H DJ

z t

×

DH J

t

0 0

0 0

J

a a a

Dx y z

y

z tH

3-17

E3.1 For

in free space

find the value(s) of k such that E satisfies both

of Maxwell’s curl equations.

Noting that E Ey (z,t)a y ,we have from

0 0, , , J = 0

– – xD

t

y

x

HJ

z

– ,

B

Et

80 cos 6 10E × ayE t kz

3-18

80

80

cos 6 10

sin 6 10

yxEB

t z

E t kzz

kE t kz

808

cos 6 106 10x

kEB t kz

– – 0 0

0 0

a a a

BE

x y z

y

t zE

3-19

Thus,

Then, noting that we have fromH Hx (z,t)ax ,

,

D

Ht

808

70

802

cos 6 106 10

4 10

cos 6 10240

B a

B BH

a

x

x

kEt kz

kEt kz

3-20

0 0

0 0

x y z

x

t z

H

a a a

D× H

2

802

sin 6 10240

y xD H

t z

k Et kz

3-21

2

803 8

cos 6 101440 10y

k ED t kz

2

803 8

cos 6 101440 10

D ay

k Et kz

90

280

2

10 36

cos 6 104

D DE

ay

k Et kz

3-22

k 2

3 108(c) m s.

Comparing with the original given E, we have

20

0 24

k EE

Sinusoidal traveling waves in free space, propagating in the z directions with velocity,

80 cos 6 10 2E ayE t z

3-23

E3.2.

3-24

3-25

Review Questions

3.1. Discuss the applicability of integral forms of Maxwell’s equations versus that of the differential forms for obtaining the solutions for the fields.3.2. State Faraday’s law in differential form for the special case of E = Ex(z, t)ax and H = Hy(z, t)ay . How is it derived from Faraday’s law in integral form?3.3. How would you derive Faraday’s law in differential form from its integral form for the general case of an arbitrary electric field?3.4. What is meant by the net right-lateral differential of the x- and y- components of a vector normal to the z- direction? Give an example in which the net right-lateral differential of Ex and Ey normal to the z-direction is zero, although the individual derivatives are nonzero.3.5. What is the determinant expansion for the curl of a vector in Cartesian coordinates?

3-26

Review Questions (Continued)

3.6. State Ampere’s circuital law in differential form for the general case of an arbitrary magnetic field. How is it obtained from its integral form?3.7. State Ampere’s circuital law in differential form for the special case of H = Hy(z, t)ay . How is it derived from the Ampere’s circuital law for the general case in differential form?3.8. If a pair of E and B at a point satisfies Faraday’s law in differential form, does it necessarily follow that it also satisfies Ampere’s circuital form and vice versa?3.9. Discuss the determination of magnetic field for one dimensional current distributions, in the static case, using Ampere’s circuital law in differential form, without the displacement current density term.

3-27

Problem S3.1. Obtaining the differential equation for a special case that satisfies both of Maxwell’s curl equations

3-28

Problem S3.2. Finding possible condition for a specified field to satisfy both of Maxwell’s curl equations

3-29

Problem S3.3. Magnetic field due to a one-dimensional current distribution for the static case

3-30

Problem S3.3. Magnetic field due to a one-dimensional current distribution for the static case (Continued)

3-31

Problem S3.3. Magnetic field due to a one-dimensional current distribution for the static case (Continued)

3-32

3.2 Gauss’ Laws and the Continuity Equation

(EEE, Sec. 3.2; FEME, Secs. 3.4, 3.5, 3.6)

3-33

GAUSS’ LAW FOR THE ELECTRIC FIELD

D • dS S dv

Vz

(x, y, z)y

x

z

y

x

x xx x x

y yy y y

z zz z z

D y z D y z

D z x D z x

D x y D x y

x y z

3-34

000

Limxyz

x y z

x y z

+Δ

000

Δ Δ

Lim

x xx x x

y yy y y

z zz z z

xyz

D D y z

D D z x

D D x y

x y z

3-35

The quantity on the left side is the net longitudinal differential of the components of D, that is, the algebraic sum of the derivatives of components of D along their respective directions. It can be written as which is known as the “divergence of D.”

Thus, the equation becomes

Longitudinal derivatives

of the components of

D

y zxD DD

x y z

• D

,D

3-36

E3.3 Given that

Find D everywhere.

0 for – a x a0 otherwise

The figure below illustrates the case of (a) zero value, and (b) nonzero value for .D

zD

yD

xD

yD yD

yD

xD xD xD

zD

zD

zDx

y

z

(a) (b)

3-37

Noting that = (x) and hence D = D(x), we set

0

x=–a x=0 x=a

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

• •

0 and 0, so that

y z

• xD

x

D y zx

D DD

x y z

3-38

Thus,

which also means that D has only an x-component. Proceeding further, we have

where C is the constant of integration. Evaluating the integral graphically, we have the following:

• D = gives

( )

xDx

x

x

xD x dx C

3-39

–a 0 a x

0

–a 0 a x

(x ) dx–x

20a

From symmetry considerations, the fields on the two sides of the charge distribution must be equal in magnitude and opposite in direction. Hence,

C = – 0a

3-40

0a

–0a

–a a x

Dx

0

0

0

for

for

for

a

D a

a

x

x

x

a x a

x a x a

a x a

3-41

B • dS = 0 = 0 dvVS

• B 0

GAUSS’ LAW FOR THE MAGNETIC FIELD

From analogy

Solenoidal property of magnetic field lines. Provides test for physical realizability of a given vector field as a magnetic field.

D • dS = dvVS

• D

• B 0

3-42

LAW OF CONSERVATION OF CHARGE

J • dS ddt

dv 0VS

• J t( ) 0

aaaa

• J t0

ContinuityEquation

3-43

SUMMARY

(4) is, however, not independent of (1), and (3) can be derived from (2) with the aid of (5).

(1)

(2)

(3)

(4)

(5)

–

•

• 0

BE

DH J

D

B

t

t

• 0

Jt

3-44

The interdependence of fields and sources through Maxwell’s equations

+

+

H ,BJ

D,E

Ampere’s Circuital Law (2)

Faraday’sLaw (1)

Gauss’ Law for E (3)

Law of Conservationof Charge (5)

3-45

Review Questions

3.10. State Gauss’ law for the electric field in differential form. How is it derived from its integral form?3.11. What is meant by the net longitudinal differential of the components of a vector field? Give an example in which the net longitudinal differential of the components of a vector field is zero, although the individual derivatives are nonzero.3.12. What is the expression for the divergence of a vector in Cartesian coordinates?3.13. Discuss the determination of electric field for one dimensional charge distributions, in the static case, using Gauss’ law for the electric field in differential form.

3-46

Review Questions (Continued)

3.14. State Gauss’ law for the magnetic field in differential form. How is it obtained from its integral form?3.15. How can you determine if a given vector field can be realized as a magnetic field?3.16. State the continuity equation.3.17. Summarize Maxwell’s equations in differential form and the continuity equation, stating which of the equations are independent.3.18. Discuss the interdependence of fields and sources through Maxwell’s equations.

3-47

Problem S3.4. Finding the electric field due to a one-dimensional charge distribution for the static case

3-48

Problem S3.5. Finding the condition for the realizability of a specified vector field as a certain type of field

3-49

Problem S3.6. Determination of the group belonging to a specified vector field, based on its physical realizability

3-50

3.3 Curl and Divergence (EEE, Sec. 3.3, App. B ; FEME, Secs. 3.3

and 3.6, App. B)

3-51

Maxwell’s Equations in Differential Form

Curl

Divergence

=

=

t

t

B×E

D× H J

D

0B

x y z

x y z

x y zA A A

a a a

× Α

=A x y zA A A

x y z

Curl and Divergence in Cylindrical Coordinates

3-52

=

a aa

A

r z

r z

r r

r z

A rA A

1 1=A zr

A Ar A

r r r z

3-53

Curl and Divergence in Spherical Coordinates

2 sinsin

=

sin

aaa

A

r

r

r rr

r

A rA r A

22

1 1= sinsin

1 sin

A rr A Ar rr

A

r

3-54

Basic definition of curl

× A is the maximum value of circulation of A per unit area in the limit that the area shrinks to the point.

Direction of is the direction of the normalvector to the area in the limit that the area shrinksto the point, and in the right-hand sense.

× A

max

Lim0

A l× A = aC

n

d

S S

3-55

Curl Meteris a device to probe the field for studying the curl of thefield. It responds to the circulation of the field.

E3.40 sin for 0 < < v az

xv x aa

3-563-56

xy

z

3-57

negative for 0

2

positive for 2

y

ax

ax a

× v

0

0 0

cos for 0 < <

x y z

zy

z

y

v

x y z xv

v xx a

a a

a a a

× v a

a

0 sin for 0 < < v azxv x a

a

3-58

Basic definition of divergence

Divergence meter

is the outward flux of A per unit volume in the limit that the volume shrinks to the point.

is a device to probe the field for studying the divergence of the field. It responds to the closed surface integral of the vector field.

Lim0

A SA S

d

v v

3-59

Divergence positive for (a) and (b), negative for (c) and (d),

and zero for (e)

3-60

E 3.5At the point (1, 1, 0)

Divergence zero

Divergence positive

Divergence negative

(a)

(b)

(c)

21 xx a

1 yy a

x

y

1

z 1

y

1

z 1

x

ayye

y

1

z 1

x

3-61

Two Useful Theorems:

Stokes’ theorem

Divergence theorem

A useful identity

C S

d dA l = × A S

S V

d dvA S = A

× A

3-62

x y z

x y z

x y zA A A

a a a

× Α

0

x y z

x y z

x y z

x y z

x y z

A A A

× A = × A × A × A

3-63

Review Questions

3.19. State and briefly discuss the basic definition of the curl of a vector.3.20. What is a curl meter? How does it help visualize the behavior of the curl of a vector field?3.21. Provide two examples of physical phenomena in which the curl of the vector field is nonzero.3.22. State and briefly discuss the basic definition of the divergence of a vector.3.23. What is a divergence meter? How does it help visualize the behavior of the divergence of a vector field?3.24. Provide two examples of physical phenomena in which the divergence of the vector field is nonzero.

3-64

Review Questions (Continued)

3.25. State Stokes’ theorem and discuss its application.3.26. State the divergence theorem and discuss its application.3.27. What is the divergence of the curl of a vector?

3-65

Problem S3.7. Investigation of the behavior of the curl of a vector field for different cases

3-66

Problem S3.8. Investigation of the behavior of the divergence of a vector field for different cases

3-67

Problem S3.9. Verification of Stokes’ theorem and an application of the divergence theorem

3-68

Problem S3.9. Verification of Stokes’ theorem and an application of the divergence theorem (Continued)

The End