EM Course Module 3 for 2009 India Programs

8/12/2019 EM Course Module 3 for 2009 India Programs

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Fundamentals of Electromagneticsfor Teaching and Learning:

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

Computer- Related Engineering Departments inEngineering Colleges in India

by

Nannapaneni Narayana RaoEdward C. Jordan Professor Emeritus

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

Amrita Vishwa Vidyapeetham, India


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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, Karnataka

June 22 July 3, 2009


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Module 3Maxwells Equations in Differential Form

3.1 Faradays law and Amperes Circuital Law 3.2 Gauss Laws and the Continuity Equation 3.3 Curl and Divergence


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Instructional Objectives16. Obtain the simplified forms of Faradays law and

Amperes circuital law in differential forms for any special cases of electric and magnetic fields, respectively,or the particular differential equation that satisfies bothlaws for a special case of electric or magnetic field

17. Determine if a given time-varying electric/magnetic fieldsatisfies Maxwells curl equations, and if so find the corresponding magnetic/electric field, and any requiredcondition, if the field is incompletely specified

18. Find the magnetic field due to one-dimensional static

current distribution using Maxwells curl equation for the magnetic field19. Find the electric field due to one-dimensional static

charge distribution using Maxwells divergence equation for the electric field


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Instructional Objectives (Continued)20. Establish the physical realizability of a static electric field

by using Maxwells curl equation for the static case, and of a magnetic field by using the Maxwells divergence equation for the magnetic field

21. Investigate qualitatively the curl and divergence of avector field by using the curl meter and divergence meterconcepts, respectively

22. Apply Stokes and divergence theorems in carrying out vector calculus manipulations


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3-5

3.1 Faradays Law and Ampres Circuital Law (EEE, Sec. 3.1; FEME, Secs. 3.1, 3.2)


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3-6

Maxwells 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 becomputed 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


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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) = 2 x + C

Thus the solution is unique to within a constant.

Ifdydx 2, what is y?

121

00

2 or 2 or 42

4

xCx dx C C

y x x


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3-8

FARADAYS LAW

First consider the special case

and apply the integral form to the rectangular pathshown, 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 )

z y

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


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3-9

,00

Lim y x z

x z

d

dt B x z

x z

y x B E

z t

SC

d d d

dt E l B S

, x x y z z z x z d E x E x B x z dt

00

Lim x x z z z

x z

E E x x z


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3-10

General Case

E

E x ( x, y, z , t )a x

E y( x, y, z , t )a y

E z ( x, y, z , t )a z H H x ( x, y, z , t )a x H y( x, y, z , t )a y H z ( x, y, z , t )a z

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


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E E x ( x, y, z , t )a x E y( x, y, z , t )a y E z ( x, y, z , t )a z

H H x ( x, y, z , t )a x H y( x, y, z , t )a y H z ( x, y, z , t )a z

Lateral spacederivatives of thecomponents of E

Time derivatives ofthe components of B

y z x

y z x

y z x

E E B

y z t B E E

z x t

E B E x y t


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The terms on the left sides are the net right-lateraldifferentials of pairs of components of E . Forexample, in the first equation, it is the net right-lateral differential of E y and E z normal to the x-direction. The figure below illustrates (a) the case of

zero value, and (b) the case of nonzero value, for thisquantity.

3-12

z E

y E

(a) (b) y E y E

y E

z E z E z E x y

z


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Combining into a single differential equation,

Differential formof Faradays Law

a a a

B x y z

x y z

x y z t

E E E

B

Et

a a a x y z x y z

Del Cross or Curl of = B

t

E E


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3-14

AMPRES CIRCUITAL LAW

Consider the general case first. Then noting that

we obtain from analogy,

E t (B )

E d l d dt

B d SS C

H

d l

J

d S d

dt D

d SS S C

H J t

(D )


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Thus

Special case:

Differential formof Amprescircuital law

E E x ( z , t )a x , H H y( z , t )a y

y x

x

H D J

z t

DH Jt

0 0

0 0

J

a a a

D x y z

y

z t H


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3-16

E3.1 For

in free spacefind the value(s) of k such that E satisfies both

of Maxwells curl equations.

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

0 0, , , J = 0

x D

t

y

x

H J z

,B

Et

80 cos 6 10E a y E t kz


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8

0

8

0

cos 6 10

sin 6 10

y x E B

t z

E t kz z

kE t kz

80 8 cos 6 106 10 xkE

B t kz

0 0

0 0

a a a

BE

x y z

y

t z E


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Thus,

Then, noting that we have fromH H x

( z , t )a x

,

,D

Ht

80 8

70

802

cos 6 106 10

4 10

cos 6 10240

B a

B BH

a

x

x

kE t kz

kE t kz


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0 0

0 0

x y z

x

t z H

a a aD

H

2

802 sin 6 10240

y x D H

t z k E

t kz


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2

803 8 cos 6 101440 10 y

k E D t kz

2

803 8 cos 6 101440 10D a y

k E t kz

90

280

2

10 36

cos 6 104

D DE

a yk E

t kz


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k 2

3 10 8 ( c) m s .

Comparing with the original given E , we have

20

0 24k E

E

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

80 cos 6 10 2E a y E t z


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E3.2.


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Review Questions

3.1. Discuss the applicability of integral forms of Maxwells equations versus that of the differential forms forobtaining the solutions for the fields.

3.2. State Faradays law in differential form for the special case of E = E x( z , t )a x and H = H y( z , t )a y . How is itderived from Faradays law in integral form?

3.3. How would you derive Faradays law in differential form from its integral form for the general case of an arbitraryelectric 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-lateraldifferential of E x and E y normal to the z -direction iszero, although the individual derivatives are nonzero.

3.5. What is the determinant expansion for the curl of a vectorin Cartesian coordinates?


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Review Questions (Continued)

3.6. State Amperes circuital law in differential form for the general case of an arbitrary magnetic field. How is itobtained from its integral form?

3.7. State Amperes circuital law in differential form for the special case of H = H y( z , t )a y . How is it derived from theAmperes circuital law for the general case in differential form?

3.8. If a pair of E and B at a point satisfies Faradays law in differential form, does it necessarily follow that it alsosatisfies Amperes circuital form and vice versa?

3.9. Discuss the determination of magnetic field for onedimensional current distributions, in the static case, usingAmperes circuital law in differential form, without the displacement current density term.


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Problem S3.1. Obtaining the differential equation for aspecial case that satisfies both of Maxwells curl equations


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Problem S3.2. Finding possible condition for a specifiedfield to satisfy both of Maxwells curl equations


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


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


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


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3.2 Gauss Laws and

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


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GAUSS LAW FOR THE ELECTRIC FIELD D d S

S dv

V z

( x, y, z ) y

x

z

y

x

x x x x x

y y y y y

z z z z z

D y z D y z

D z x D z x

D x y D x y

x y z


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000

Lim x y z

x y z x y z

+

000

Lim

x x x x x

y y y y y

z z z z z x y z

D D y z

D D z x

D D x y

x y z


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The quantity on the left side is the net longitudinaldifferential of the components of D , that is, thealgebraic sum of the derivatives of components of Dalong their respective directions. It can be written as which is known as the divergence of D .Thus, the equation becomes

Longitudinal derivativesof the components of

D

y z x D D D

x y z

D

,D


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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

z D

y D

x D

y D y D

y D

x D x D x D

z D

z D

z D x

y

z

(a) (b)


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

0

x = a x =0 x = a

0 and 0, so that

y z

x D x

D y z x D D D

x y z


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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 thefollowing:

D = gives

( ) x D

x x

x

x D x dx C


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a 0 a x

0

a 0 a x

( x) dx x

2 0 a

From symmetry considerations, the fields onthe two sides of the charge distribution must

be equal in magnitude and opposite indirection. Hence,

C = 0a

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0 a

0a

a a x

D x

0

0

0

for

for

for

a

D a

a

x

x

x

a x a

x a x a

a x a


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B d S = 0 = 0 dvV S

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 magneticfield.

D

d S = dvV S

D

B 0


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LAW OF CONSERVATION OF CHARGE

J d S d dt dv 0V S

J t ( ) 0

J t 0 ContinuityEquation


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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

B

E

DH J

D

B

t

t

0

J t


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The interdependence of fields and sourcesthrough Maxwells equations

+

+H ,BJ

D,E

Amperes

CircuitalLaw (2)

Faradays Law (1)

Gauss Law for E (3)

Law of Conservationof Charge (5)


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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 thecomponents of a vector field? Give an example in

which the net longitudinal differential of thecomponents of a vector field is zero, although theindividual derivatives are nonzero.

3.12. What is the expression for the divergence of a vector inCartesian coordinates?

3.13. Discuss the determination of electric field for onedimensional charge distributions, in the static case,using Gauss law for the electric field in differential form.

3 4


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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 berealized as a magnetic field?

3.16. State the continuity equation.3.17. Summarize Maxwells equations in differential form and the continuity equation, stating which of theequations are independent.

3.18. Discuss the interdependence of fields and sourcesthrough Maxwells equations.

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

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

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

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

and 3.6, App. B)

3 50


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Maxwells Equations in Differential Form

Curl

Divergence

=

=

t

t

B E

D H J

D

0B

x y z

x y z

x y z

A A A

a a a

=A x y z A A A

x y z

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Curl and Divergence in Cylindrical Coordinates

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=

a aa

A

r z

r z

r r

r z A rA A

1 1=A z r A A

r Ar r r z

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Curl and Divergence in Spherical Coordinates

2 sinsin

=

sin

aaa

A

r

r

r r r

r A rA r A

221 1= sinsin1

sin

A r r A Ar r r A

r

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Basic definition of curl

A is the maximum value of circulation of A perunit 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

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Curl Meter

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

E3.4 0 sin for 0 <


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negative for 02

positive for2

y

a x

a x a

v

0

0 0

cos for 0

Documents

EM Course Module 3 for 2009 India Programs