Section 3 1 TRƯỜNG ĐIỆN TỪ

No Slide Title Slide Presentations for ECE 329, Introduction to Electromagnetic Fields, to supplement “Elements of Engineering Electromagnetics, Sixth Edition” by Nannapaneni Narayana Rao Edward C Jor[.]

Introduction to Electromagnetic Fields,

to supplement “Elements of Engineering

Electromagnetics, Sixth Edition”

by

Nannapaneni Narayana Rao

Edward C Jordan Professor of Electrical and Computer Engineering

University of Illinois at Urbana-Champaign, Urbana, Illinois, USA

Distinguished Amrita Professor of Engineering Amrita Vishwa Vidyapeetham, Coimbatore, Tamil Nadu, India

Faraday’s Law and

Ampère’s Circuital Law

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

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

Cx dx 2 or C x2

2





 

01

 10 2 or C 4

 y(x) 4x.

If dy

dx 2, what is y?

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

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

(x, z)

z (x, z + z)

x

(x + x, z) (x + x, z + z)

z y

dt





 E x z z x E x z x d  B y x z, x z

dt

         

   

0

0

Lim x z z x z

x

z

x z



 

 

0 0

Lim y x z

x z

d

dt B x z

x z

 

 

   

 



 

y

E



 

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

E z

y –

E y

B x

t

E x

z –

E z

B y

t

E y

E x

B z

t

Lateral space derivatives of the

components of E

Time derivatives of

the components of B

ax ay az



x



y



z

E x E y E z

– B

t

 E – B

t

Differential form

of Faraday’s Law

 ax 

x  ay



y  az



z

Del Cross E or Curl of E = – B

t

AMPÈRE’S CIRCUITAL LAW

Consider the general case first Then noting that

we obtain from analogy,

E –

t (B)

E • dl  – d

dt S B • dS

C







 

H • dl  J • dS  ddt D • dSS S C



H J  t(D)





Special case:

 H J 

t of Ampère’s circuital law

E  E x (z,t)a x , H  H y (z,t)a y

ax a y az

z

J  D

t

– H y

z  J x 

D x

t

 8 

0 cos 6 10 y

H y

z  – J x –

D x

t

Ex For

in free space

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

of Maxwell’s curl equations.

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

 E – B

t ,

      0,  J = 00,  ,

 

8 0

8 0

y

B

z











B

t –  E  – 0 0



z

0





Then, noting that we have fromH  H x (z,t)a x ,

 H D

t ,

8

0 8

7 0

8

0 2

cos 6 10

6 10

4 10

cos 6 10 240

x

x















H

a

0 0

x

H

a a a

D × H

2

8

0

2 sin 6 10 240

y H x







D

 

2

8

0

3 8 cos 6 10

y

k E





2

8

0





9 0

2

8

0 2

10 36 cos 6 10







E

a

 8 

k 2

 3 108(c) m s.

Comparing with the original given E, we have

2 0

4

k E E





Sinusoidal traveling waves in free space, propagating in the

z directions with velocity,