Section 1 5 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

The Electric Field

The Electric Field

is a force field acting on charges by virtue of the property of charge

Coulomb’s Law

R

F1 a21Q1

Q2 a12

F2 F1  Q1Q2

40R2 a21

F2  Q2Q1

40R2 a12

0 permittivity of free space

10–9

36 F / m

a Q

2a

D1.13(b)

From the construction, it is evident that the resultant force is directed away from the center of the square The magnitude of this resultant force is given by

Q2/40(2a2)

Q2/40(4a2)

Q2/40(2a2)

Q  40

   

2 2

2

4

2

0.957

a a

a



Electric Field Intensity, E

is defined as the force per unit charge experienced by

a small test charge when placed in the region of the field

Thus

Units:

E Lim

q 0

F

q

Fe qE

N

C N – mC • m  Vm

qE

q

E

–q

–qE

Sources: Charges;

Time-varying magnetic field

 

2 0

2 0

4

4 due to

R

R

Qq R Q q

R









a E

aR

q R

Q

2 0

due to

Q Q

R



Electric Field of a Point Charge

(Coulomb’s Law)

Constant magnitude surfaces

are spheres centered at Q.

Direction lines are radial lines

emanating from Q.

E due to charge distributions

(a) Collection of point charges

Q n

Q3

Q2

Q1 R1

R2

R3

R n

aR n

aR3

aR2

aR1

40R2j aRj

j 1

n



E

Q

aR

R

Q (> 0) d

x

e

d Q (> 0)

y

z

d 2 + x 2  d 2 + x 2

Electron (charge e and mass m) is displaced from the origin by  (<< d) in the +x-direction and released from rest at t = 0 We wish to obtain differential

equation for the motion of the electron and its

solution

For any displacement x,

is directed toward the origin,

and x   d.

 F – Q e x

20d3 ax

2 2 0

3 2

2 2 0

4 2

x

x

Q e

d x

Q e x

d x















a

The differential equation for the motion of the electron is

Solution is given by

m d2 x

dt2 –

Qe x

20d3

d2 x

dt2 

Q e

2m0d3 x 0

2m0d3t  B

Using initial conditions and at t = 0,

we obtain

which represents simple harmonic motion about the origin with period

x  dx

dt 0

2m0d3 t

2 Qe

2m0d3 .

(b) Line Charges

Line charge density, L (C/m)

(c) Surface Charges

Surface charge density, S (C/m2)

(d) Volume Charges

Volume charge density,  (C/m3)

P dl

dS

dv

z a

dz

–a

ar

y

E

r





L L0 40 C m

 2 2 

0

For a  , E  L0

20r ar

 0 3 2

0 2 2 0

0

2 2 2

0

0

2 4

2

2 2

r

r

a

r z

a L

r z

L

dz

r z

z

r r z

a

r r a r r a

























a

Infinite Plane Sheet of Charge

of Uniform Surface Charge Density

z

z

y

dy x

y

 z2

y2

S0



dE z 2 S0 dy

20 y2  z2 cos 

 S0

0

z dy

y2  z2

E z  S0 z

0

dy

y2  z2

y0



 S0 z

0

1

z 0 d

 2

 S0

20

E  S0

20 az for z  0

 S0

20 an

+ + + + +

z < 0

z = 0 z

z > 0

S0

S0

– S0

2 0 az

S1 S2 S3

E(3,5,1) 0 V m

S

1 2 3 0

1 2 3 0

1 2 3 0

0 2

2

2







2

2 6 C m0

S

Solving, we obtain

2

3 2 C m0

S

   (d) E 2,1, 6   4 V maz

(a)

(c)

(b)