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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

Faraday’s Law

E • dl  – d

dt C

dS

S C

B

Wb m2  m , or Wb.2

E • dl =

C

 Voltage around C, also known as

electromotive force (emf) around C (but not really a force),

B • dS

S

 = Magnetic flux crossing S,

– d

dt S B • dS = Time rate of decrease of

magnetic flux crossing S,

Wb s, or V

V m m, or V.

Important Considerations

(1) Right-hand screw (R.H.S.) Rule

The magnetic flux crossing the

surface S is to be evaluated toward that side of S a right-hand screw

advances as it is turned in the sense of

C.

C

z R C

y Q

P x

O

(2) Any surface S bounded by C.

The surface S can be any surface bounded

by C For example:

This means that, for a given C, the values of

magnetic flux crossing all possible surfaces bounded by it is the same, or

the magnetic flux bounded by C is unique.

z C

y R

x P

Q O

(3) Imaginary contour C versus loop of wire.

There is an emf induced around C in either

case by the setting up of an electric field A loop of wire will result in a current flowing in the wire

(4) Lenz’s Law

States that the sense of the induced emf is

such that any current it produces, if the closed path were a loop of wire, tends to oppose the change in the magnetic flux that

produces it

Thus the magnetic flux produced by the

induced current and that is bounded

by C must be such that it opposes the change in

the magnetic flux producing the induced emf

(5) N-turn coil.

For an N-turn coil, the induced emf is N

times that induced in one turn, since the surface

bounded by one turn is bounded N

times by the N-turn coil Thus

emf – N d

dt

 

0 sin x cos y

0 sin

S d B  t

 B  S =

z

1

1

x

y C

D2.5

where  is the magnetic flux linked by one turn

0

sin cos V

C

d

dt









emf < 0

emf > 0

B0

0

–B0

emf

0

Lenz’s law is verified

 inc.



 dec

–B0

t

t

B0

(b) S B • dS

1

2 B0 sin t – 12 B0 cos t

 1

2 B0 sin t –  4

z

C

y x

1

1

1

E • dl

C



– d

dt

1

2 B0 sin t –  4





– B0

2 cos t –  4V

(c) S B • dS

 B0 sint  B0 cos t

 2 B0 sin t  

4



 

z

C

y x

1

E • dl

C



– d

dt  2 B0 sin t   4

– 2 B0 cos t  

4



 V

 

0

0 0 0

S d B ly

B l y v t

Motional emf concept

conducting bar

y  y0  v0t

C

B

v0ay

y

0 z

B



E • dl  – d

dt C

This can be interpreted as due to an electric field

induced in the moving bar, as viewed by an observer moving with the bar, since

E  F

Q v0B0ax

v0B0l x0 l v0B0a x • dx a x

x0 l E • dl

0 0

d B l y v t

dt

B lv

   



is the magnetic force on a charge Q in the bar

Hence, the emf is known as motional emf.

F Qv B

Qv0ay  B0az

Qv0B0a x