Lecture 11

1ECE 303 Fall 2007 Farhan Rana Cornell University

Lecture 11

Faradays Law and Electromagnetic Inductionand

Electromagnetic Energy and Power Flow

In this lecture you will learn:

More about Faradays Law and Electromagnetic Induction

The Non-uniqueness of Voltages in Magnetoquasistatics

Electromagnetic Energy and Power Flow

Electromagnetic Energy Stored in Capacitors and Inductors

Appendix (some proofs)

ECE 303 Fall 2007 Farhan Rana Cornell University

Faradays Law Revisited

adHt

adBt

sdE orrrrrr ...

==

A closed contour

BFaradays Law: The line integral of E-field over a closed contour is equal to ve of the time rate of change of the magnetic flux that goes through any arbitrary surface that is bounded by the closed contour

Important Note: In electroquasistatics the line integral of E-field over a closed contour was always zero

In magnetoquasistatics this is NOT the case

( ) == 0.. sdsdE rrr


Electromagnetic Induction and Kirchoffs Voltage Law

Now consider a circuit through which the magnetic flux is changing with time (Kirchoffs voltage law is violated)

+

+-

-

dtdadB

tsdE =

= rrrr

..

dtdVRIRIsdE =+= 21. r

r

1R

2R

I

( ) ( ) dtd

RRRRVI

2121

1++=

V+-( )t

+

+-

-

1R

2R

I V+-

Kirchoffs voltage law comes from the electroquasistaticapproximation: 0. = sdE r

r

0. 21 =+= VRIRIsdE rr

( )21 RRVI +=


Lenz Law

+

+-

-

Suppose an induced current I is flowing through the wire:

( ) dtd

RRI

21

1+=

1R

2R

IThe induced current in the wire produces its own magnetic field

Lenz Law is just an easy way to remember in which direction the induced current flows

The law states that the induced current will flow in a direction such that its own magnetic field opposes the time variation of the magnetic field that produced it

Example: Suppose the magnetic flux through the wire loop shown above wasincreasing with time (so that d/dt > 0). Lenz would tell us that the induced current would flow in the clockwise direction so that its own magnetic field would oppose the increasing magnetic flux through the loop

In the equation above this fact comes out from the negative sign on the right hand side

( )t


Non-Uniqueness of Voltages in Magnetoquasistatics - I

+

+-

-

( )dtdRRIsdE =+= 21. r

r

1R

2R

I

2V1V

Question: What is the voltages difference V2-V1 ?

One may be tempted to write .

112

221

VRIVVRIV

==

00

12 ==

VVI

.. which cannot be correct since we know that:

We have:

What went wrong? Our usual concepts of circuit theory and potentials which are based on conservative E-fields are not valid when time varying magnetic fields are present

( ) dtd

RRI

21

1+=

( )t


Non-Uniqueness of Voltages in Magnetoquasistatics - II

+

+-

-

1R

2R

I

2V1V

Lets do some real experimental measurements and see what we get for V2-V1

+-

V

C1

C2

Faradays Law for contour C1:

voltmeter

dtdRIRI

tadBsdE

=+

=

21

..rrrr


112

1 0

..

IRVVVRIV

tadBsdE

===

=

rrrr

+== dt

dRR

RVVV 21

112

This value for V2-V1 would actually be measured experimentally if the measurement is done as shown above

( )t


Non-Uniqueness of Voltages in Magnetoquasistatics - III

+

+-

-

1R

2R

I

2V1V

Lets do another real experimental measurement and see what we get for V2-V1

+-

V

C1

C3


voltmeter

dtdRIRI

tadBsdE

=+

=

21

..rrrr


212

2 0

..

IRVVVRIV

tadBsdE

===

=rrrr

+== dt

dRR

RVVV 21

212

This value for V2-V1 would actually be measured experimentally if the measurement is done as shown above

Lesson: The result of a voltage measurement depends on how exactly you do the measurement when time varying magnetic fields are present

( )t


Inductors and Faradays Law - I

Consider an inductor with a time varying current:

+_

( )tIInductance = L

C

The magnetic flux is given by:

( ) ( )tILt =

Consider Faradays law over the closed contour C:

adHt

adBt

sdE orrrrrr

... =

=

( )tV

( ) ( ) ( )dt

tIdLdt

tdtV ==


Inductors and Faradays Law - II

+_

( )tIInductance = L

CConsider Faradays law over the new closed contour C:

( ) ( )( )2

.

21.

...

vertical

vertical

tVsdE

dttdsdEtV

adHt

adBt

sdE o

=

=+

=

=

rr

rr

rrrrrr

( )tV

Question: is there electric field anywhere else besides at the terminals??

There must be E-field inside the loop !!

This E-field is not very easy to determine analytically.


The Current-Charge Continuity Equation in Electromagnetism - I

Start from:tEJH

+=rrr

Take divergence on both sides: ( ) ( )t

EJH +=

rrr ...0

( )t

EJ +=

rr ..0

Use Gauss law: = Er.

To get:t

J = r. Current-charge continuity equation

What does this equation mean??


Suppose the current density is spatially varying as shown below

x

x xx +( )xJx ( )xxJx +The difference in current density at x and x+x must result in piling up of charges in the infinitesimal strip .

( ) ( ) ( )t

xtxtxxJtxJ xx =+ ,,,

( ) ( ) ( )t

xtxx

txJtxxJ xx

=+ ,,,

( ) ( )t

txx

txJx

= ,,

Now generalize to 3D and get the desired result:

( ) ( )t

trtrJ = ,,.

rrr

The Current-Charge Continuity Equation in Electromagnetism - II


A continuity equation of the form:

( ) ( )t

trtrJ = ,,.

rrr

establishes a relation between the flow rate of a quantity (in the present case the flow rate of charge density) and the time rate of change of the quantity

In the next few slides we will try to find a power-energy continuity equation for the electromagnetic field

A continuity equation is in fact a conservation law

For example, the current-charge continuity equation expresses charge conservation. It says that charge cannot just disappear. In the integral form it becomes:

( ) ( ) ( )dt

tdQdVtrt

adtrJ == ,., rrrr

If there is a net inflow of charge into a closed volume then that must result in an increase in the total charge Q inside that closed volume

J

closed volume

The Current-Charge Continuity Equation in Electromagnetism - III


The Electromagnetic Power-Energy Continuity Equation - I

We know that electric and magnetic fields have energy. Let W be the energy density (i.e. energy per unit volume) of the electromagnetic field

( ) =trS ,rr

( ) =trW ,r Suppose we had a vector that expressed the energy flow rate (or energy flux) in Joules/(m2-sec) for the electromagnetic field

Energy density (units: Joules/m3)

Energy flow rate (units: Joules/(m2-sec))

Then the conservation of electromagnetic energy must be expressed in a continuity equation of the form:

( ) ( )t

trWtrS = ,,.

rrr

Compare with the current-charge continuity equation:

( ) ( )t

trtrJ = ,,.

rrr


The Electromagnetic Power-Energy Continuity Equation - II

Start from the two Maxwells equations:

tEJH

tHE o

+=

=rrr

rr

Take the dot-product of the first equation with and of the second equation with to get:

Er

Hr

( )( )

tEEEJEH

tHHHE o

+=

=

rrrrrr

rrrr

.21..

.21.

Subtract the above two to get:

( ) ( )t

EEEJt

HHEHHE o

=rrrrrrrrrr .

21..

21..


The Electromagnetic Power-Energy Continuity Equation - II

Use the vector identity on the left hand side: ( ) ( ) ( ) A.BB.ABA. rrrrrr = ( ) EJEEHH

tHE o

rrrrrrvr.

2.

2..- +

+

=

( ) ( )t

EEEJt

HHEHHE o

=rrrrrrrrrr .

21..

21..

Define: ( ) ( ) ( )trHtrEtrS ,,, rrrrrr =

( ) ( ) ( ) ( ) ( )

+=

2,.,

2,.,, trEtrEtrHtrHtrW o

rrrrrrrr

To get:

And get:

( ) ( ) ( ) ( )trEtrJt

trWtrS ,.,,,.rrrrrrr +

=This is what we wanted (other than the last term on the right hand side)


The Electromagnetic Power-Energy Continuity Equation - III

is called the Poynting vector and it is the energy flow per second per unit area

points in the direction of power flow

has the units of Joules/(m2-sec) or Watt/m2

( )trS ,rr

is the energy density of electromagnetic field

has the units of Joules/m3

has two parts:

( )trW ,r

Poynting Vector:

Energy Density: ( ) ( ) ( ) ( ) ( )

+=

2,.,

2,.,, trEtrEtrHtrHtrW o

rrrrrrrr

( )trW ,r

( ) ( )2

,., trHtrHorrrr

( ) ( ) ( )trHtrEtrS ,,, rrrrrr =

( )trS ,rr( )trS ,rr

( )trW ,r

( ) ( )2

,., trEtrErrrr

Energy density of the magnetic field

Energy density of the electric field


( ) ( ) ( ) ( )trEtrJt

trWtrS ,.,,,.rrrrrrr +

=

The Electromagnetic Power-Energy Continuity Equation - IVPower dissipation:

The last term in the continuity equation J.E indicates power dissipation

Its units are Joules/(m3-sec)

This term represents the energy that is lost from the electromagnetic field

Essentially what is happening is that if the medium is conductive then in the presence of electric fields currents will be produced and the last term represents the usual I 2R losses in conductors or resistors

Integral Form: ( ) ( ) ( ) ( )+= dVtrEtrJdVtrWtadtrS ,.,,.,

rrrrrrrr

S

closed volume

Net power flow into a closed volume

=Rate of increase of the total electromagnetic energy inside the closed volume

+Rate of the total energy loss through dissipation


Example: Parallel Plate Capacitor

V+ -

= V = 0area = A

( )

( ) ( )dV

xxxE

dxVx

x ==

=

21

++++++++

--------

0 x

d

Electric field energy density = ( ) ( ) 222

.

=dVxExE xx

Total electric field energy = ( ) ( ) ( ) ( )2

2

21

22.

CV

dAdVdAxExE xx

=

=

dAC =

For all (linear) capacitors the total stored electric field energy equals CV 2/2

Recall that:

10


Example: Parallel Plate Inductory

x

zK

d

W

zK

xH

WIK =

WIKHx ==

Wd

IdHL oxo ll ==

Magnetic field energy density =2

22.

=WIHH oxxo

Total magnetic field energy = ( ) ( )2

2

21

22.

LI

WdWIdWHH oxxo

=

= ll

The structure has length in the z-directionRecall that:

For all (linear) inductors the total stored magnetic field energy equals LI 2/2

(L is the total inductance)


Power Flow in Electroquasistatics and Circuit Theory - I

Consider a complex electrical circuit shown below:

+_1V

1I

+_2V 2I

+ _3V

3I

A complex electrical circuit

If you wanted to calculate the total electrical power P going into the circuit you would do the following sum:

44332211 IVIVIVIVIVPn

nn ++ ==

+_4V

4I

Where does this formula comes from?

11


Power Flow in Electroquasistatics and Circuit Theory - II

+_1V

1I

+_2V 2I

+ _3V

3I

A complex electrical circuit+_

4V4I

Start from the Poynting Vector: ( ) ( ) ( )trHtrEtrS ,,, rrrrrr =Total power flow into the closed volume

closed volume

( ) === adHEadSP rrrrr ..In electroquasistatics: =Er

Therefore: ( ) = adHP rr .Use the vector identity: ( ) ( ) ( )FFF rrr +=

( ) ( ) = adHadHP rrrr .. To get:

In magnetoquasistatics: JHrr =


Power Flow in Electroquasistatics and Circuit Theory - III

+_1V

1I

+_2V 2I

+ _3V

3I

A complex electrical circuit+_

4V4I

closed volume

( ) ( ) = adHadHP rrrr .. 0

( )==

=

nnn IVadJP

adHP

rr

rr

.

.

=n

nn IVP

curl of a vector integrated over a closed surface is always zero

Just what we wanted to prove

12


( ) ( )

QV

VQVQ

adEadE

adEadEadE

dVadE

dVEEdVE

dVEEU

SS

SSS

SSSSSS

SSSSSS

SSS

21

21

21

.21.

21

.21.

21.

21

21.

21

..21.

21

2.

2211

21

21

2121

2121

21

=

+=

=

=

+=

= =

=

++++

++++

++

rrrr

rrrrrr

rr

rrr

rr

V

S

S1

S2

0

0

V1=V

V2=0

Appendix: Energy Stored in Capacitors is QV/2


I

S

( )( )

I

sdAI

dVJA

dVHA

dVHAadAH

dVHAAHdVAH

dVHHU

C

S

S

SS

SS

S

o

21

.21

.21

.21

.21.

21

..21.

21

2.

=

=

=

=

+ =

+= =

=

rr

rr

rr

rrrrr

rrrrrr

rr

C

0

Appendix: Energy Stored in Inductors is I/2

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Lecture 11