Induction

Induction - Spring 2006Induction - Spring 2006 11

InductionInduction

March 29, 2006March 29, 2006


Calendar…Calendar…

Today we finish up some material from Today we finish up some material from the last chapter and begin the chapter on the last chapter and begin the chapter on induction.induction.

Friday – Quiz on Friday – Quiz on LASTLAST chapter (30) chapter (30) Next Friday … still likely date for the next Next Friday … still likely date for the next

exam.exam. If you have a problem with this date please If you have a problem with this date please

email me with reason and we will try to email me with reason and we will try to figure out how to deal with it.figure out how to deal with it.


Let’s Finish Some Let’s Finish Some DetailsDetailsDisplacement Current Displacement Current

Induction - Spring 20064

Magnetic Flux

SurfaceOpen - Gauss Like AB d

For a CLOSED Surface we might expect this to be equal to some constant times the

enclosed poles … but there ain’t no such thing!

0 AB d


Examples

S N


Consider the poor little capacitor…

i i

CHARGING OR DISCHARGING …. HOW CAN CURRENTFLOW THROUGH THE GAP??


Through Which Surface Do we measure the current for Ampere’s Law?

I=0


In the gap… DISPLACEMENT CURRENT

dt

dI

ntDisplaceme

I

dt

dq

Let

dt

dq

dt

d

qEA

d

d

Current

1

S through FLUX The

0

0

0

2

Induction - Spring 2006 9


From The Demo ..


Faraday’s Experiments

??


Insert Magnet into Coil


Remove Coil from Field Region


That’s Strange …..

These two coils are perpendicular to each otherThese two coils are perpendicular to each other


Definition of TOTAL Definition of TOTAL ELECTRIC FLUX through a ELECTRIC FLUX through a surface:surface:

dA

is surface aLEAVING Field

Electric theofFlux Total

out

surfaced

nE


Magnetic Flux:

THINK OFMAGNETIC FLUX

as the“AMOUNT of Magnetism”

passing through a surface.Don’t quote me on this!!!


Consider a Loop Magnetic field passing

through the loop is CHANGING.

FLUX is changing. There is an emf

developed around the loop.

A current develops (as we saw in demo)

Work has to be done to move a charge completely around the loop.

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Faraday’s Law (Michael Faraday)

For a current to flow around the circuit, there must be an emf.

(An emf is a voltage) The voltage is found to

increase as the rate of change of flux increases.



Faraday’s Law (Michael Faraday)

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demf

Law sFaraday'

We will get to the minus sign in a short time.


Faraday’s Law (The Minus Sign)


Using the right hand rule, wewould expect the directionof the current to be in thedirection of the arrow shown.


Faraday’s Law (More on the Minus Sign)


The minus sign means that the current goes the other way.

This current will produce a magnetic field that would be coming OUT of the page.

The Induced Current therefore creates a magnetic field that OPPOSES the attempt to INCREASE the magnetic field! This is referred to as Lenz’s Law.


How much work?


dt

ddVqW

sE/

ChargeWork/Unit

A magnetic field and an electric field areintimately connected.)

emf


The Strange World of Dr. Lentz


MAGNETIC FLUX

This is an integral over an OPENOPEN Surface. Magnetic Flux is a Scalar

The UNIT of FLUX is the weber 1 weber = 1 T-m2

AB dB


We finally stated

dt

ddVemf

sE

FARADAY’s LAW


From the equation

dt

ddVemf

sE

AB dB

LentzLentz


Flux Can Change

If B changes If the AREA of the loop changes Changes cause emf s and currents and

consequently there are connections between E and B fields

These are expressed in Maxwells Equations

AB dB


Maxwell’s Equations(Next Course .. Just a Preview!)

Gauss

Faraday


Another View Of That damned minus sign again …..SUPPOSE that B begins to INCREASE its MAGNITUDE INTO THE PAGE

The Flux into the page begins to increase.

An emf is induced around a loop

A current will flow That current will create a

new magnetic field. THAT new field will change

the magnetic flux.



Lenz’s Law

Induced Magnetic Fields always FIGHT to stop what you are trying to do!i.e... Murphy’s Law for Magnets


Example of Nasty Lenz

The induced magnetic field opposes thefield that does the inducing!


Don’t Hurt Yourself!

The current i induced in the loop has the directionsuch that the current’s magnetic field Bi opposes thechange in the magnetic field B inducing the current.


Let’s do theLentz Warp

again !


Lenz’s Law

An induced current has a directionsuch that the magnetic field due tothe current opposes the change in the magnetic flux that induces thecurrent. (The result of the negative sign!) …

OR

The toast will always fall buttered side down!


An Example

The field in the diagramcreates a flux given byB=6t2+7t in milliWebersand t is in seconds.

(a)What is the emf whent=2 seconds?

(b) What is the directionof the current in the resistor R?


This is an easy one …

mVemf

tdt

demf

ttB

31724

seconds 2at t

712

76 2

Direction? B is out of the screen and increasing.Current will produce a field INTO the paper (LENZ). Therefore current goes clockwise and R to left in the resistor.


Figure 31-36 shows two parallel loops of wire having a common axis. The smaller loop (radius r) is above the larger loop (radius R) by a distance x >> R. Consequently, the magnetic field due to the current i in the larger loop is nearly constant throughout the smaller loop. Suppose that x is increasing at the constant rate of dx/dt = v. (a) Determine the magnetic flux through the area bounded by the smaller loop as a function of x. (Hint: See Eq. 30-29.) In the smaller loop, find (b) the induced emf and (c) the direction of the induced current.

v


B is assumed to be constant through the center of the small loop and caused by the large one.


The calculation of Bz

2/322

20

2/122220

2/122

220

2

4

cos

4coscos

xR

iRB

Rdds

xR

R

xR

idsdB

xR

R

xR

idsdBdB

z

z

z


More Work

In the small loop:

Vx

iRr

dt

demf

x

iRr

xR

iRrBrAB zz

4

20

2

3

20

2

2/322

20

22

2

3

2

)prescribed asAway (Far RFor x

2

dx/dt=v


Which Way is Current in small loop expected to flow??


What Happens Here?

Begin to move handle as shown.

Flux through the loop decreases.

Current is induced which opposed this decrease – current tries to re-establish the B field.


moving the bar

R

BLv

R

emfi

BLvdt

dxBL

dt

demf

BLxBAFlux

sign... minus theDropping


Moving the Bar takes work

v

R

vLBP

vR

vLBP

FvFxdt

d

dt

dWPOWER

R

vLBF

orR

BLvBLBiLF

222

22

22


What about a SOLID loop??

METAL Pull

Energy is LOSTBRAKING SYSTEM


Back to Circuits for a bit ….


Definition

Current in loop produces a magnetic fieldin the coil and consequently a magnetic flux.

If we attempt to change the current, an emfwill be induced in the loops which will tend tooppose the change in current.

This this acts like a “resistor” for changes in current!


Remember Faraday’s Law

dt

ddVemf

sE

Lentz


Look at the following circuit:

Switch is open NO current flows in the circuit. All is at peace!


Close the circuit…

After the circuit has been close for a long time, the current settles down.

Since the current is constant, the flux through

the coil is constant and there is no Emf. Current is simply E/R (Ohm’s Law)



When switch is first closed, current begins to flow rapidly.

The flux through the inductor changes rapidly. An emf is created in the coil that opposes the

increase in current. The net potential difference across the resistor is the

battery emf opposed by the emf of the coil.



dt

demf

0

)(

dt

diRV

notationVEbattery


Moving right along …

0

solonoid, aFor

N. turns,ofnumber the toas wellas

current the toalproportion isflux The

0

)(

dt

diLiRV

dt

diL

dt

d

NLii

dt

diRV

notationVE

B

battery


Definition of Inductance L

i

NL B

UNIT of Inductance = 1 henry = 1 T- m2/A

is the flux near the center of one of the coilsmaking the inductor


Consider a Solenoid

n turns per unit lengthniB

or

nliBl

id enclosed

0

0

0

sBl


So….

AnlL

or

AlnL

ori

niAnl

i

nlBA

i

NL B

2

20

0

lengthunit

inductance/

Depends only on geometry just like C andis independent of current.


Inductive Circuit

Switch to “a”. Inductor seems like a

short so current rises quickly.

Field increases in L and reverse emf is generated.

Eventually, i maxes out and back emf ceases.

Steady State Current after this.

i


THE BIG INDUCTION

As we begin to increase the current in the coil The current in the first coil produces a

magnetic field in the second coil Which tries to create a current which will

reduce the field it is experiences And so resists the increase in current.


Back to the real world…

i

0

equationcapacitor

theas form same

0

:0 drops voltageof sum

dt

dqR

C

qE

dt

diLiRE

Switch to “a”


Solution

R

L

eR

Ei LRt

constant time

)1( /


Switch position “b”

/

0

0

teR

Ei

iRdt

diL

E


Max Current Rate ofincrease = max emfVR=iR

~current


constant) (time

)1( /

R

L

eR

Ei LRt

Solve the lo

op equation.


IMPORTANT QUESTION

Switch closes. No emf Current flows for a

while It flows through R Energy is conserved

(i2R)

WHERE DOES THE ENERGY COME FROM??


For an answerReturn to the Big C

We move a charge dq from the (-) plate to the (+) one.

The (-) plate becomes more (-)

The (+) plate becomes more (+).

dW=Fd=dq x E x d+q -q

E=0A/d

+dq


The calc

2

0

2

020

2

00

22

0

2

00

00

2

1

eunit volum

energy

2

1

2

1

2

1)(

2

2

)()()(

E

E

u

AdAdAd

AA

dW

or

q

A

dqdq

A

dW

dA

qdqddqEddqdW

The energy is inthe FIELD !!!


What about POWER??

Ridt

diLiiE

i

iRdt

diLE

2

:

powerto

circuit

powerdissipatedby resistor

Must be dWL/dt


So

2

2

2

12

1

CVW

LiidiLW

dt

diLi

dt

dW

C

L

L

Energystoredin the

Capacitor


WHERE is the energy??

l

Al

NiBA

l

Ni

niB

nilBll

id enclosed

0

0

0

0

0

B

or

0

sB


Remember the Inductor??

turn.onegh flux throu MagneticΦ

current.

inductorin turnsofNumber

i

Ni

NL

?????????????


So …

l

AiN

l

NiANiW

l

NiA

iNi

NiLiW

L

Ni

i

NL

2220

0

0

0

22

2

1

2

1

2

1

2

1

2

1


2

0

2

0

22

0

0

2220

0

2

1

or

(volume) 2

1

2

1

B

:before From

2

1

BV

Wu

VBl

AlBW

l

Ni

l

AiNW

ENERGY IN THEFIELD TOO!


IMPORTANT CONCLUSION

A region of space that contains either a magnetic or an electric field contains electromagnetic energy.

The energy density of either is proportional to the square of the field strength.

Induction

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