Induction and Inductance Chapter 30 Magnetic Flux

Induction and Inductance Chapter 30

Magnetic Flux

Gauss Like AB d

Insert Magnet into Coil

Remove Coil from Field Region

From The Demo ..

First experiment Second experiment

Faraday’s Experiments

??

That’s Strange …..

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

Definition of TOTAL ELECTRIC FLUX through a 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.

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

MAGNETIC FLUX

This is an integral over an OPEN Surface.

Magnetic Flux is a Scalar

The UNIT of FLUX is the weber1 weber = 1 T-m2

AB dB

We finally stated

dt

demf

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(chapter 32 .. Just a Preview!)

Gauss

Faraday

Another View Of That hopeless 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!

Example of 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.

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!) …

#1 CHAPTER 30

The field in the diagramcreates a flux given byFB=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.

#21 Figure 30-50 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. 29-27.) 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 theCoil

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.

10. A uniform magnetic field B increases in magnitude with time t as given by Fig. 30-43b, where the vertical axis scale is set by Bs=9 mT and the horizontal scale is set by ts=3 s . A circular conducting loop of area A= 8x10-4 m2 lies in the field, in the plane of the page. The amount of charge q passing point A on the loop is given in Fig. 30-43c as a function of t, with the vertical axis scale set by qs=3 mC and the horizontal axis scale again set by ts=3 s. What is the loop's resistance?

29. If 50.0 cm of copper wire (diameter=1mm ) is formed into a circular loop and placed perpendicular to a uniform magnetic field that is increasing at the constant rate of 10.0 mT/s, at what rate is thermal energy generated in the loop?

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Induction and Inductance Chapter 30 Magnetic Flux