magnetism1 prelude to an exam allegro con brio next friday – examination #2 next friday –...

MagnetismMagnetism 11

Prelude to an ExamPrelude to an ExamAllegro con brio Allegro con brio

Next Friday – EXAMINATION #2Next Friday – EXAMINATION #2 Watch those WebAssigns .. no more Watch those WebAssigns .. no more

extensions.extensions. Monday will beMonday will be

• A Quiz on CircuitsA Quiz on Circuits• A review of circuits and some other problems.A review of circuits and some other problems.

Wednesday, more on Magnetism. Only Wednesday, more on Magnetism. Only day 1 on the exm. Watch for a new day 1 on the exm. Watch for a new Webassign.Webassign.

MagnetismMagnetism 22

MagnetismMagnetism

A Whole New TopicA Whole New Topic

Magnetism 3

DEMO

Magnetism 4

Lodestone (Mineral)

• Lodestones attracted iron filings.

• Lodestones seemed to attract each other.

• Used as a compass.– One end always

pointed north.

• Lodestone is a natural magnet.

Magnetism 5

Magnetism

• Refrigerators are attracted to magnets!

Magnetism 6

ApplicationsApplications

• Motors• Navigation – Compass• Magnetic Tapes

– Music, Data

• Television– Beam deflection Coil

• Magnetic Resonance Imaging• High Energy Physics Research

Magnetism 7

Magnets

• Like Poles Repel• Opposite Poles

Attract• Magnetic Poles

are only found in pairs.– No magnetic

monopoles have ever been observed.

Shaded End is NORTH PoleShaded End of a compass points

to the NORTH.

S N

Magnetism 8

Observations

• Bring a magnet to a charged electroscope and nothing happens. No forces.

• Bring a magnet near some metals (Co, Fe, Ni …) and it will be attracted to the magnet.– The metal will be attracted to both the N and S

poles independently.– Some metals are not attracted at all.– Wood is NOT attracted to a magnet.– Neither is water.

• A magnet will force a compass needle to align with it. (No big Surprise.)

+

++

Magnetism 9

Magnets

Cutting a bar magnet in half produces TWO bar magnets, each with N and S poles.

Magnetic Field

Magnetism 10

Consider a Permanent Magnet

N S

B

Magnetism 11

Introduce Another Permanent Magnet

N S

B

N

S

The bar magnet (a magnetic dipole) wants to align with the B-field.

pivot

Magnetism 12

The south pole of the small bar magnet is attracted towards the north pole of the big magnet. Also, the small bar magnet (a magnetic dipole) wants to align with the B-field. The field attracts and exerts a torque on the small magnet.

Field of a Permanent Magnet

N S

B

N

S

Magnetism 13

Field of a Permanent Magnet

N S

B

N S

The field exerts a torque on the dipole

The bar magnet (a magnetic dipole) wants to align with the B-field.

Magnetism 14

The Magnetic Field

• Similar to Electric Field … exists in space.– Has Magnitude AND Direction.

• The “stronger” this field, the greater is the ability of the field to interact with a magnet.

Magnetism 15

Convention For Magnetic Fields

X

Field INTO Paper Field OUT of Paper

B

Magnetism 16

Experiments with Magnets Show

• Current carrying wire produces a circular magnetic field around it.

• Force on Compass Needle (or magnet) increases with current.

Magnetism 17

Current Carrying Wire

Current intothe page.

B

Right hand Rule-Thumb in direction of the currentFingers curl in the direction of B

Magnetism 18

Current Carrying WireCurrent Carrying Wire

• B field is created at ALL POINTS in space surrounding the wire.

• The B field had magnitude and direction.

• Force on a magnet increases with the current.

• Force is found to vary as ~(1/d) from the wire.

Magnetism 19

Compass and B Field

• Observations– North Pole of magnets

tend to move toward the direction of B while S pole goes the other way.

– Field exerts a TORQUE on a compass needle.

– Compass needle is a magnetic dipole.

– North Pole of compass points toward the NORTH.

MagnetismMagnetism 2020

Planet EarthPlanet Earth

MagnetismMagnetism 2121

Inside it all.Inside it all.

8000Miles

MagnetismMagnetism 2222

On the surface it looks like this..On the surface it looks like this..

MagnetismMagnetism 2323

Inside: Warmer than Inside: Warmer than FloriduhFloriduh

MagnetismMagnetism 2424

Much Warmer than FloriduhMuch Warmer than Floriduh

MagnetismMagnetism 2525

FinallyFinally

MagnetismMagnetism 2626

In BetweenIn Between

The molten iron core exists in a magnetic The molten iron core exists in a magnetic field that had been created from other field that had been created from other sources (sun…).sources (sun…).

The fluid is rotating in this field.The fluid is rotating in this field. This motion causes a current in the molten This motion causes a current in the molten

metal.metal. The current causes a magnetic field.The current causes a magnetic field. The process is self-sustaining.The process is self-sustaining. The driving force is the heat (energy) that The driving force is the heat (energy) that

is generated in the core of the planet.is generated in the core of the planet.

MagnetismMagnetism 2727

After molten lava emerges from a volcano, it solidifies to a rock. In most cases it is a black rock known as basalt, which is faintly magnetic, like iron emerging from a melt. Its magnetization is in the direction of the local magnetic force at the time when it cools down.

Instruments can measure the magnetization of basalt. Therefore, if a volcano has produced many lava flows over a past period, scientists can analyze the magnetizations of the various flows and from them get an idea on how the direction of the local Earth's field varied in the past. Surprisingly, this procedure suggested that times existed when the magnetization had the opposite direction from today's. All sorts of explanation were proposed, but in the end the only one which passed all tests was that in the distant past, indeed, the magnetic polarity of the Earth was sometimes reversed.

MagnetismMagnetism 2828

Ancient NavigationAncient Navigation

MagnetismMagnetism 2929

This planet is really screwed up!This planet is really screwed up!NORTHPOLE

SOUTH POLE

Magnetism 30

CompassDirection

Repeat

NavigationDIRECTION

N

S

If N directionis pointed to bythe NORTH poleof the Compass Needle, then the

pole at the NORTHof our planet must

be a SOUTH MAGNETICPOLE!

NavigationDIRECTION

S

N

And it REVERSES from time to time.

MagnetismMagnetism 3131

Magnetism 32

Rowland’s Experiment

RotatingINSULATING

Diskwhich is CHARGED

+ or –on exterior.

xxxxxx Bxxx

Field is created byany moving charge.

Increases with charge on thedisk.

Increases withangular velocity ofthe disk.

Electrical curent is a moving charge.

++ + + ++

Magnetism 33

A Look at the Physics

B

q

There is NO force ona charge placed into amagnetic field if thecharge is NOT moving.

Bq

v

• If the charge is moving, thereis a force on the charge,perpendicular to both v and B.

F = q v x B

There is no force if the chargemoves parallel to the field.

Magnetism 34

WHAT THE HECK IS THAT???

• A WHAT PRODUCT?• A CROSS PRODUCT – Like an angry

one??• Alas, yes ….

• F=qv X B

Magnetism 35

The Lorentz Force

This can be summarized as: F q B v

vF

Bqm

or:

F qvBsin

is the angle between B and V

Magnetism 36

Note

B is sort of the Force per unit (charge-velocity)

Whatever that is!!

Magnetism 37

Practice

Which way is the Force???

B and v are parallel.Crossproduct is zero.So is the force.

Magnetism 38

Units

m)-N/(A 1 T 1 tesla1

mAmp

N

/

:

)

sCm

N

qv

FB

Units

Bqv Sin(θF

Magnetism 39

teslas are

At the Surface of the Earth 3 x 10-5 T

Typical Refrigerator Magnet 5 x 10-3 T

Laboratory Magnet 0.1 T

Large Superconducting Magnet 10 T

Magnetism 40

The Magnetic Force is Different From the Electric Force.

Whereas the electric force acts in the same direction as the field:

The magnetic force acts in a direction orthogonal to the field:

And --- the charge must be moving !!

F q B v

F qE

(Use “Right-Hand” Rule to determine direction of F)

MagnetismMagnetism 4141

So…So…A moving charge can create a A moving charge can create a magnetic field.magnetic field.A moving charge is acted upon by a A moving charge is acted upon by a magnetic field.magnetic field.

In Magnetism, things move.In Magnetism, things move.In the Electric Field, forces and the In the Electric Field, forces and the field can be created by stationary field can be created by stationary charges.charges.

Magnetism 42

Trajectory of Charged Particlesin a Magnetic Field

+ + + +

+ + + +

+ + + +

+ + + +

+ + + +

vB

F

+ + + +

+ + + +

+ + + +

+ + + +

+ + + +

vB

F

(B field points into plane of paper.)

Magnetism 43

Trajectory of Charged Particlesin a Magnetic Field

+ + + +

+ + + +

+ + + +

+ + + +

+ + + +

+ + + +

+ + + +

+ + + +

+ + + +

+ + + +

vvB B

FF

(B field points into plane of paper.)

Magnetic Force is a centripetal force

Magnetism 44

Review of Rotational Motion

atar

at = r tangential acceleration

ar = v2 / r radial acceleration

The radial acceleration changes the direction of motion,while the tangential acceleration changes the speed.

r s = s / r s = r ds/dt = d/dt r v = r

= angle, = angular speed, = angular acceleration

Uniform Circular Motion

= constant v and ar constant but direction changes

ar = v2/r = 2 rF = mar = mv2/r = m2r

KE = ½ mv2 = ½ mw2r2

v

ar

Magnetism 45

Magnetism 46

+ + + +

+ + + +

+ + + +

+ + + +

+ + + +

Radius of a Charged ParticleOrbit in a Magnetic Field

vB

F

r

mr

q B

r mqB

v v

v

2

Centripetal Magnetic Force Force =

Note: as , the magneticforce does no work!

Fv

Magnetism 47

Cyclotron Frequency

+ + + +

+ + + +

+ + + +

+ + + +

+ + + +

vB

F

r

The time taken to complete one orbit is:

qB

m

rT

v

v

2v

2

m

qBf

m

qB

Tf

c

2

2

1

Magnetism 48

More Circular Type Motion in a Magnetic Field

Magnetism 49

Mass Spectrometer

Smaller Mass

Magnetism 50

Magnetism 51

Cyclotron Frequency

+ + + +

+ + + +

+ + + +

+ + + +

+ + + +

vB

F

r

The time taken to complete one orbit is:

qB

m

rT

v

v

2v

2

m

qBf

m

qB

Tf

c

2

2

1

Magnetism 52

An ExampleA beam of electrons whose kinetic energy is K emerges from a thin-foil “window” at the end of an accelerator tube. There is a metal plate a distance d from this window and perpendicular to the direction of the emerging beam. Show that we can prevent the beam from hitting the plate if we apply a uniform magnetic field B  such that

22

2

de

mKB

Magnetism 53

Problem Continued

r

22

22

2

e

2mKB

:Bfor Solvee

22

eB

m

2 so

2

1

qB

mvr

Before From

d

dB

mK

m

Kr

m

KvmvK

Magnetism 54

Some New Stuff

Magnetism and Forces

Magnetism 55

Let’s Look at the effect of crossed E and B Fields:

x

x xx

xx

•q , m

B

v

E

Magnetism 56

What is the relation between the intensities of the electric and magnetic fields for the particle to move in a straight line ?.

x

x xx

xx

•q m

B

v

E

•FEFB

FE = q E and FB = q v B

If FE = FB the particle will movefollowing a straight line trajectory

q E = q v B

v = E / B

MagnetismMagnetism 5757

What does this mean??What does this mean??

This equation only This equation only contains the E and contains the E and B fields in it.B fields in it.

Mass is missing!Mass is missing!

Charge is missing!Charge is missing!

This configuration This configuration is a velocity filter!is a velocity filter!

v = E / B

Magnetism 58

“Real” Mass Spectrometer

Create ions from injected species.This will contain various masses,

charges and velocities.These are usually accelerated to a

certain ENERGY (KeV) by an applied electric field.

The crossed field will only allow a selected velocity to go forward into the MS.

From before: R=mv/Bq

Magnetism 59

Components of MS:

applied

applied

Vv

m

q

So

qVmv

1

2

1

2

1

.difference potentialknown a through ions theAccelerate

2

2

The velocity can be selected via an E x B field and the MS willseparate by:

Bq

mvR Unknown is mass to charge ratio

which can be sorted from the spectrum

Magnetism 60

Magnetism 61

VECTOR CALCULATIONS

zyx

zyx

bbb

aaa

kji

ba

Magnetism 62

Problem: A Vector ExampleA proton of charge +e and mass m is projected into a uniform magnetic field B=Bi with an initial velocity v=v0xi +v0yj. Find the velocity at a later time.

yz

xx

yzzyx

vm

eBv

m

eB

dt

dvmma

mvveB

B

vvv

kji

e

dt

dv and

dt

dv 0

dt

dv

:components Equating

z. andy for etc.

)(

00

zyx

akjF

BvF

vx is constant

Magnetism 63

More

z.for thingSame

)cos(v

motion!circular Simple

0

m

eBLet

0y

22

2

22

2

tv

vdt

vd

vdt

dv

dt

vd

y

yy

yzy

Magnetism 64

Magnetism 65

Wires

• A wire with a current contains moving charges.

• A magnetic field will apply a force to those moving charges.

• This results in a force on the wire itself.– The electron’s sort of

PUSH on the side of the wire.

F

Remember: Electrons go the “other way”.

Magnetism 66

The Wire in More Detail

B out of plane of the paper

Assume all electrons are moving with the same velocity vd.

(i). charge POSITIVE of

motion theofdirection in the

:

L

BLF

i

vector

iLBBvv

LiBqvF

v

Litiq

dd

d

d

Magnetism 67

Magnetic Levitation

Current = img

Magnetic Force

Where does B point???? Into the paper.

iL

mgB

mgiLB

Magnetism 68

MagLev

Magnetism 69

Magnetic Repulsion

Magnetism 70

Detail

Magnetism 71

Moving Right Along ….

Magnetism 72

Acceleration

MagnetismMagnetism 7373

Don’t Buy A Ticket Quite Yet..Don’t Buy A Ticket Quite Yet..

This is still experimental.This is still experimental.

Much development still required.Much development still required.

Some of these attempts have been Some of these attempts have been abandoned because of the high cost abandoned because of the high cost of building a MagLev train.of building a MagLev train.

Probably 10-20 years out.Probably 10-20 years out.

Or More.Or More.

Magnetism 74

Current Loop

Loop will tend to rotate due to the torque the field applies to the loop.

What is forceon the ends??

Magnetism 75

The Loop

pivot

OBSERVATION

Force on Side 2 is outof the paper and that onthe opposite side is into the paper. No net forcetending to rotate the loopdue to either of these forces.

The net force on the loop is also zero,

Magnetism 76

An ApplicationThe Galvanometer

Magnetism 77

The other sides

1=F1 (b/2)Sin() =(B i a) x (b/2)Sin()

total torque on the loop is: 21

Total torque:

=(iaB) bSin() =iABSin()

(A=Area)

MagnetismMagnetism 7878

Watcha Gonna DoWatcha Gonna Do

Quiz TodayQuiz Today

Return to Magnetic MaterialReturn to Magnetic Material

Exams not yet returned. Sorry.Exams not yet returned. Sorry.

Magnetism 79

Wires

• A wire with a current contains moving charges.

• A magnetic field will apply a force to those moving charges.

• This results in a force on the wire itself.– The electron’s sort of

PUSH on the side of the wire.

F

Remember: Electrons go the “other way”.

Magnetism 80

The Wire in More Detail

B out of plane of the paper

Assume all electrons are moving with the same velocity vd.

(i). charge POSITIVE of

motion theofdirection in the

:

L

BLF

i

vector

iLBBvv

LiBqvF

v

Litiq

dd

d

d

Magnetism 81

Current Loop

Loop will tend to rotate due to the torque the field applies to the loop.

What is forceon the ends??

Magnetism 82

Last Time

1=F1 (b/2)Sin() =(B i a) x (b/2)Sin()

total torque on the loop is: 21

Total torque:

=(iaB) bSin() =iABSin()

(A=Area)

Magnetism 83

A Coil

)

: thereforeis coil on the torque

net the turns,N of COIL aFor

NiABSin(θτ

Normal to thecoil

RIGHT HAND RULE TO FIND NORMALTO THE COIL:

“Point or curl you’re the fingers of your right hand in the direction of the current and yourthumb will point in the direction of the normalto the coil.

Magnetism 84

Dipole Moment Definition

Define the magneticdipole moment ofthe coil as:

=NiA

We can convert thisto a vector with Aas defined as being normal to the area asin the previous slide.

Magnetism 85

Current Loop

Bμ

Aμ

and

Moment Magnetic Define

wire.of turnsN with coil aConsider

sin

Ni

iAB

Magnetism 86

A length L of wire carries a current i. Show that if the wire is formed into a circular coil, then the maximum torque in a given magnetic field is developed when the coil has one turn only, and that maximum torque has the magnitude … well, let’s see.

Circumference = L/N

N

Lr

N

Lr

2

2

Magnetism 87

Problem continued…

4

and 1N when Maximum

(BiA) 42

2

90 is angle the when

maximum is B),sin( since

2

2

22

2

2

o

iBL

N

LiB

N

NLiB

N

LNiB

rA

NiAB

Magnetism 88

Energy

aligned! be want toB and

-)U(

dipole electric theLike

B

Similar to Electric Dipole Moment

Similar to Electric Dipole Moment

Magnetism 89

The Hall Effect

Magnetism 90

What Does it Do?

• Allows the measurement of Magnetic Field if a material is known.

• Allows the determination of the “type” of current carrier in semiconductors if the magnetic field is known.• Electrons• Holes

Magnetism 91

Hall Geometry (+ Charge)

Current is moving to the right. (vd)

Magnetic field will force the charge to the top.

This leaves a deficit (-) charge on the bottom.

This creates an electric field and a potential difference.

Magnetism 92

Negative Carriers

Carrier is negative. Current still to the

right. Force pushes negative

charges to the top. Positive charge builds

up on the bottom. Sign of the potential

difference is reversed.

Magnetism 93

Hall Math

• Eventually, the field due to the Hall effect will allow the current to travel un-deflected through the conductor.

net

iB

newt

iwBV

wtAneA

iwBBwvV

neA

iv

AinevJ

BwvV

orw

VqqEBqv

balance

Hall

dHall

d

d

dHall

HallHalld

/

:

Magnetism 94

Magnetic Fields Due to Currents

Chapter 30

Magnetism 95

Try to remember…

30

20 4

1

4

1

r

dq

r

dq

rd

rrE

VECTOR UNITr

r

Magnetism 96

For the Magnetic Field,current “elements” create the

field.

DEFINITION BY

1026.1/104

44

:field fashion tosimilar aIn

770

30

20

TmATm

typermeabilir

id

r

id unit

rsrs

B

E

This is the Law ofBiot-Savart

Magnetism 97

Magnetic Field of a Straight Wire

• We intimated via magnets that the Magnetic field associated with a straight wire seemed to vary with 1/d.

• We can now PROVE this!

Magnetism 98

From the Past

Using Magnets

Magnetism 99

Right-hand rule: Grasp the element in your right hand with your extended thumb pointing in the direction of the current. Your fingers will then naturally curl around in the direction of the magnetic field lines due to that element.

Magnetism 100

Let’s Calculate the FIELD

Note:

For ALL current elements

ds X r

is into the page

Magnetism 101

The Details

02

0

20

)sin(

2B

it. DOUBLE and to0 from integrate

wesoamount equalan scontribute

wire theofportion Negative

)sin(

4

r

dsi

r

idsdB

Magnetism 102

Moving right along

R

i

Rs

rdsiB

SoRs

R

Rsr

22

)sin(sin

0

02/322

0

22

22

1/d

Magnetism 103

A bit more complicatedA finite wire

Magnetism 104

P1

)sin()sin(: NOTE

r

ds

2/122

20 )sin(

4

)sin(

)sin(

Rsr

r

dsidB

r

R

rdsrds

Magnetism 105

More P1

R

i

whenRL

L

R

iB

and

Rs

dsiB

L

L

2B

,L 42

4

0

22

0

2/

2/2/322

0

Magnetism 106

P2

22

0

0

2/322

0

4

4

Rs

L

R

iB

or

Rs

dsiRB

L

Magnetism 107

APPLICATION:Find the magnetic field B at point P in for i = 10 A and a = 8.0 cm.

Magnetism 108

Circular Arc of Wire

Magnetism 109

More arc…

Cpoint at 4

44

44

0

0

0

02

0

20

20

R

iB

dR

i

R

iRddBB

R

iRd

R

idsdB

Rdds ds

Magnetism 110

Howya Do Dat??

0rsdNo Field at C

Magnetism 111

Force Between Two Current Carrying Straight Parallel

ConductorsWire “a” createsa field at wire “b”

Current in wire “b” sees aforce because it is movingin the magnetic field of “a”.

Magnetism 112

The Calculation

d

iLi

iFd

iB

ba

b

a

2F

angles...right at are and Since

2

:calculatedjust what weis a"" wire

todue b"" at wire FIELD The

0

b""on

0b""at

BL

BL

Magnetism 113

Definition of the AmpereThe force acting between currents in parallel wires is the basis for the definition of the ampere, which is one of the seven SI base units. The definition, adopted in 1946, is this: The ampere is that constant current which, if maintained in two straight, parallel conductors of infinite length, of negligible circular cross section, and placed 1 m apart in vacuum, would produce on each of these conductors a force of magnitude 2 x 10-7 newton per meter of length.

MagnetismMagnetism 114114

TRANSITIONTRANSITION

AMPEREAMPERE

MagnetismMagnetism 115115

Welcome to Welcome to Andre’ Marie Ampere’s LawAndre’ Marie Ampere’s Law

Normally written as a “circulation” vector Normally written as a “circulation” vector equation.equation.

We will look at another form, but first…We will look at another form, but first…

Magnetism 116

Remember GAUSS’S LAW??

0enclosedq

d AESurfaceIntegral

Magnetism 117

Gauss’s Law

• Made calculations easier than integration over a charge distribution.

• Applied to situations of HIGH SYMMETRY.• Gaussian SURFACE had to be defined

which was consistent with the geometry.

• AMPERE’S Law is the Gauss’ Law of Magnetism! (Sorry)

Magnetism 118

The next few slides have been lifted from Seb Oliver

on the internet

Whoever he is!

Magnetism 119

Biot-Savart

• The “Coulombs Law of Magnetism”

20 ˆ

4 r

rdsB

id

Magnetism 120

Invisible Summary

• Biot-Savart Law – (Field produced by wires)– Centre of a wire loop radius R– Centre of a tight Wire Coil with N turns– Distance a from long straight wire

• Force between two wires

• Definition of Ampere

a

II

l

F

2

210

a

IB

2

0

R

IB

20

R

NIB

20

20 ˆ

4 r

rdsB

id

Magnetism 121

Magnetic Field from a long wire

r

IB

2

0

cosdsBdsB

I

B

r

ds1cos0 dsBdsB

dsr

I

2

0 dsB

Using Biot-Savart Law

Take a short vector on a circle, ds

Thus the dot product of B & the short vector ds is:

Magnetism 122

Sum B.ds around a circular path

I

B

r

ds

dsr

I

2

0 dsB

dsB

Sum this around the whole ring

dsr

I

2

0 dsr

I

2

0

rds 2 Ιrr

I0

0 22

dsB

Circumference of circle

Magnetism 123

Consider a different path

• Field goes as 1/r

• Path goes as r.• Integral

independent of r

0 dsB

i

Magnetism 124

SO, AMPERE’S LAWby SUPERPOSITION:

We will do a LINE INTEGRATIONAround a closed path or LOOP.

Magnetism 125

Ampere’s Law

enclosedid 0 sB

USE THE RIGHT HAND RULE IN THESE CALCULATIONS

Magnetism 126

The Right Hand Rule

Magnetism 127

Another Right Hand Rule

Magnetism 128

COMPARE

enclosedid 0 sB

0enclosedq

d AE

Line Integral

Surface Integral

Magnetism 129

Simple Example

Magnetism 130

Field Around a Long Straight Wire

enclosedid 0 sB

r

iB

irB

2

2

0

0

Magnetism 131

Field INSIDE a WireCarrying UNIFORM Current

Magnetism 132

The Calculation

rR

iB

andR

rii

irBdsBd

enclosed

enclosed

20

2

2

0

2

2

sB

Magnetism 133

R r

B

R

i

2

0

Magnetism 134

Procedure

• Apply Ampere’s law only to highly symmetrical situations.

• Superposition works.– Two wires can be treated separately and the

results added (VECTORIALLY!)

• The individual parts of the calculation can be handled (usually) without the use of vector calculations because of the symmetry.

• THIS IS SORT OF LIKE GAUSS’s LAW WITH AN ATTITUDE!

Magnetism 135

The figure below shows a cross section of an infinite conducting sheet carrying a current per unit x-length of l; the current emerges perpendicularly out of the page. (a) Use the Biot–Savart law and symmetry to show that for all points P above the sheet, and all points P´ below it, the magnetic field B is parallel to the sheet and directed as shown. (b) Use Ampere's law to find B at all points P and P´.

Magnetism 136

FIRST PART

Vertical ComponentsCancel

Magnetism 137

Apply Ampere to Circuit

Infinite Extent

B

B

Li

: thereforeis loop theinsideCurrent

lengthunit per current

L

Magnetism 138

The “Math”

Infinite Extent

B

B

20

0

0

B

LBLBL

id enclosedsB

Bds=0

Magnetism 139

A Physical Solenoid

Magnetism 140

Inside the Solenoid

For an “INFINITE” (long) solenoid the previous problem and SUPERPOSITION suggests that the field OUTSIDE this

solenoid is ZERO!

Magnetism 141

More on Long Solenoid

Field is ZERO!

Field is ZERO

Field looks UNIFORM

Magnetism 142

The real thing…..

Weak Field

Stronger - Leakage

Fairly Uniform field

Finite Length

Magnetism 143

Another Way

0

:

0

0

0

niB

nihBhh

id

Ampere

enclosed

sB

Magnetism 144

Application

• Creation of Uniform Magnetic Field Region

• Minimal field outside– except at the ends!

Magnetism 145

Two Coils

Magnetism 146

“Real” Helmholtz Coils

Used for experiments.

Can be aligned to cancelout the Earth’s magneticfield for critical measurements.

Magnetism 147

The Toroid

Slightly lessdense than

inner portion

Magnetism 148

The Toroid

r

NiB

so

totalNirBd

Ampere

2

turns)# (N 2

:nintegratio ofpath the

in contained coil INNER about the

only worry need Weagain.

0

0

sB