Chapter 28 New

2006 Physics 2112 Fundamentals of Physics Chapter 28 1

Fundamentals of Physics

Chapter 29 Magnetic Fields

1. The Magnetic Field2. B3. Crossed Fields: Discovery of the Electron4. Crossed Fields: the Hall Effect

5. A Circulating Charged Particle6. Cyclotrons & Synchrotrons

7. Magnetic Force on a Current-Carrying Wire8. Torque on a Current Loop9. The Magnetic Dipole Moment

Review & SummaryChapter QuestionsExercises & Problems


Magnetism in Ancient Times

>2000 years ago Ancient Greeks - magnetite orenamed for the Magnesia region of Asia Minor

Attracts small bits of iron

Magnetite can magnetize an iron bar.

Ancient Chinese discovered the compass.

The Chinese compass seen at the right uses a piece of magnetite shaped like a spoon. The handle of the spoon points south.



11th Century

An elongated piece of magnetite suspended from a thread turns to the earth s north-south direction.

lodestone - a leading stone

1269 ad Peter Peregrinus (Pierre de Maricourt) was an engineer in the army of the King of Sicily.

A piece of magnetite has 2 poles (Latin polus)Every magnet has a north and a south pole.How the north and south poles attract or repel

He also described how to make a perpetual motion machine using magnets.

People are still trying this (unsuccessfully) today!



William Gilbert, 1544-1603The Earth is a magnet

William Gilbert published De Magnete (On the Magnet) in 1600.

first to distinguish the electric force (named for the Greek word for amber) from the magnetic force.

debunked many folk myths about the curative properties of magnets.

Based on his observations of the preferred directions of thin iron needles near spherical lodestones, and the similarity of this phenomenon to the tendency of compass needles to tilt with respect to the horizontal plane as well as point north-south, Gilbert deduced that the earth itself must be a giant magnet.

The Earth s North Pole is actually a south magnetic pole.


Ben Franklin again!: lightning can magnetize iron.

Oersted, Ampere, Gauss, Weber & Faraday (1810-1830)

Force of a Magnetic Field

Right-Hand Rule for Cross Products

sinBvqF

BvqF

B

B

Magnets are surrounded by a magnetic field:

B = Magnetic Field (aka magnetic-flux density &magnetic induction )

Experimentally, the force on charge q moving with velocity v :


The Magnetic Field B

Units: Tesla (T) = 1 (N/C) / (m/s) = 1 N/(A m)Gauss = T / 10,000

Earth s magnetic field ~ 0.6 gauss

Iron magnet ~ 0.5 TFermilab electro-magnets ~ 1-2 T

BvqFB


Checkpoint 1

Direction of B ?


Example 1

B = 1.2 mT vertically outward

proton KE = 5.3 MeV enters moving South to NorthWhat deflecting force acts on proton(m = 1.67 x 10-27 kg)


The Definition of B

A magnetic field surrounds magnets.Magnetic Field Lines

Tangent gives the direction of the B fieldSpacing of the lines is proportional to the magnitude of B

Magnets have a North Pole and a South PoleOpposite poles attract each other, and like poles repel.No magnetic monopoles have been observed (a la +/- charges).


Charged Particle Moving in Crossed Electric & Magnetic Fields

BE FFF

BvqEqF

B

Ev

Velocity Selector: adjust the fields so that there is no deflection (F = 0):


Crossed Fields: Discovery of the Electron

Cathode Ray Tube:

Discovery of the Electron - J. J. Thomson (Cambridge) 1897


q /m for Electrons - J. J. Thomson (1897)

p. 864

Add a crossed magnetic field adjusted to cancel out the deflection y:

B

Ev

tvtvL

a

x

x 0

221 tay

EqamF

y

yy

Consider the deflection of a charged particle moving through an electric field:

2

2

2 vm

LEqy

Ey

LB

q

m

2

22


Checkpoint 2

positively charged particle velocity Vrank 1, 2, 3 for net forcewhich direction could lead to zero deflection?


Crossed Fields: the Hall Effect

Use a magnetic field to deflect the charge carriers inside a conductor:

This creates an electric field across the conductor, generating a potential difference:

V = E dE grows until it balances the effect of the B-field.

e E = e vd BCurrent in a conductor

i = n e A vd = n e d l vd

Number density of the charge carriers:

elV

iBn

BvqFB


Crossed Fields: the Hall Effect

Positive Charge Carriers Negative Charge Carriers

A calibrated Hall effect probe is also used to measure magnetic fields.

Move the sample in the opposite direction of the drift until a velocity is found that makes the potential difference disappear -

the magnitude of the drift velocity!

The sign of the side-to-side potential difference gives the sign of the drifting charges inside the conductor.


Example 2

d =1.5 cmmoving uniformly in +y directionv = 4.0 ms

B = .050 T towards +za) which face is at lower electric potential

b) what is potential difference


A magnetic field does no work on moving charged particles, but it does deflect their velocity:

A Circulating Charged Particle

BvqFB

Bq

vmr

r

vmBvqF

2

Circular Motion:

Period:Bq

m

v

rT

22

(r ~ momentum)

(independent of v!)


Example 3

Mass Spectrometer:

Energy:

x = 2 r

momentum:

Bq

vmr

Vqvm 22

1

V

xqBm

8

22

B = 80.000 mT

V = 1000.0 Vq = 1.6 x 10-19 C

x = 1.6254 mmass m in amu s (1 u = 1.6605 x 10-27 kg)

Mass

Momentum



Helical Paths:

z

y

x

BvqF xyxy

constantzv0zF

kBB z Uniform circular motion

in xy-plane


29-5 A Circulating Charged Particle

Magnetic Bottle: Charged particles spiraling in a non-uniform magnetic field.

reflectionreflection



Van Allen Belts - Earth s magnetic bottle


Van Allen Belts

Protons & electrons boiling off the Sun are trapped in the Earth s magnetic bottle:

The Northern (& Southern) Lights!


The Cyclotron

Constant BElectric potential difference between the Dees accelerates particles across the gap.

larger momentum larger radiusNo E inside the Dee.Orbital period in B is independent of energy.

Constant frequency oscillator.

Acceleration continues until particles reach the radius of the Dee; then deflected out.

Bq

m

v

rT

22


Magnetic Force on a Current-Carrying Wire

outB


Magnetic Force on a Current-Carrying Wire

The force that a magnetic field exerts on the charge carriers inside a conductor is transmitted to a conducting wire itself.

BLiF

eqAvqni

LAnBvqF

Bvqf

B

d

dB

dB

)626(

Or: and integrate.BdLiFd B BdLiFd B


Example 6

I = 28 A

Magnitude and direction of minimum magnetic field B to suspend the wire.

= 46.6 g/m


Torque on a Current Loop

BLiF



Forces on all 4 sides of a current loop in a magnetic field.

Suppose that the current loop can only rotate about a horizontal axis.



BLiF

cos90sin 02 BbiBbiF

BaiF1

24 FF

Sides 1 & 3 are perpendicular to B:

31 FF

Sides 2 & 4 are not:



cos24 BbiFF

sin

sinsin2

2 1

BAiN

BAib

F

Produce a net torque.

BaiFF 31

No torque produced.



A Motor: Reverse the current every half turn.

Force reversesConstant torque direction


Simple DC Motor


Magnetic Dipole Moment

AiNB where

sinBAiN

Magnetic Dipole Moment


The Magnetic Dipole Moment

Magnetic Moment:

AiNB where

BU


Example 8

circular coil with 250 turnsA = 2.52 x 10-4 m2

current = 100 mAB = 0.85 T, with dipole moment m initially aligned with Ba) direction of current in coil?b) how much work would the torque from an external agent have to do on the coil to rotate it 900 from its original orientation so the m is perpendicular to B and the coil is at rest again?

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