department of physics and astronomy pa113/unit 3 electricity and magnetism course pa113 – unit 3

DEPARTMENT OF PHYSICS AND ASTRONOMY

PA113/Unit 3

Electricity and MagnetismElectricity and Magnetism

Course PA113 – Unit Course PA113 – Unit 33

PA113/Unit 3

UNIT UNIT 33 – – IntroductoryIntroductory Lecture Lecture

The Magnetic FieldThe Magnetic Field – Chapter 2Chapter 288

Sources of the Magnetic FieldSources of the Magnetic Field– Chapter 2Chapter 299

PA113/Unit 3

Importance of Importance of Magnetic FieldsMagnetic Fields

Practical UsesPractical Uses– Electric motors, Loud speakers, Navigation Electric motors, Loud speakers, Navigation

(Earth’s magnetic field)(Earth’s magnetic field) In Experimental PhysicsIn Experimental Physics

– Mass spectrometers, Particle accelerators, Mass spectrometers, Particle accelerators, Plasma confinementPlasma confinement

In the UniverseIn the Universe– Stars (e.g. the Sun), Interstellar space, Stars (e.g. the Sun), Interstellar space,

Intergalactic structure, JetsIntergalactic structure, Jets

PA113/Unit 3


PA113/Unit 3


Units – SI Tesla (T) = (N CUnits – SI Tesla (T) = (N C-1-1)/(m s)/(m s-1-1) or N A) or N A-1 -1 m m -1 -1

– 1 Gauss (G) = 101 Gauss (G) = 10-4-4 T T

ExamplesExamples– Terrestrial B field ~ 4x10Terrestrial B field ~ 4x10-5 -5 T T– Solenoid ~ 10Solenoid ~ 10-3 -3 T T– Permanent magnet ~ 10Permanent magnet ~ 10-1 -1 T T– Atomic interactions ~ 10 TAtomic interactions ~ 10 T– Superconducting magnet ~ 10Superconducting magnet ~ 102 2 T T– White dwarfs ~ 10White dwarfs ~ 102 2 - 10 - 103 3 T T– Neutron stars < 10Neutron stars < 108 8 T T

PA113/Unit 3

Ch2Ch288 – – The Magnetic FieldThe Magnetic Field

2288-1 -1 Force exerted by a Magnetic FieldForce exerted by a Magnetic Field 2288-2 -2 Motion of a point charge in a Motion of a point charge in a

Magnetic FieldMagnetic Field 2288-3 -3 Torques on current loops and Torques on current loops and

magnetsmagnets 2288-4 -4 The Hall EffectThe Hall Effect

PA113/Unit 3

Vector NotationVector Notation

The DOT productThe DOT product

The CROSS productThe CROSS product

sinABC BAC

cosABC BAC

PA113/Unit 3

28-1 28-1 The Force Exerted by a The Force Exerted by a Magnetic FieldMagnetic Field

Key Concept –Key Concept – Magnetic fields apply a Magnetic fields apply a force to moving chargesforce to moving charges

BvF q

BdldF ICurrent element

PA113/Unit 3


PA113/Unit 3

Representation of Magnetic FieldRepresentation of Magnetic Field

Like electric field, can be represented by field Like electric field, can be represented by field lineslines– Field direction indicated by direction of linesField direction indicated by direction of lines– Field strength indicated by density of linesField strength indicated by density of lines

But, unlike electric fieldBut, unlike electric field– Magnetic field lines perpendicular to forceMagnetic field lines perpendicular to force– No isolated magnetic poles, so no points in space No isolated magnetic poles, so no points in space

where field lines begin or endwhere field lines begin or end

PA113/Unit 3

28-2 28-2 Motion of a Point Charge in a Motion of a Point Charge in a Magnetic FieldMagnetic Field

Key Concept – Key Concept – Force is perpendicular to Force is perpendicular to field direction and velocityfield direction and velocity

Therefore, magnetic fields do no work Therefore, magnetic fields do no work on particleson particles

There is no change in magnitude of There is no change in magnitude of velocity, just directionvelocity, just direction

PA113/Unit 3

Motion of a Point Charge in a Motion of a Point Charge in a Magnetic FieldMagnetic Field

PA113/Unit 3


Radius of circular orbitRadius of circular orbit

Cyclotron periodCyclotron period

Cyclotron frequencyCyclotron frequency

qB

mvr

qB

mT

2

m

qB

Tf

21

PA113/Unit 3

28-3 Torques on Current Loops 28-3 Torques on Current Loops and Magnetsand Magnets

Key concept – a current loop Key concept – a current loop experiences no net force in a uniform B experiences no net force in a uniform B field but does experience a torquefield but does experience a torque

PA113/Unit 3


IaBFF 21

sin2bF sinIaBb sinIAB sinNIAB

Bμτ Magnetic dipole moment

nμ NIA

PA113/Unit 3

Potential Energy of a Magnetic Potential Energy of a Magnetic Dipole in a Magnetic FieldDipole in a Magnetic Field

Potential energyPotential energy Work done…..Work done….. dBddW sin

dBdWdU sin

IntegrateoUBU cos

Zero atθ = 90o

Bμ cosBU

PA113/Unit 3

28-28-4 The Hall Effect4 The Hall Effect

Vh = vdBw

PA113/Unit 3

Ch2Ch299 – – Sources of the Magnetic Sources of the Magnetic FieldField

2299-1 -1 The Magnetic Field of moving point The Magnetic Field of moving point chargescharges

2299-2 -2 The Magnetic Field of CurrentsThe Magnetic Field of Currents– Biot-Savart LawBiot-Savart Law

2299-3 -3 Gauss’ Law for MagnetismGauss’ Law for Magnetism 2299-4 -4 AmpAmpèère’s Lawre’s Law 2299-5 -5 Magnetism in matterMagnetism in matter

PA113/Unit 3

229-19-1 The Magnetic Field of Moving The Magnetic Field of Moving Point ChargesPoint Charges

Point charge q moving with velocity v Point charge q moving with velocity v produces a field B at point Pproduces a field B at point P

2

ˆ

4 r

qo rvB

μo= permeability of free space μo= 4 x 10-7 T·m·A-1

PA113/Unit 3

29-2 The Magnetic Field of 29-2 The Magnetic Field of Currents: The Biot-Savart LawCurrents: The Biot-Savart Law

Key concept – current as a series of Key concept – current as a series of moving charges – replace moving charges – replace qqvv by by IIdldl

2

ˆ

4 r

Idd

o rlB

Add each element to get total B field

PA113/Unit 3

Key Key cconcept –oncept – The net flux of magnetic The net flux of magnetic field lines through a closed surface is field lines through a closed surface is zero (i.e. no magnetic monopoles)zero (i.e. no magnetic monopoles)

snnetm dAB 0,

Magnetic flux

229-3 Gauss’ Law for Magnetism9-3 Gauss’ Law for Magnetism

PA113/Unit 3


Electric dipole Magnetic dipole (or current loop)

PA113/Unit 3

29-4 Amp29-4 Ampère’s Lawère’s Law

Key concept – like Gauss’ law for electric Key concept – like Gauss’ law for electric field, uses symmetry to calculate B field field, uses symmetry to calculate B field around a closed curve Caround a closed curve C

c

coId lB.

N.B. This version assumes the currents are steady

PA113/Unit 3

29-5 Magnetism in Matter29-5 Magnetism in Matter Magnetization, M = Magnetization, M = m m BBappapp//0 0

m m is the magnetic susceptibilityis the magnetic susceptibility ParamagneticParamagnetic

– M in same direction as B, dipoles weakly add to B field M in same direction as B, dipoles weakly add to B field (small +ve (small +ve m m ))

DiamagneticDiamagnetic– M in opposite direction to B, dipoles weakly oppose B M in opposite direction to B, dipoles weakly oppose B

field (small -ve field (small -ve m m )) FerromagneticFerromagnetic

– Large +ve Large +ve mm, dipoles strongly add to B-field. Can result , dipoles strongly add to B-field. Can result in permanent magnetic field in material.in permanent magnetic field in material.

PA113/Unit 3

End of lecture 1End of lecture 1


PA113/Unit 3


Course 113 – Unit Course 113 – Unit 33

PA113/Unit 3

UNIT UNIT 33 – – Problem solvingProblem solving Lecture Lecture



PA113/Unit 3

Problem SolvingProblem Solving

Read the book!!!!!Read the book!!!!! Look at some examplesLook at some examples Try out some questionsTry out some questions Draw a diagram – include vector nature Draw a diagram – include vector nature

of the field (of the field (rr and and vv or or dl dl ))

PA113/Unit 3

You must know how to…You must know how to…

Calculate force on a moving chargeCalculate force on a moving charge– Or current elementOr current element

Understand the properties of a dipoleUnderstand the properties of a dipole– Torque and magnetic momentTorque and magnetic moment

Calculate the B field usingCalculate the B field using1.1. The Biot-Savart lawThe Biot-Savart law

2.2. AmpAmpère’s Lawère’s Law

Understand Gauss’ Law for MagnetismUnderstand Gauss’ Law for Magnetism

PA113/Unit 3

29-2 Example – the Biot-Savart 29-2 Example – the Biot-Savart Law applied to a current loopLaw applied to a current loop

PA113/Unit 3

Field due to a current loopField due to a current loop

2

ˆ

4 r

Idd

o rlB

2

ˆ

4 r

dId

orl

B

224 Rx

Idld

o

B

PA113/Unit 3


224 Rx

Idld

o

B

22sin

Rx

RdBdBdBx

22224 Rx

R

Rx

IdldB

o

x

PA113/Unit 3


22224 Rx

R

Rx

IdldB

o

x

dlRx

IRdBB

o

xx 2/3224

dlRx

IRdBB o

xx 2/3224

2πR

2/322

22

4 Rx

IRB ox

PA113/Unit 3

Magnetic field lines of 2 loopsMagnetic field lines of 2 loops

PA113/Unit 3

Many loops – a solenoidMany loops – a solenoid

PA113/Unit 3

The B field in a very long solenoidThe B field in a very long solenoid

Can use the Biot-Savart Law or Ampère’s Law

Length L

N turns

n = N/L

Radius R

Current Idi=nIdx

Field in a very long solenoid: B =0nI

PA113/Unit 3

Field around and inside a wireField around and inside a wire

Classic example of the use of Ampère’s Law

R

IB o 2

4

22 R

IrB o

c

coId lB.

PA113/Unit 3

Direction of field around a wireDirection of field around a wire

PA113/Unit 3

End of lecture 2


PA113/Unit 3


Course 113 – Unit Course 113 – Unit 33

PA113/Unit 3

UNIT UNIT 33 – – Follow-upFollow-up Lecture Lecture



PA113/Unit 3

Ch2Ch288 – – The Magnetic FieldThe Magnetic Field

2288-1 -1 Force exerted by a Magnetic FieldForce exerted by a Magnetic Field 2288-2 -2 Motion of a point charge in a Motion of a point charge in a

Magnetic FieldMagnetic Field 2288-3 -3 Torques on current loops and Torques on current loops and

magnetsmagnets 2288-4 -4 The Hall EffectThe Hall Effect

PA113/Unit 3


Key Concept –Key Concept – Magnetic fields apply a Magnetic fields apply a force to moving chargesforce to moving charges

BvF q

BdldF ICurrent element

PA113/Unit 3


Radius of circular orbitRadius of circular orbit

Cyclotron periodCyclotron period

Cyclotron frequencyCyclotron frequency

qB

mvr

qB

mT

2

m

qB

Tf

21

PA113/Unit 3


IaBFF 21

sin2bF sinIaBb sinIAB sinNIAB

Bμτ Magnetic dipole moment

nμ NIA

PA113/Unit 3

Ch2Ch299 – – Sources of the Magnetic Sources of the Magnetic FieldField

2299-1 -1 The Magnetic Field of moving point The Magnetic Field of moving point chargescharges

2299-2 -2 The Magnetic Field of CurrentsThe Magnetic Field of Currents– Biot-Savart LawBiot-Savart Law

2299-3 -3 Gauss’ Law for MagnetismGauss’ Law for Magnetism 2299-4 -4 AmpAmpèère’s Lawre’s Law 2299-5 -5 Magnetism in matterMagnetism in matter

PA113/Unit 3

29-2 The Magnetic Field of 29-2 The Magnetic Field of Currents: The Biot-Savart LawCurrents: The Biot-Savart Law

Key concept – current as a series of Key concept – current as a series of moving charges – replace moving charges – replace qqvv by by IIdldl

2

ˆ

4 r

Idd

o rlB

Add each element to get total B field

PA113/Unit 3

Key Key cconcept –oncept – The net flux of magnetic The net flux of magnetic field lines through a closed surface is field lines through a closed surface is zero (i.e. no magnetic monopoles)zero (i.e. no magnetic monopoles)

snnetm dAB 0,

Magnetic flux


PA113/Unit 3

29-4 Amp29-4 Ampère’s Lawère’s Law

Key concept – like Gauss’ law for electric Key concept – like Gauss’ law for electric field, uses symmetry to calculate B field field, uses symmetry to calculate B field around a closed curve Caround a closed curve C

c

coId lB.

N.B. This version assumes the currents are steady

PA113/Unit 3

ExampleExample

PA113/Unit 3

Field of a tightly wound toroidField of a tightly wound toroid

)(,2

brar

NIB o

arB ,0

brB ,0

If b-a < r then B varies little – principle of fusion reactors

PA113/Unit 3

Why use fusion?Why use fusion?

• Chemical reaction C+02 CO2 (e.g. Coal) goes at ~700 K and gives ~107 J kg-1

• Fission, such as U235 + n Ba143 + Kr91 + 2n goes at ~103 K and gives ~1012 J kg-1

• Fusion, such as in the Sun, H2 + H3 He4 + n goes at ~108 K and gives ~1014 J kg-1

PA113/Unit 3

Conditions requiredConditions required

PA113/Unit 3

Typical Fusion Reaction ChainsTypical Fusion Reaction Chains

The Sun The laboratory

PA113/Unit 3

Tokamak Fusion Test ReactorTokamak Fusion Test Reactor

Operated from 1982 – 1997

Max Temp = 510 million K; Max power = 10.7 MW

PA113/Unit 3

Reactor ResultsReactor Results

PA113/Unit 3

End of lecture 3End of lecture 3

PA113/Unit 3

Definition of the AmpDefinition of the Ampèreère

Force between 2 straight parallel Force between 2 straight parallel conducting wiresconducting wires

122 BlF Idd

department of physics and astronomy pa113/unit 3 electricity and magnetism course pa113 – unit 3

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