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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
Importance of Importance of Magnetic FieldsMagnetic Fields
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
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
28-2 28-2 Motion of a Point Charge in a Motion of a Point Charge in a Magnetic FieldMagnetic Field
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
28-3 Torques on Current Loops 28-3 Torques on Current Loops and Magnetsand Magnets
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
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
229-3 Gauss’ Law for Magnetism9-3 Gauss’ Law for Magnetism
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.
DEPARTMENT OF PHYSICS AND ASTRONOMY
PA113/Unit 3
Electricity and MagnetismElectricity and Magnetism
Course 113 – Unit Course 113 – Unit 33
PA113/Unit 3
UNIT UNIT 33 – – Problem solvingProblem solving 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
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
Field due to a current loopField due to a current loop
224 Rx
Idld
o
B
22sin
Rx
RdBdBdBx
22224 Rx
R
Rx
IdldB
o
x
PA113/Unit 3
Field due to a current loopField due to a current loop
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
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.
DEPARTMENT OF PHYSICS AND ASTRONOMY
PA113/Unit 3
Electricity and MagnetismElectricity and Magnetism
Course 113 – Unit Course 113 – Unit 33
PA113/Unit 3
UNIT UNIT 33 – – Follow-upFollow-up 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
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
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
28-2 28-2 Motion of a Point Charge in a Motion of a Point Charge in a Magnetic FieldMagnetic Field
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
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
229-3 Gauss’ Law for Magnetism9-3 Gauss’ Law for Magnetism
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
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
Tokamak Fusion Test ReactorTokamak Fusion Test Reactor
Operated from 1982 – 1997
Max Temp = 510 million K; Max power = 10.7 MW