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Moving charge produces a curly magnetic field
B units: T (Tesla) = kg s-2A-1
Single Charge:
Biot-Savart Law
The Biot-Savart law for a short length of thin wire
Current:
π0
4π=10β7 T β m
A
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Four-step approach:
1. Cut up the current distribution into pieces and draw B
2. Write an expression for B due to one piece
3. Add up the contributions of all the pieces
4. Check the result
Magnetic Field of Current Distributions
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Step 1:Cut up the current distribution into pieces and draw B.
Origin: center of wire
Vector r:
Magnitude of r:
A Long Straight Wire
Unit vector:
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Step 2:Write an expression for B due to one piece.
:
B field due to one piece:
A Long Straight Wire
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need to calculate only z component
A Long Straight Wire
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Step 3:Add up the contribution of all the pieces.
A Long Straight Wire
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Special case: x<<L
A Long Straight Wire
What is the meaning of βxβ?
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Step 4: Check results
direction
far away: r>>L
units:
A Long Straight Wire
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For Infinite Wire
Semi-infinite Straight Wire
0
β
β β
β β
+β
0
+β
π΅π πππ=π0
4 ππΌπ₯
π΅β=π0
4 π2 πΌπ₯
For Semi-Infinite Wire
Even Function: Half the integral β¦
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Off-axis for Long Straight Wire
y
x
a Angle betweenβ οΏ½οΏ½
πβ οΏ½οΏ½=
π0
4π1π2 β π¦ sinπΌ (β οΏ½οΏ½ )
Rewrite in terms of
See Quest Course Resources for details (offaxisline.pdf)
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Right-hand Rule for Wire
Conventional Current Direction
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QuestionCurrent carrying wires below lie in X-Y plane.
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Question
π΅π€πππ=π΅ hππππ‘ tan (π)ΒΏ (2Γ1 0β5 T) tan (12Β° )T
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Step 1:Cut up the distribution into pieces
Make use of symmetry!
Need to consider only Bz due to one dl
Magnetic Field of a Wire Loop
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Step 2: B due to one piece
Origin: center of loop
Vector r:
Magnitude of r:
Unit vector:
l:
Magnetic field due to one piece:
Magnetic Field of a Wire Loop
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Step 2: B due to one piece
need only z component:
Magnetic Field of a Wire Loop
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Step 3: Sum the contributions of all pieces
Magnetic field of a loop along its axis:
Magnetic Field of a Wire Loop
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Step 4: Check the results
units:
direction:
Magnetic Field of a Wire Loop
Check several pieces with the right hand rule
Note: Weβve not calculated or shown the βrestβ of the magnetic field
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Using general form (z=0) :
Special case: center of the loopMagnetic Field of a Wire Loop
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for z>>R:
Magnetic Field of a Wire LoopSpecial case: far from the loop
The magnetic field of a circular loop falls off like 1/z3
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For whole loop
Special case: at center of the semicircle
Magnetic Field of a Semicircle
β«0
π
ΒΏ 12β«0
2π
β
π΅π§ , π πππ=π0
4ππ πΌπ
π΅π§ , β π=π0
4π2π πΌπ
β π2π
What is for 1.5 loops?
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What if we had a coil of wire?
For N turns:
single loop:
A Coil of Wire
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far from coil: far from dipole:
magnetic dipole moment: - vector in the direction of B
Magnetic Dipole Moment
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The magnetic dipole moment acts like a compass needle!
In the presence of external magnetic field a current-carrying loop rotates to align the magnetic dipole moment along the field B.
Twisting of a Magnetic Dipole
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What are the directions of the magnetic fields at the center of the loop?
Exercise: a loop of radius R and a long straight wire. The center of the loop is 2R from the wire.
XI
I
What is the net magnetic field at the center of the loop?
|οΏ½οΏ½ ππππ|=π0
4π2π πΌπ
|οΏ½οΏ½π€πππ|=π0
4π2 πΌπ
οΏ½οΏ½ππππβ οΏ½οΏ½π€πππ=π0
4π2π πΌπ
βπ0
4π2 πΌ2π
ΒΏπ0 πΌ
4 π π (2π β1 )
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How does the magnetic field around a bar magnet look like?
The Magnetic Field of a Bar Magnet
N S
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How do magnets interact with each other?Magnets interact with iron or steel, nickel, cobalt.
Does it interact with charged tape?
Does it work through matter?
Does superposition principle hold?Similarities with E-field:
β’ can repel or attractβ’ superpositionβ’ works through matter
Differences with E-field:β’ B-field only interacts with some objects β’ curly patternβ’ only closed field lines
Magnets and Matter
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Horizontal component of magnetic field depends on latitude
Maine: ~1.5.10-5 TTexas: ~2.5x10-5 T
Can use magnetic field of Earth as a reference to determine unknown field.
Magnetic Field of EarthThe magnetic field of the earth has a pattern that looks like that of a bar magnet
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An electric dipole consists of two opposite charges β monopoles
NS
Break magnet:
S N
There are no magnetic monopoles!
Magnetic Monopoles
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The magnetic field of a current loop and the magnetic field of a bar magnet look the same.
Batom 0
42z3
, R2I
What is the direction?
SNWhat is the average current I?
current=charge/second: I e
t
T 2 R
v R
evI
2
One loop:
eRvR
evR
2
1
22
The Atomic Structure of Magnets
Electrons
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eRv2
1Magnetic dipole moment of 1 atom:
Method 1: use quantized angular momentum
Orbital angular momentum: RmvL
Lm
eRmv
m
eeRv
2
1
2
1
2
1
Quantum mechanics: L is quantized:
sJ , 341005.1nL
If n=1: 1
2
e
mL 0.9 10 23 A m2 per atom
Magnetic Dipole Moment
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eRv2
1Magnetic dipole moment of 1 atom:
Method 2: estimate speed of electron
Momentum principle: netFdt
pd
Circular motion:
drp
dt p
v
Rmv Fnet
w β angular speed
2
2
0
2
4
1
R
e
R
mv
m/s 62
0
106.14
1
Rm
ev
1.3 10 23 A m2 /atom
Magnetic Dipole Moment
p p const
v / R
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Magnetic dipole moment of 1 atom: /atommA 2 2310
Mass of a magnet: m~5g
Assume magnet is made of iron: 1 mole β 56 g
6.1023 atoms
number of atoms = 5g/56g . 6.1023 ~ 6.1022
magnet 6 1022 10 23 0.6 A m2
Magnetic Dipole Moment
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1. Orbital motion
There is no βmotionβ, but a distribution
Spherically symmetric cloud (s-orbital)has no
Only non spherically symmetric orbitals (p, d, f) contribute to
There is more than 1 electron in an atom
Modern Theory of Magnets
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2. Spin
Electron acts like spinning charge- contributes to
Electron spin contribution to is of the same order as one due to orbital momentum
Neutrons and proton in nucleus also have spin but their βs are much smaller than for electron
same angular momentum: m
e
2
1
NMR, MRI β use nuclear
Modern Theory of Magnets
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Alignment of atomic magnetic dipole moments:
most materialsferromagnetic materials:iron, cobalt, nickel
Modern Theory of MagnetsWhy are only some materials magnetics?
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Magnetic domains
Hitting or heating while in a magnetic field can magnetize the iron
Hitting or heating can also demagnetize
Modern Theory of Magnets
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Magnetic domains
Why are there Multiple Domains?
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Multiplier effect:
ironcoilnet BBB
coilnet BB
Electromagnet:
Iron Inside a Coil
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Step 1: Cut up the distributioninto pieces
B
origin: center of the solenoid
Step 2: Contribution of one piece
Bz 0
42 R2I
R2 d z 2 3/2one loop:
Number of loops per meter: N/L
Number of loops in z: (N/L) z
Field due to z: Bz 0
42 R2I
R2 d z 2 3/2
N
Lz
Magnetic Field of a Solenoid
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Step 3: Add up the contributionof all the pieces
B
dBz 0
42 R2I
R2 d z 2 3/2
N
Ldz
Bz 0
42 R2NI
L
dz
R2 d z 2 3/2 L /2
L /2
β«
Bz 0
42 NI
L
d L / 2
d L / 2 2 R2
d L / 2
d L / 2 2 R2
Magnetic field of a solenoid:
Magnetic Field of a Solenoid
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Bz 0
42 NI
L
d L / 2
d L / 2 2 R2
d L / 2
d L / 2 2 R2
Special case: R<<L, center of the solenoid:
Bz 0
42 NI
L
L / 2
L / 2 2
L / 2
L / 2 2
0
42 NI
L2
L
NIBz
0 in the middle of a long solenoid
Magnetic Field of a Solenoid