Download - Biot-Savart Law
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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:
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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
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Step 2:Write an expression for B due to one piece.
Unit vector:
:
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 “x”?
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Step 4: Check results
directionfar away: r>>L
units:
A Long Straight Wire
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For Infinite Wire
Semi-infinite Straight Wire
0
∞
− ∞
− ∞
+∞
0
+∞
𝐵𝑠𝑒𝑚𝑖=𝜇0
4 𝜋𝐼𝑥
𝐵∞=𝜇0
4 𝜋2 𝐼𝑥
For Semi-Infinite Wire
Half the integral …
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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
𝐵𝑤𝑖𝑟𝑒=𝐵 h𝑒𝑎𝑟𝑡 tan (𝜃)¿ (2 ×1 0−5 T) tan (12° )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: We’ve not calculated or shown the “rest” 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 semicircleMagnetic Field of a Semicircle
∫0
𝜋
¿ 12 ∫
0
2𝜋
❑
𝐵𝑧 , 𝑠𝑒𝑚𝑖=𝜇0
4𝜋𝜋 𝐼𝑅
𝐵𝑧 , ∆ 𝜃=𝜇0
4𝜋2𝜋 𝐼𝑅
∆ 𝜃2𝜋 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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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:
• can repel or attract• superposition• works through matter
Differences with E-field:• B-field only interacts with some objects • curly pattern• 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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Current is flowing to the right in a wire. The magnetic field at the position P points
A. B.C. D.
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What is the direction of the magnetic field inside the solenoid?
A. B. C. D.
Current upward on side nearest you
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A current in the loop has created the magnetic field, B, shown below. What is the current direction in this loop if you look from the top? And which side of the loop is the north pole?(To get the pole, you need to replace the loop with a bar magnet that has the same field direction)
A. Current clockwise; north pole on top
B. Current counterclockwise, north pole on top
C. Current clockwise; north pole on bottom
D. Current counterclockwise, north pole on bottom
B
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An electric dipole consists of two opposite charges – 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 et
T 2 R
v RevI2
One loop:
eRvR
evR21
22
The Atomic Structure of Magnets
Electrons
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eRv21
Magnetic dipole moment of 1 atom:
Method 1: use quantized angular momentum
Orbital angular momentum: RmvL
LmeRmv
meeRv
21
21
21
Quantum mechanics: L is quantized:
sJ , 341005.1nL
If n=1: 12
em
L 0.9 10 23 A m2 per atom
Magnetic Dipole Moment
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eRv21
Magnetic dipole moment of 1 atom:
Method 2: estimate speed of electron
Momentum principle: netFdtpd
Circular motion:
drpdt
p vR
mv Fnet – angular speed
2
2
0
2
41
Re
Rmv
m/s 62
0
106.14
1
Rmev
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 – 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 ‘motion’, 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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Alignment of atomic dipole moments:
most materialsferromagnetic materials:iron, cobalt, nickel
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 ‘s are much smaller than for electron
same angular momentum: me
21
NMR, MRI – use nuclear
Modern Theory of Magnets
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Magnetic domains
Very pure iron – no residual magnetism spontaneously disordersHitting or heating can also demagnetize
Modern Theory of Magnets
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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
NL
z
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
NL
dz
Bz 0
42 R2NI
Ldz
R2 d z 2 3/2 L /2
L /2
∫
Bz 0
42 NI
Ld 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
Ld 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
LL / 2
L / 2 2
L / 2
L / 2 2
0
42 NI
L2
LNIBz
0 in the middle of a long solenoid
Magnetic Field of a Solenoid