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:
๐0
4๐=10โ7 T โ m
A
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
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:
Step 2:Write an expression for B due to one piece.
:
B field due to one piece:
A Long Straight Wire
need to calculate only z component
A Long Straight Wire
Step 3:Add up the contribution of all the pieces.
A Long Straight Wire
Special case: x<<L
A Long Straight Wire
What is the meaning of โxโ?
Step 4: Check results
direction
far away: r>>L
units:
A Long Straight Wire
For Infinite Wire
Semi-infinite Straight Wire
0
โ
โ โ
โ โ
+โ
0
+โ
๐ต๐ ๐๐๐=๐0
4 ๐๐ผ๐ฅ
๐ตโ=๐0
4 ๐2 ๐ผ๐ฅ
For Semi-Infinite Wire
Even Function: Half the integral โฆ
Off-axis for Long Straight Wire
y
x
a Angle betweenโ ๏ฟฝ๏ฟฝ
๐โ ๏ฟฝ๏ฟฝ=
๐0
4๐1๐2 โ ๐ฆ sin๐ผ (โ ๏ฟฝ๏ฟฝ )
Rewrite in terms of
See Quest Course Resources for details (offaxisline.pdf)
Right-hand Rule for Wire
Conventional Current Direction
QuestionCurrent carrying wires below lie in X-Y plane.
Question
๐ต๐ค๐๐๐=๐ต h๐๐๐๐ก tan (๐)ยฟ (2ร1 0โ5 T) tan (12ยฐ )T
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
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
Step 2: B due to one piece
need only z component:
Magnetic Field of a Wire Loop
Step 3: Sum the contributions of all pieces
Magnetic field of a loop along its axis:
Magnetic Field of a Wire Loop
Step 4: Check the results
units:
direction:
Magnetic Field of a Wire Loop
Check several pieces with the right hand rule
Note: Weโve not calculated or shown the โrestโ of the magnetic field
Using general form (z=0) :
Special case: center of the loopMagnetic Field of a Wire Loop
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
For whole loop
Special case: at center of the semicircle
Magnetic Field of a Semicircle
โซ0
๐
ยฟ 12โซ0
2๐
โ
๐ต๐ง , ๐ ๐๐๐=๐0
4๐๐ ๐ผ๐
๐ต๐ง , โ ๐=๐0
4๐2๐ ๐ผ๐
โ ๐2๐
What is for 1.5 loops?
What if we had a coil of wire?
For N turns:
single loop:
A Coil of Wire
far from coil: far from dipole:
magnetic dipole moment: - vector in the direction of B
Magnetic Dipole Moment
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
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?
|๏ฟฝ๏ฟฝ ๐๐๐๐|=๐0
4๐2๐ ๐ผ๐
|๏ฟฝ๏ฟฝ๐ค๐๐๐|=๐0
4๐2 ๐ผ๐
๏ฟฝ๏ฟฝ๐๐๐๐โ ๏ฟฝ๏ฟฝ๐ค๐๐๐=๐0
4๐2๐ ๐ผ๐
โ๐0
4๐2 ๐ผ2๐
ยฟ๐0 ๐ผ
4 ๐ ๐ (2๐ โ1 )
How does the magnetic field around a bar magnet look like?
The Magnetic Field of a Bar Magnet
N S
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:
โข can repel or attractโข superpositionโข works through matter
Differences with E-field:โข B-field only interacts with some objects โข curly patternโข only closed field lines
Magnets and Matter
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
An electric dipole consists of two opposite charges โ monopoles
NS
Break magnet:
S N
There are no magnetic monopoles!
Magnetic Monopoles
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
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
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 โ 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
Magnetic dipole moment of 1 atom: /atommA 2 2310
Mass of a magnet: m~5g
Assume magnet is made of iron: 1 mole โ 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
1. Orbital motion
There is no โmotionโ, 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
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 โs are much smaller than for electron
same angular momentum: m
e
2
1
NMR, MRI โ use nuclear
Modern Theory of Magnets
Alignment of atomic magnetic dipole moments:
most materialsferromagnetic materials:iron, cobalt, nickel
Modern Theory of MagnetsWhy are only some materials magnetics?
Magnetic domains
Hitting or heating while in a magnetic field can magnetize the iron
Hitting or heating can also demagnetize
Modern Theory of Magnets
Magnetic domains
Why are there Multiple Domains?
Multiplier effect:
ironcoilnet BBB
coilnet BB
Electromagnet:
Iron Inside a Coil
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
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
โซ
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
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