Download Lecture 2 - Purdue Physics

Last Time
• (Cross Products: Mathematically)
• Electron Current and Conventional
Current
• Calculating the Electron Current
• True vs. Useful
• Biot-Savart Law in a Wire
• Relativity??
1
Biot-Savart Law
BIOT-SAVART LAW
point charge
BIOT-SAVART LAW
current in a wire
= length of this
chunk of wire
2
Frame of Reference
Biot-Savart Law is an Approximation
Electric fields: produced by charges
Magnetic fields: produced by moving charges
B=
m0 qv ´ rˆ
=0
2
4p r
Any magnetic field?
m0 qv ´ rˆ
B=
¹0
2
4p r
Must use the velocities of the charges as you observe them in
charged tape
your reference frame!
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Retardation
Biot-Savart Law is an Approximation
If we suddenly change the current in a wire:
Magnetic field will not change instantaneously.
Electron and positron collide:
Produce both electric and magnetic field, these fields exist
even after annihilation.
Changes propagate at speed of light
m0 qv ´ rˆ
B=
4p r 2
Why is there no time in Biot-Savart law?
 We assume speed of charges is small
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Today
•
•
•
•
•
Magnetic Field of a Straight Wire
(Magnetic Field of a Current Loop)
Magnetic Dipole Moment
Bar Magnet
Atomic Dipoles
5
Magnetic Field of a Straight Wire
BIOT-SAVART LAW
current in a wire
Calculate B in the
bisecting plane
We want to calculate B in the bisecting plane.
How did we do this for E?
Break the rod up into point charges.
Add up E due to all point charges.
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Electric Field in the Bisecting Plane
for a point charge
Δqi
We only need the x component
θ
Break the rod up into point charges.
Add up E due to all point charges.
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Magnetic Field of a Straight Wire
BIOT-SAVART LAW
current in a wire
Break the rod up into point charges.
Add up B due to all point charges.
Calculate B in the
bisecting plane
-
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Magnetic Field of a Straight Wire
Calculate B in the
bisecting plane
Does x change
during the integration?
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Magnetic Field of a Straight Wire
Calculate B in the
bisecting plane
Observation point x
does not change
during the integration
Look up the integral
B of a Long
Straight Wire
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Magnetic Field of a Straight Wire
B in the
bisecting plane
I
Which direction does B point?
 Always along concentric circles
r
In cylindrical coordinates,
it points in the " " direction
Will the y axis look different from the x axis?
Cylindrical
Coordinates
No, so we can trade x  r
B of a Long
Straight Wire
(Cylindrical
Coordinates)
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Very Close to the Wire
Very close to the wire: r << L
CLOSE TO
THE WIRE
Another Right-Hand Rule
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iClicker Question
Current carrying wires below lie in X-Y plane.
Direction of magnetic field at the
observation location?
A) +y
B) –y
C) +z
D) –z
E) zero
magnitude
13
B along Axis of Circular Loop of Wire
BIOT-SAVART LAW
current in a wire
1) Draw
NET
MAGNETIC
FIELD
for one piece
 Symmetry: only z component survives!
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B along Axis of Circular Loop of Wire
BIOT-SAVART LAW
current in a wire
1) Draw
NET
MAGNETIC
FIELD
for one piece
 Symmetry: only z component survives!
2) Write
z
due to one piece
(lots of math!)
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B along Axis of Circular Loop of Wire
BIOT-SAVART LAW
current in a wire
1) Draw
NET
MAGNETIC
FIELD
for one piece
 Symmetry: only z component survives!
2) Write
z
due to one piece
(lots of math!)
3) Integrate
to find total B
4) Doulbecheck
✔ UNITS
Right-hand Rule
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✔
B along Axis of Circular Loop of Wire
m0 2p R 2 I
Bz =
4p ( R 2 + z 2 )3/2
Yet Another Right Hand Rule!
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B along Axis of Circular Loop of Wire
m0 2p R 2 I
Bz =
4p ( R 2 + z 2 )3/2
Far from the loop: z>>R
(z
2
+R
)
2 3/2
B Along Axis
Far from Loop
» (z 2 )3/2 = z 3
B for dipole
Both fall off like 1/r3 far from the dipole
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Electric and Magnetic Dipole Moments
B Along Axis
Far from Loop
B for dipole
Electric Dipole Moment:
E Along Axis
Far from Dipole
Magnetic Dipole Moment:
B Along Axis
Far from Loop
Magnetic field of a solenoid
L
• A constant magnetic field could be
produced by an infinite sheet of current.
In practice, however, it is easier and
more convenient to use a solenoid.
• A solenoid is defined by a current I
flowing through a wire that is wrapped n
turns per unit length on a cylinder of
radius R and length L.
R
Magnetic field of a solenoid
(continued)
Contribution to B at origin from length dx
0 I
2 R 2
dBx 
 2
 n  dx
2 3/ 2
4 ( R  x )
one turn
# turns in
length dx
x
0
dx
2
 Bx 
 2 R nI  
x ( R 2  x 2 ) 3/ 2
4
2
1

1
x2
x1

 0nI

2
2
2
2

2
x

R
x

R
1
 2




As x1  , x2   (or for L>>R)
Bx  0nI
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(half at ends)
Solenoid’s B field synopsis
• Long solenoid (R<<L):
B inside solenoid
// to axis
B outside solenoid
nearly zero
(not very close to the ends or wires)
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RHIC STAR Experiment
STAR
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Atomic Structure of Magnets
Assume: Electrons make circular current loops
Batom
m0 2 m
=
,
3
4p z
m = pR I
2
Electrons
What is the direction?
These are electrons
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Atomic Structure of Magnets
Assume: Electrons make circular current loops
Batom
m0 2 m
=
,
3
4p z
Electrons
m = pR I
2
What is the direction?
What is the average current I per loop?
e
current=charge/second: I =
t
ev
2p R
I=
T=
2pR
v
1
2 ev
m = pR
= eRv
2pR 2
S
N
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Atomic Structure of Magnets
Assume: Electrons make circular current loops
1
m = eRv Dipole Moment of 1 Atom
2
Angular Momentum is Quantized:
Orbital angular momentum: L = Rmv
1
1 e
1 e
m = eRv =
Rmv =
L
2
2m
2m
Quantum mechanics: L is quantized:
L=n ,
= 1.05 ´ 10-34 J × s
1 e
If n=1: m =
L = 0.9 ´ 10 -23 A × m 2 per atom 26
2m
Quantum Magnetism
Magnetic Moment = Orbital Motion + "Spin"
ORBITAL MOTION:
Fuzzy electron cloud.
Shapes are set by "spherical harmonics"
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
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Quantum Magnetism
Magnetic Moment = Orbital Motion + "Spin"
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
1 e
same angular momentum: m »
2m
NMR, MRI – use nuclear 
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Refrigerator Magnets
Alignment of atomic dipole moments:
ferromagnetic materials:
iron, cobalt, nickel
most materials
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Reality Physics - Domains
Magnetic domains
Apply
B-Field
to Align
Domains
Hitting or heating can also demagnetize
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Why are there Multiple Domains?
Magnetic domains
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iClicker Question
Top view, looking down on the table: The compass deflects 12 degrees
West. What is the direction of
electron current in the wire?
A)
B)
C)
D)
North
South
East
West
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Today
•
•
•
•
•
Magnetic Field of a Straight Wire
(Magnetic Field of a Current Loop)
Magnetic Dipole Moment
Bar Magnet
Atomic Dipoles
33
34
Magnetic Dipole Moment
1
Magnetic dipole moment of 1 atom: m = eRv
2
Method 2: estimate speed of electron
dp
Momentum principle:
= Fnet
dt
Circular motion: p = p = const
dp
v
= w p = mv = Fnet
dt
R
mv 2
1 e2
=
R
4pe 0 R 2
e2
v=
» 1.6 ´ 106 m/s
4pe 0 Rm
1
 – angular speed
w =v/R
m » 1.3 ´ 10-23 A × m2 /atom
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