Transcript Lect13

Forces & Magnetic Dipoles
B
x
q
F
F
q
.
m
m  AI
  
  mB

U  m  B

Today...
• Application of equation for trajectory of charged
particle in a constant magnetic field: Mass
Spectrometer
• Magnetic Force on a current-carrying wire
• Current Loops
• Magnetic Dipole Moment
• Torque (when in constant B field)  Motors
• Potential Energy (when in constant B field)
• Appendix: Nuclear Magnetic Resonance Imaging
Solar Flare/Aurora Borealis Pictures
Last Time…
• Moving charged particles are deflected in magnetic
fields F  q v  B
mv
• Circular orbits R 
qB
• If we use a known voltage V to accelerate a particle
q
2V
2
1
 2 2
qV  2 mv
m R B
• Several applications of this
• Thomson (1897) measures q/m ratio for “cathode rays”
•
• All have same q/m ratio, for any material source
• Electrons are a fundamental constituent of all matter!
Accelerators for particle physics
• One can easily show that the time to make an orbit does not
depend on the size of the orbit, or the velocity of the particle
→ Cyclotron
Mass Spectrometer
• Measure m/q to identify substances
m R2 B2

q
2V
1. Electrostatically accelerated electrons knock
electron(s) off the atom  positive ion (q=|e|)
2. Accelerate the ion in a known potential U=qV
3. Pass the ions through a known B field
–
Deflection depends on mass: Lighter deflects more, heavier less
Mass Spectrometer, cont.
4. Electrically detect the ions which “made it through”
5. Change B (or V) and try again:
Applications:
Paleoceanography: Determine relative abundances of isotopes
(they decay at different rates  geological age)
Space exploration: Determine what’s on the moon, Mars, etc.
Check for spacecraft leaks.
Detect chemical and biol. weapons (nerve gas, anthrax, etc.).
See http://www.colby.edu/chemistry/OChem/DEMOS/MassSpec.html
Yet another example
• Measuring curvature of
charged particle in
magnetic field is usual
method for determining
momentum of particle in
modern experiments:
e.g.
e+
mv
2mK
R

qB
qB
- charged
particle
+ charged
particle
B
e-
End view: B into screen
Magnetic Force on a Current
• Consider a current-carrying wire in the
N
presence of a magnetic field B.
• There will be a force on each of the charges
moving in the wire. What will be the total force
dF on a length dl of the wire?
• Suppose current is made up of n charges/volume
each carrying charge q and moving with velocity
v through a wire of cross-section A.
 
• Force on each charge = qv  B

 
• Total force = dF  nA(dl )qv  B
• Current =
dq nAv(dt )q
I

 nAvq 
dt
dt
Yikes! Simpler: For a straight length of wire L
carrying a current I, the force on it is:
 

dF  Idl  B

 
F  IL  B
S
Magnetic Force on a Current Loop
• Consider loop in magnetic field as
on right: If field is ^ to plane of
loop, the net force on loop is 0!
– Force on top path cancels force
on bottom path (F = IBL)
x
x
Fx
x
x
x
x
x
x
x
x
x
x
x
x
– Force on right path cancels
force on left path. (F = IBL)
• If plane of loop is not ^ to field, there
will be a non-zero torque on the loop!
x
x
x
x
x
B
x
x
x
x F
x
B
x
F
1
F
x x
x x
x x
x x
x xI
F
F
.
Lecture 13, Act 1
• A current I flows in a wire which is
formed in the shape of an isosceles
right triangle as shown. A constant
magnetic field exists in the -z direction.
– What is Fy, net force on the wire in
the y-direction?
(a) Fy < 0
(b) Fy = 0
y
B
(c) Fy > 0
x
x
x
x
x
x
x
x
x x
x x
x x
x Lx
x x
x2L x
x x
x x
x
x
x
Lx
x
x
x
x
x
x
x
x
x
x
x
x
x
Calculation of Torque
• Suppose the coil has width w (the
side we see) and length L (into the
screen). The torque is given by:
  
τ  r F

w

τ  2 F sin q 
2

F = IBL 
where
  AIB sinq
B
x
q
w
F
F
.
r
A = wL = area of loop
F
• Note: if loop ^ B, sinq = 0   = 0
maximum  occurs when loop parallel to B
r ×F
Magnetic Dipole Moment
• We can define the magnetic dipole moment of a
current loop as follows:
magnitude:
m  AI
q
direction: ^ to plane of the
loop in the direction the thumb
of right hand points if fingers
curl in the direction of current.
B
x
F
q
F
m
• Torque on loop can then be rewritten as:
  
τ  μ B
  AIB sinq 
• Note: if loop consists of N turns, m = NAI
.
Bar Magnet Analogy
• You can think of a magnetic dipole moment as a bar
magnet:
μ
N
=
– In a magnetic field they both experience a torque trying to line
them up with the field
– As you increase I of the loop  stronger bar magnet
– N loops  N bar magnets
• We will see next lecture that such a current loop does produce
magnetic fields, similar to a bar magnet. In fact, atomic scale
current loops were once thought to completely explain magnetic
materials (in some sense they still are!).
Applications: Galvanometers
(≡Dial Meters)
We have seen that a magnet can exert a torque on a loop of
current – aligns the loop’s “dipole moment” with the field.
– In this picture the loop (and hence the
needle) wants to rotate clockwise
– The spring produces a torque in the
opposite direction
– The needle will sit at its equilibrium
position
Current increased
 μ = I • Area increases
 Torque from B increases
 Angle of needle increases
Current decreased
 μ decreases
 Torque from B decreases
 Angle of needle decreases
This is how almost all dial meters work—voltmeters, ammeters,
speedometers, RPMs, etc.
Motors
Free rotation
of spindle
B
Slightly tip the loop
Restoring force from the magnetic torque
Oscillations
Now turn the current off, just as the loop’s μ is
aligned with B
Loop “coasts” around until its μ is ~antialigned with B
Turn current back on
Magnetic torque gives another kick to the loop
Continuous rotation in steady state
Motors, continued
Even better
Have the current change directions every half rotation
Torque acts the entire time
Two ways to change current in loop:
1. Use a fixed voltage, but change the circuit (e.g., break
connection every half cycle
 DC motors
2. Keep the current fixed, oscillate the source voltage
AC motors
VS I
t
2
Lecture 13, Act 2
• How can we increase the
speed (rpm) of a DC
2A
motor?
(a) Increase I
(b) Increase B
B
(c) Increase number of loops
Example: Loop in a B-Field
A circular loop has radius R = 5 cm and carries current I = 2 A in the
counterclockwise direction. A magnetic field B =0.5 T exists in the
negative z-direction. The loop is at an angle q = 30 to the xy-plane.
B
y
x
x
x
x
x
x
x
z
x x
x x
x x
xI x
x x
x x
x x
What is the magnetic moment m of the loop?
x
x
x
x
x
x
x
m = p r2 I = .0157 Am2
The direction of m is perpendicular to the
plane of the loop as in the figure.
Find the x and z components of m :
x
z
m
mx = –m sin 30 = –.0079 Am2
X
B
mz = m cos 30 = .0136 Am2
q
yX
x
Electric Dipole Analogy
+q
F
E
q
p
F
  
τ  r F


F  qE
B
x
.
-q
F
q
F
.
m
  
τ  r F
  
F  IL  B (per turn)


p  2qa
μ  NAI
  
τ  p E
  
τ  μ B
Potential Energy of Dipole
• Work must be done to change the
orientation of a dipole (current loop)
in the presence of a magnetic field.
q
F
• Define a potential energy U (with zero at
position of max torque) corresponding to
this work.
θ
U
 τdθ
90
B
x
F
q
.
m
θ

Therefore,
U  μB cos θ 
θ
90
U
 μB sin θdθ
90
 
 U   μB cos θ  U   μ  B
Potential Energy of Dipole
m
B
m
x
B
x
m
x
=0
 = mB X
=0
U = -mB
U=0
U = mB
negative work
positive work
B
Summary
• Mass Spectrometer
• Force due to B on I dF  I dl  B
• Magnetic dipole m  AIN
  
– torque
  mB
– potential energy U   m  B
zero defined at q  90
• Applications: dials, motors, NMR, …
• Next time: calculating B-fields from
currents
MRI (Magnetic Resonance Imaging) ≡
NMR (Nuclear Magnetic Resonance)
[MRI invented by UIUC Chem. Prof. Paul Lauterbur,
who shared 2003 Nobel Prize in Medicine]
A single proton (like the one in
every hydrogen atom) has a charge
(+|e|) and an intrinsic angular
momentum (“spin”). If we (naively)
imagine the charge circulating in a
loop  magnetic dipole moment μ.
In an external B-field:
– Classically: there will be torques unless m is aligned along B or
against it.
– QM: The spin is always ~aligned along B or against it
Aligned: U1  m B
Anti-aligned: U 2  m B
Energy Difference: U  U 2  U1  2m B
MRI / NMR Example
Aligned: U1  m B
Anti-aligned: U 2  m B
Energy Difference: U  U 2  U1  2m B
μproton = 1.41•10-26 Am2
B = 1 Tesla (=104 Gauss)
U  2m B  2.82 10 J
26
(note: this is a big field!)
In QM, you will learn that photon
energy = frequency • Planck’s constant
h ≡ 6.63•10-34 J s
2.82 1026 J
f 
 42.5 MHz
-34
6.63 10 J s
What does this have to do with
?
MRI / NMR continued
If we “bathe” the protons in radio waves at this frequency, the
protons can flip back and forth.
If we detect this flipping  hydrogen!
The presence of other molecules can partially shield the applied B,
thus changing the resonant frequency (“chemical shift”).
Looking at what the resonant frequency is  what molecules are
nearby.
Finally, because f  U  B, if we put a strong magnetic field
gradient across the sample, we can look at individual slices, with
~millimeter spatial resolution.
B
Small B
low freq.
Bigger B
high freq.
Signal at the right frequency only from this slice!
See it in action!
Thanks to
Solar Flare/Aurora Borealis links
http://cfa-www.harvard.edu/press/soolar_flare.mov
http://science.nasa.gov/spaceweather/aurora/gallery_01oct03_page2.html