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B. x. q. F. F. q. m. Forces & Magnetic Dipoles. 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

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forces magnetic dipoles

B

x

q

F

F

.

q

m

Forces & Magnetic Dipoles

today
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

  • Circular orbits
  • If we use a known voltage V to accelerate a particle
Last Time…
  • 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

Mass Spectrometer
  • Electrostatically accelerated electrons knock electron(s) off the atom  positive ion (q=|e|)
  • Accelerate the ion in a known potential U=qV
  • Pass the ions through a known B field
    • Deflection depends on mass: Lighter deflects more, heavier less
mass spectrometer cont

Electrically detect the ions which “made it through”

  • Change B (or V) and try again:
Mass Spectrometer, cont.

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.).

Seehttp://www.colby.edu/chemistry/OChem/DEMOS/MassSpec.html

slide6

- charged particle

+ charged particle

Yet another example

  • Measuring curvature of charged particle in magnetic field is usual method for determining momentum of particle in modern experiments: e.g.

B

e+

e-

End view: B into screen

magnetic force on a current
Consider a current-carrying wire in the

presence of a magnetic fieldB.

There will be a force on each of the charges moving in the wire. What will be the total force

dFon a lengthdlof the wire?

Suppose current is made up ofncharges/volume each carrying chargeqand moving with velocityvthrough a wire of cross-sectionA.

  • Force on each charge =
  • Total force =

Þ

  • Current =

Yikes! Simpler: For a straight length of wire L carrying a current I, the force on it is:

Magnetic Force on a Current

N

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!

F

B

x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

F

F

  • Force on top path cancels force on bottom path(F = IBL)

I

F

  • Force on right path cancels force on left path.(F = IBL)

B

x

  • If plane of loop is not ^ to field, there will be a non-zero torque on the loop!

F

F

.

Magnetic Force on a Current Loop
lecture 13 act 1
A currentIflows 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?

y

x x x x x

x x x x x

x x x x x

L

L

x x x x x

B

x x x x x

x x x x x

L

x x x x x

(a) Fy < 0

x x x x x

(c) Fy > 0

(b) Fy = 0

x

Lecture 13, Act 1
calculation of torque
Note: if loop^B, sinq = 0Þt = 0

maximumtoccurs when loopparallel toB

B

x

q

w

.

Þ

Þ

t = AIBsinq

F = IBL

where

A = wL = area of loop

Calculation of Torque
  • Suppose the coil has widthw(the side we see) and lengthL(into the screen). The torque is given by:
magnetic dipole moment
direction:^to plane of the loop in the direction the thumb of right hand points if fingers curl in the direction of current.

m = AI

magnitude:

B

x

q

.

q

  • Torque on loop can then be rewritten as:

Þ

t = AIBsinq

  • Note: if loop consists of N turns,m= NAI
Magnetic Dipole Moment
  • We can define the magnetic dipole moment of a current loop as follows:
bar magnet analogy

μ

N

=

  • You can think of a magnetic dipole moment as a bar magnet:
Bar Magnet Analogy
    • 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

Current increased

  • μ=I • Area increases
  • Torque fromBincreases
  • Angle of needle increases

Current decreased

  • μdecreases
  • Torque fromBdecreases
  • Angle of needle decreases
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

This is how almost all dial meters work—voltmeters, ammeters, speedometers, RPMs, etc.

slide14

B

Free rotation of spindle

Motors

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 itsμ is ~antialigned withB

Turn current back on

Magnetic torque gives another kick to the loop

Continuous rotation in steady state

slide15

VS I

t

Motors, continued

  • Even better
  • Have the current change directions every half rotation
  • Torque acts the entire time
  • Two ways to change current in loop:
    • 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
lecture 13 act 2
How can we increase the speed (rpm) of a DC motor?

B

2A

(a)IncreaseI

(b) IncreaseB

(c)Increase number of loops

Lecture 13, Act 2
example loop in a b field

B

y

x x x x

x x x x

x x x x

x x x x

x x x x

x x x x

x x x x

I

The direction ofmis perpendicular to the plane of the loop as in the figure.

Find thexandzcomponents ofm:

x

z

X

mx=–m sin 30=–.0079 Am2

mz=mcos 30=.0136 Am2

A circular loop has radiusR = 5 cmand carries currentI = 2 Ain the

counterclockwise direction. A magnetic fieldB =0.5 T exists in the

negative z-direction. The loop is at an angleq= 30to the xy-plane.

Example: Loop in a B-Field

What is themagnetic momentmof the loop?

m = pr2 I = .0157 Am2

z

m

B

q

X

y

x

electric dipole analogy

E

B

+q

x

q

.

.

q

-q

(per turn)

Electric Dipole Analogy
potential energy of dipole
Work must be done to change the orientation of a dipole (current loop) in the presence of a magnetic field.

B

x

q

.

q

  • Define a potential energy U (with zero at position of max torque) corresponding to this work.

Þ

Therefore,

Þ

Þ

Potential Energy of Dipole
potential energy of dipole1

m

x

B

B

B

m

m

t = mB

x

x

X

negative work

positive work

Potential Energy of Dipole

t = 0

U = -mB

t = 0

U = mB

U = 0

summary
Mass Spectrometer

Force due to B on I

Magnetic dipole

torque

potential energy

Applications: dials, motors, NMR, …

Nexttime:calculatingB-fieldsfrom currents

Summary
slide22

In an external B-field:

  • Classically: there will be torques unless is aligned along B or against it.
  • QM: The spin is always ~aligned along B or against it

Aligned:

Anti-aligned:

Energy Difference:

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.

mri nmr example

h ≡ 6.63•10-34 J s

Aligned:

Anti-aligned:

?

What does this have to do with

Energy Difference:

MRI / NMR Example

μproton=1.41•10-26 Am2

B = 1 Tesla (=104 Gauss)

(note: this is a big field!)

In QM, you will learn that photon

energy = frequency • Planck’s constant

mri nmr continued

B

Small B

low freq.

Bigger B

high freq.

Signal at the right frequency only from this slice!

  • 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 , if we put a strong magnetic field gradient across the sample, we can look at individual slices, with ~millimeter spatial resolution.
MRI / NMR continued
solar flare aurora borealis links
Solar Flare/Aurora Borealis links

http://cfa-www.harvard.edu/press/soolar_flare.mov

http://science.nasa.gov/spaceweather/aurora/gallery_01oct03_page2.html