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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)
  • Nuclear Magnetic Resonance Imaging

Text Reference: Chapter 28.1 and 28.3

Examples: 28.2, 28.3, and 28.6 through 28.10

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
slide9

Preflight 13:

A square loop of wire is carrying current in the counterclockwise direction. There is a horizontal uniform magnetic field pointing to the right.

2) What is the force on section a-b of the loop? a) zero b) out of the page c) into the page

3) What is the force on section b-c of the loop? a) zero b) out of the page c) into the page

4) What is the net force on the loop? a) zero b) out of the page c) into the page

slide10

By symmetry:

ab:Fab = 0 = Fcdsince the wire is parallel toB.

bc:Fbc = ILBRHR:Iis up,Bis to the right, soF

points into the screen.

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
lecture 13 act 11
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

  • The forces on each segment are determined by:
  • From symmetry,Fx = 0
  • For the y-component:

45˚

F1

F2

  • Therefore:

F3

Lecture 13, Act 1

There is never a net force on a loop in a uniform field!

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:
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.

slide15

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

slide16

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
What should we do to increase the speed (rpm) of a DC motor?

B

2A

(a)IncreaseI

(a)IncreaseI

(b) IncreaseB

(b) IncreaseB

(c)Increase number of loops

(c)Increase number of loops

2B

Lecture 13, Act 2
  • What should we do to increase the speed of an AC motor?
lecture 13 act 21
What should we do to increase the speed (rpm) of a DC motor?

B

2A

(a)IncreaseI

(b) IncreaseB

(c)Increase number of loops

Lecture 13, Act 2

All of the above, though for variable speed motors, it is always the current that is adjusted.

lecture 13 act 22
What should we do to increase the speed (rpm) of an AC motor?

B

2B

(a)IncreaseI

(b) IncreaseB

(c)Increase number of loops

Lecture 13, Act 2

Again, all of the above!

BUT, while the answers above are still true, the main change is to vary the drive frequency—this must match the rotation rate.

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!).
electric dipole analogy

E

B

+q

x

q

.

.

q

-q

(per turn)

Electric Dipole Analogy
slide23

Preflight 13:

A square loop of wire is carrying current in the counterclockwise direction. There is a horizontal uniform magnetic field pointing to the right.

6) What is the net torque on the loop?

a) zero b) up c) down d) out of the page e) into the page

slide24

The torque due toFcbis up, sincer(from the center of the loop) is to the

right, andFcbis into the page. Same goes forFda.

mpoints out of the page (curl your fingers in the direction of the current around the loop, and your thumb gives the direction of m). Use the RHR to find the direction of t to be up.

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

slide27

Preflight 13:

Two current carrying loops are oriented in a uniform magnetic field. The loops are nearly identical, except the direction of current is reversed.

8) What direction is the torque on loop 1?

a) clockwise b) counter-clockwise c) zero

9) How does the torque on the two loops compare?

a) τ1 > τ2 b) τ1 = τ2 c) τ1 < τ2

10) Which loop occupies a potential energy minimum, and is therefore stable?

a) loop 1 b) loop 2 c) the same

slide28

Loop 1: U1 = +mB

Loop 2: U2 = -mBU2is a minimum.

Loop 1: m points to the left, so the angle between mand B is equal to 180º, hence t = 0.

Loop 2: m points to the right, so the angle between mand B is equal to 0º, hence t = 0.

lecture 13 act 3
A circular loop of radiusRcarriescurrentI as shown in the diagram. A constantmagnetic fieldBexists in the+xdirection.Initially the loop is in the x-y plane.

The coil will rotate to which of the following positions?

y

B

R

3A

b

I

a

x

3B

Whatis the potential energyU0of the loop in its initial position?

(c)neither

(b)U0 is maximum

(a)U0 is minimum

Lecture 13, Act 3

y

y

w

w

(b)

(a)

(c)It will not rotate

b

a

b

a

z

z

lecture 13 act 31
A circular loop ofradius R carriescurrent Ias shown in the diagram. A constantmagnetic field Bexists in the+x direction. Initially the loop is in the x-y plane.

The coil will rotate to which of the following positions?

y

B

R

3A

b

I

a

x

y

y

w

w

b

a

b

a

z

z

  • The coil will rotate if the torque on it is non-zero:
  • The magnetic momentm is in+z direction.
  • Therefore the torquet is in the+y direction.
  • Therefore the loop will rotate as shown in (b).
Lecture 13, Act 3

(b)

(a)

(c)It will not rotate

lecture 13 act 32
A circular loop ofradius Rcarriescurrent Ias shown in the diagram. A constantmagnetic field Bexists in the+x direction. Initially the loop is in the x-y plane.

What is the potential energy U0 of the loop in its initial position?

y

B

R

b

I

a

x

  • The potential energy of the loop is given by:
  • In its initial position, the loop’s magnetic moment vector points in the +z direction, so initial potential energy is ZERO
    • This does NOT mean that the potential energy is a minimum!!!
    • When the loop is in the y-z plane and its magnetic moment points in the same direction as the field, its potential energy is NEGATIVE and is in fact the minimum.
  • Since U0 is not minimum, the coil will rotate, converting potential energy to kinetic energy!

3B

(c)neither

(b)U0 is maximum

(a)U0 is minimum

Lecture 13, Act 3
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

z

m

X

mx=–m sin45=–.0111 Am2

B

mz=mcos45=.0111 Am2

q

X

y

x

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= 45to the xy-plane.

Example: Loop in a B-Field

What is themagnetic momentmof the loop?

m = pr2 I = .0157 Am2

summary
Mass Spectrometer

Force due to B on I

Magnetic dipole

torque

potential energy

Applications: dials, motors, NMR, …

Nexttime:calculatingB-fieldsfrom currents

Summary

Reading assignment: Chapter 29.1 through 29.3

mri magnetic resonance imaging nmr nuclear magnetic resonance

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)

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 ≡ 6.6•10-34 J s

Aligned:

Anti-aligned:

?

What does this have to do with

Energy Difference:

MRI / NMR Example

μproton=1.36•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