Forces & Magnetic Dipoles

Bx

.FF

BU

B

AI

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

F q v B

mvR

qB

212qV mv

2 2

2q V

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

2 2

2

m R B

q V

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.

End view: B into screen

e+

e-

B

2mv mKR

qB qB

+ charged particle

- charged particle

Magnetic Force on a Current

• Consider a current-carrying wire in thepresence of a magnetic field B.

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

BlIdFd

BvqdlnAFd

)(

nAvqdt

qdtnAv

dt

dqI

)(

Bvq

• Force on each charge =

• Total force =

• Current =

N S

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

BLIF

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!

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

B

x

.FF

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

B

I

F

F

FF– Force on top path cancels force

on bottom path (F = IBL)

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

1

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

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

Preflight 13:

BLIF

ab: Fab = 0 = Fcd since the wire is parallel to B. bc: Fbc = ILB RHR: I is up, B is to the right, so Fpoints into the screen.

By symmetry:da bcF F

net ab bc cd da 0F F F 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 (c) Fy > 0

x x x x xx x x x xx x x x xx x x x xx x x x xx x x x xx x x x xx x x x x

LL

L

B

x

y

2

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 (c) Fy > 0

x x x x xx x x x xx x x x xx x x x xx x x x xx x x x xx x x x xx x x x x

LL

L

B

x

y

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

• The forces on each segment are determined by: BLIF

• From symmetry, Fx = 0• For the y-component: 1 2

2y y

BILF F

3 ( 2 )yF BI L

• Therefore: 0321 yyyy FFFF

F1 F2

F3

45˚

2

Calculation of Torque

• Suppose the coil has width w (the side we see) and length L (into the screen). The torque is given by:

• Note: if loop B, sin = 0 = 0maximum occurs when loop parallel to B

Frτ

AIBsinF = IBL

where A = wL = area of loop

sin

22 F

wτ

r × F

F

r

B

x

.

w

F F

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

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

BFree rotation of spindle

Motors, continuedEven 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 It

2

Lecture 13, Act 2• What should we do to

increase the speed (rpm) of a DC motor?2A

2B • What should we do to increase the speed of an AC motor?

(a) Increase I (b) Increase B (c) Increase number of loops

B

(a) Increase I (b) Increase B (c) Increase number of loops

Lecture 13, Act 2• What should we do to

increase the speed (rpm) of a DC motor?2A

(a) Increase I (b) Increase B (c) Increase number of loops

B

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

Lecture 13, Act 2• What should we do to

increase the speed (rpm) of an AC motor?2B

B

(a) Increase I (b) Increase B (c) Increase number of loops

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

• We can define the magnetic dipole moment of a current loop as follows:

direction: to plane of the loop in the direction the thumb of right hand points if fingers curl in the direction of current.

magnitude: AI

• Torque on loop can then be rewritten as:

• Note: if loop consists of N turns, = NAI

AIBsin Bμτ

Bx

.

F F

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

magnet:

– 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!).

=N

μ

Electric Dipole Analogy

Frτ

Frτ

(per turn)EqF

BLIF

aqp

2 NAIμ

Epτ

Bμτ

Bx

.

E

.

+q

-qF F

F

F

p

6) What is the net torque on the loop?

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

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

Preflight 13:

Frτ

The torque due to Fcb is up, since r (from the center of the loop) is to the right, and Fcb is into the page. Same goes for Fda.

Bμτ

points out of the page (curl your fingers in the direction of the current around the loop, and your thumb gives the direction of ). Use the RHR to find the direction of 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.

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

θ

τdθU90

θ

θdθμBU90

sin

Therefore,

θθμBU 90cos θμBU cos BμU

Bx

.

F

F

Potential Energy of Dipole

B

x

Bx

B

x

= 0

U = -B

= 0

U = B

negative work positive work3

=B

X

U = 0

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

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?

8) What direction is the torque on loop 1?

a) clockwise b) counter-clockwise c) zero

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

Preflight 13:

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

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

Loop 1: U1 = B Loop 2: U2 = B U2 is a minimum.

BμU

Lecture 13, Act 3• A circular loop of radius R carries current I as

shown in the diagram. A constant magnetic field B exists in the +x direction. Initially the loop is in the x-y plane.

– The coil will rotate to which of the following positions?

3A

3B

(a) U0 is minimum (b) U0 is maximum (c) neither

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

I

R

x

y B

a b

(a) (b) (c) It will not rotate

z

y

a b

z

y

a b

Lecture 13, Act 3• A circular loop of radius R carries current I as

shown in the diagram. A constant magnetic field B exists in the +x direction. Initially the loop is in the x-y plane.

– The coil will rotate to which of the following positions?

3A

(a) (b) (c) It will not rotate

• The coil will rotate if the torque on it is non-zero: Bμτ

• The magnetic moment is in +z direction.• Therefore the torque is in the +y direction.• Therefore the loop will rotate as shown in (b).

z

y

a b

z

y

a b

I

R

x

y B

a b

Lecture 13, Act 3• A circular loop of radius R carries current I as shown

in the diagram. A constant magnetic field B exists 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?

3B

(a) U0 is minimum (b) U0 is maximum (c) neither

– The potential energy of the loop is given by: BμU

– 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!

I

R

x

y B

a b

Example: Loop in a B-FieldA 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 = 45 to the xy-plane.

= r2 I = .0157 Am2

What is the magnetic moment of the loop?x x x xx x x xx x x xx x x xx x x xx x x xx x x x

I

x

yB

z

y x

zX

B

X

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

Find the x and z components of :

x = – sin45 = –.0111 Am2

z = cos45 = .0111 Am2

Summary

• Mass Spectrometer• Force due to B on I

• Magnetic dipole– torque

– potential energy

• Applications: dials, motors, NMR, …• Next time: calculating B-fields from

currents

dF I dl B

AIN

B

90U B zero defined at

Reading assignment: Chapter 29.1 through 29.3

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.

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: 1U B Anti-aligned: 2U B

Energy Difference: 2 1 2U U U B

MRI / NMR Example

262 2.7 10 JU B

26

-34

2.7 10 J41 MHz

6.6 10 J sf

Aligned: 1U B Anti-aligned: 2U BEnergy Difference: 2 1 2U U U B

B = 1 Tesla (=104 Gauss) (note: this is a big field!)

μproton = 1.36•10-26 Am2

In QM, you will learn that photonenergy = frequency • Planck’s constant

h ≡ 6.6•10-34 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 , if we put a strong magnetic field gradient across the sample, we can look at individual slices, with ~millimeter spatial resolution.

f U B

B

Small Blow freq.

Bigger Bhigh freq.

Signal at the right frequency only from this slice!

See it in action!

Thanks to