26. Magnetism: Force & Field

1. What is Magnetism?

2. Magnetic Force & Field

3. Charged Particles in Magnetic Fields

4. The Magnetic Force on a Current

5. Origin of the Magnetic Field

6. Magnetic Dipoles

7. Magnetic Matter

8. Ampere’s law

This ultraviolet image shows delicate loops of million-kelvin ionized gas in the Sun’s atmosphere.

What force shapes the gas into such intricate structures, and why don’t we see similar things in Earth’s atmosphere?

26.1. What is Magnetism?

Magnets exert (magnetic) forces on each other & materials like irons.

Iron filings align with the magnetic field, tracing out the field of a bar magnet.

Field description:A magnet produces a magnetic field B.Another interacts with B at its position.

Ampere :Moving charges can produce B.Magnetic dipoles ~ current loops.

Quantum mechanics:Intrinsic magnetic moment related to spin.

26.2. Magnetic Force & Field

Magnetic force on a charge q moving with velocity v in a field B:

B q F v B

[B] = N s / (C m)

= T (Tesla)

= 10,000 G (Gauss)

BEarth ~ 1 G

Bsmall magnet ~ 100 G

BMRI ~ 1 T

Bmagnetar ~ 1011 T

v B: greatest F

v // B:

sinBF q v B

GOT IT? 26.1

Figure shows a proton in a magnetic field.

For which of the three proton velocities shown will the magnetic force be greatest?

What will be the direction of the force in all three cases? & into paper

Example 26.1. Steering Protons

Figure shows 3 protons entering a 0.10-T magnetic field.

All three are moving at 2.0 Mm/s.

Find the magnetic force on each.

velocity selector.

q q F E v B Electromagnetic force

B

1

2

3

1ˆq v BF i 19 6 ˆ1.6 10 2.0 10 / 0.10C m s T i

14 ˆ3.2 10 N i ˆ32 f N i

3ˆq v BF i ˆ32 f N i

2 0F

q F v B

FB + FE = 0 when v = E/B

26.3. Charged Particles in Magnetic Fields

,B dF v x WB = 0

For v B, FB = q v B.

BF qvB2v

mr

m vr

q B

Circular motion

v = const CW about B for q > 0.

GOT IT? 26.2

A uniform magnetic field points out of this page.

Will an electron that’s moving in the plane of the page circle

(a) clockwise or

(b) counterclockwise

as viewed from above the page?

Example 26.2. Mass Spectrometer

A mass spectrometer separates ions according to their ratio of charge to mass.

Such devices are widely used to analyze unknown mixtures,

and to separate isotopes of chemical elements.

Figure below shows ions of charge q & mass m being

first accelerated from rest through a potential difference V,

then entering a region of uniform B pointing out of the page.

Find an expression for the horizontal distance x.

2 2m v

x rq B

22

m q V

q B m

21

2q V m v

2 2 m V

B q

2vm q v B

r

m vr

q B

The Cyclotron Frequency

Period of particle in circular orbit in uniform B:

2 rT

v

2 m v

v q B

2 m

q B

q B

m

2

T

Cyclotron frequency

Trajectory in 3-DCharged particles frozen to B field lines.

Motion // B not affected by it

Application: The Cyclotron

Whole device in vacuum chamber.Small V across the dee’s, which alternates polarities at the cyclotron frequency.

Particle injected at center of gap & spirals out.

E ~ MeVs.

Applications:Manufacture of radioactive isotopes.e.g., PET (Positron Emission Tomography).

Higher energies: Relativity effects Synchrotron (B also varies)

26.4. The Magnetic Force on a Current

dq f v BForce on carrier in wire:

dn A L q F v B

Force on straight wire of cross section A & length L:

I L B dI n A q v ˆ dLL v

Prob 58

F out of paper

e moving left deflected upward …

… resulting charge separation leads to upward force on whole wire

Fmag on + charge is also upward

GOT IT? 26.3

Figure shows a flexible wire passing through a magnetic field that points out of page.

The wire is deflected upward, as shown.

Is the current flowing

(a) to the left or

(b) to the right?

Example 26.3. Power Line

A power line runs along Earth’s equator, where B points horizontally from north to south and has strength 50 T.

The current in the line is 500 A, flowing from west to east.

Find magnitude & direction of the magnetic force on each kilometer of the power line.

ˆ25 N k

I F L B

upward

c.f. weight of wire is ~ 10 kN

3 6ˆ ˆ500 1.0 10 50 10A m T i j

The Hall Effect

Direction of FB depends on I, not on sign of charged carriers.

Carriers of both signs are deflected upwards

Direction of E due to built-up charges depends on signs of charged carriers:Hall effect.

Steady state, Fz = 0 :

H dE v B

H dV v B hI

B hn q A

I

Bn q t

Hall potential:

Hall coefficient: 1HR

n q

e moves to left & deflected upward

p moves to right & deflected upward

EH is upward

EH is downward

Hd

E Jv

B nq HR J

x

y

z0H dq E q v B

26.5. Origin of the Magnetic Field

Biot-Savart Law: 02

ˆ

4

I dd

r

L rB

7 20 10 /4

N A

710 /T m A permeability constant

02

ˆ

4

I d

r

L rB

Thumb // I

Curl of finger gives B.

302

ˆ

4d r

r

J r

C.f.3

20

1ˆ

4d r

r

E r

Example 26.4. Current Loop

Find the magnetic field at an arbitrary point P on the axis of a circular loop of radius a carrying current I.

02

ˆ

4

I d

r

L rB

ˆ cosx

d dL L r

By symmetry, only Bx 0.

adL

r

034x

I aB dL

r

03

24

I aa

r

2

032

I a

r

2

03/22 22

I a

x a

dL r

cos = a / r

Example 26.5. Straight Line

Find the magnetic field produced by an infinitely long straight wire carrying current I.

02

ˆ

4

I d

r

L rB

ˆ ˆsind dL L r z ˆy

dLr

z

034z

I yB dL

r

0

2

I

y

0

3/22 2

1

4I y d x

x y

0 ˆ2

I

d

B φ

3/2 2 2 22 2

1 xd x

a x ax a

22 2 2lim lim

x x

x x

a xa x a

2

1

a

The Magnetic Force Between Conductors

I F L B

0 11 1

ˆ2

I

d

B bField of I1 at I2 :

2 2 1I F L B

Force on length L of I2 :

0 1 2 ˆ2

I IL

d

d

2 0 1 22

ˆ2

I I

L d

F

f dForce per unit length on I2 : points toward I1

Hum from electric equipments are vibrations of transformers in response to AC.

Definition: 1 A is the current in two long, parallel wire 1 m apart & exerting 2107 N per meter of length.1C is the charge passing in 1s through a wire carrying 1A.

GOT IT? 26.4

A flexible wire is wound into a flat spiral as shown in figure.

If a current flows in the direction shown, will the coil

(a) tighten or

(b) become looser?

Does your answer depend on the current direction?No

26.6. Magnetic Dipoles

Field on axis of current loop of radius a :

2

03/22 22x

I aB

x a

20

32

I ax a x

0

32

I A

x

C.f. electric dipole:3

2p

E kx

Setting 0

4k

p I A magnetic dipole

N-turn current loop:

N Iμ A

Detailed calculations show:• = I A valid for arbitrary loop.• Vector behavior of similar to that of p for r >> a.

Far away, fields look similar …

… but close in, they’re different.

Sunspots are regions of intense magnetic field. The large sunspot group below the solar equator was responsible for some of the most energetic solar outbursts ever recorded, which occurred in late 2003.

Multi-turn loops = electromagnets

Very strong field requires superconducting wires, e.g., MRI scanners.

Orbiting e in atom .

Currents flows in Earth liquid outer core geomagnetic field,

which deflects harmful high E solar particles.

Magnetic reversal:

Earth (period ~ millions of yrs) :

map for sea-flooe spreading over.

Sun (period ~ 11 yrs.) : sunspots.

Dipoles & Monopoles

Some elementary particle theories suggest existence of magnetic monopoles.

But none was ever observed.

Microscopic origin of B:1.Charged current.2.Intrinsic spin.

No magnetic charges : B lines always form loops,either encircling moving charges, or joining the 2 poles of a magnet.

0d B A Gauss Law

GOT IT? 26.5

The figure shows two fields.

Which could be a magnetic field?

Torque on a Magnetic Dipole

τ μ BTorque on dipole:

Associated energy: U μ B

Forces on top & bottom cancel.

Forces on sides also cancel; but give net torque.

sideF a I B

1sin

2side sideb F

1sin

2b a I B

2 side sinb a I B

sinB

beneath planeabove plane

Application: Electric Motors

Rotating loop

Brushes

Commutator

Battery

CCW

CW

Always CCW

Example 26.6. Hybrid Car Motor

Toyota’s Prius gas-electric hybrid car uses a 50-kW electric motor that develops a maximum torque of 400 Nm.

Suppose you want to produce this same torque in the motor just discussed,

with a 950-turn rectangular coil of wire measuring 30 cm by 20 cm,

in a uniform field of 50 mT.

How much current does the motor need?

max B

τ μ B

Max torque at sin = 1 :

N I A Rectangular coil :

max N I A B

maxIN A B

400

950 0.30 0.20 0.050

N m

m m T

140 A

950-turn coil

26.7. Magnetic Matter

Ferromagnetism:

e.g., Fe, Ni, Co, & their alloys.

Material divided into magnetic domains in which ’s are aligned.

T > TC : orientations of in different domains random

< > B (paramagnetic)

T < TC : orientations of in different domains aligned

< > 0 even when B = 0 (ferromagnetic)

Atomic current loops are CCW.

Adjacent loops cancel

No cancellation on boundaries:net I on surface.

// Bapplied

p // Eapplied

Distant field similar. Internal field opposite.

Field inside dipole < Eapplied

Field inside dipole > Bapplied

Paramagnetism :Materials with randomly oriented permanent and very weak - interaction.

< > = B with > 0 = magnetic susceptibility

Diamagnetism :Materials with no intrinsic .

< > = B with < 0( induced )

26.8. Ampere’s law

0 enclosedd I B r

Field around long wire carrying current I :

0 ˆ2

I

r

B φ from Biot-Savart law

20

0 2

Id r d

r

B r 0 I

True for arbitrary closed paths & steady currents:

Ampere’s law

c.f. Gauss’ law

net field from ALL sources

Bdr = 0

Example 26.7. Solar Currents

The long dimension of the rectangular loop in figure is 400 Mm,

and B near the loop has a constant magnitude of 2 mT.

Estimate the total current enclosed by the rectangle.

3 62 2 10 400 10T m rectangle

2d B L B r

B dr on the short segment B dr = 0 there.

B // dr on the long segment

61.6 10 T m

0

2 B LI

121.3 10 A

6

7

1.6 10

4 10 /

T m

T m A

Direction: into the page3D: around equator

GOT IT? 26.6

The figure shows three parallel wires carrying current of the same magnitude I, but in one of them the current is opposite to that of the other two.

If 0d B r around loop 2,

(a) what is dB r around loop 1, and

(b) which current is the opposite one? A

0

Using Ampere’s Law

Base on symmetry, choose the amperian loop such that B is either // or to it.

STRATEGY 26.1 Ampère’s Law :

Example 26.8. Outside & Inside a Wire

A long, straight wire of radius R carries a current I distributed uniformly over its cross section.

Find the magnetic field

(a) outside and

(b) inside the wire.

enclosedI I

2d r B B r

Amperian loop is a circle.

0

2

IB

r

(a)

2

2enclosed

rI I

R

022

I rB

R

(b)

By symmetry, B is azimuthal.

Example 26.9. Current Sheet

An infinite flat sheet carries current out of this page.

The current is distributed uniformly along the sheet, with current per unit width given by JS .

Find the magnetic field of this sheet.

enclosed SI J L

2d B L B r

Amperian loop is a rectangle.

0

1

2 SB J

By symmetry, B is // to sheet & I.

Far out, B lines nearly circular

Close in, B lines ~ infinite sheet

Fields of Simple Current Distributions

For arbitrary distributions:

Far away from any loop ~ dipole.

Very near any wire ~ infinite straight wire.

Very near any current sheet ~ infinite flat sheet.

Solenoids

Solenoid: a long, tightly wounded coil.

d B L B r enclosedI n L I

0B n IBoutside = 0

solenoid field

Application: magnetic switches, valves, ….

n turns per unit length.n L loops in L.encircled current = nLI.

Table 26.1. Fields of Ssome Simple Charge & Current Distributions

Field(r) Q E J B

r 3

r 2

r 1

const

Dipole

Point / spherical

Line

Sheet

Dipole

NA

Line

Sheet

Example 26.10. MRI Scanner

The solenoid used in an MRI scanner is 2.4 m long & 95 cm in diameter.

It’s wounded from superconducting wire 2.0 mm in diameter,

with adjacent turns separated by an insulating layer of negligible thickness.

Find the current that will produce a 1.5-T magnetic field inside the solenoid.

7 1

1.5

4 10 / 500

T

T m A m

0

BI

n

2.4 kA

wire diameter = 1/500 m

n = 500 turns/m

Field of a solenoid is very similar to that of a bar magnet.

Toroids

Toroid : a solenoid wrapped around on itself.

Used for confining plasma for fusion.If successful: 1 L of sea water ~ 300 L of gasoline

0

2

N IB

r

toroid field

2d r B B r

Field lines are circles inside toroid.

enclosedI N I

Amperian loop coincide with B