PHY2049: Chapter 28 1

Chapter 28 Subjects Bar magnets

Poles, magnetic field lines, etc.

Effects of magnetic fields on charges and currentsForce on a moving chargePath followed by charged particle in magnetic fieldForce on a currentTorque on a current loop

ApplicationsMass spectrometersCyclotrons and synchrotronsHall effectElectric motor

PHY2049: Chapter 28 2

Bar Magnets

Two poles: “north” and “south”

Like poles repel

Unlike poles attract

So far, the poles are like electric charges

PHY2049: Chapter 28 3

Magnetic Monopoles?

Can any isolated magnetic charge exist?We would call this a “magnetic monopole”It would have a + or – magnetic charge

How can we isolate this magnetic charge?Cut a bar magnet in half? NO!

No one has ever found magnetic monopoles in nature.

What you getis a bunch oflittle magnets!

PHY2049: Chapter 28 4

PHY2049: Chapter 28 5

Bar Magnets (2)

NS

Similar to electric field produced by electric dipole

Sign convention:Field lines from North pole to South pole (for outside only)When used as a compass, North pole points north

PHY2049: Chapter 28 6

Earth is a big magnet!!

Earth’s magnetic poles have reversed very frequently on a geological time scale:http://science.nasa.gov/headlines/y2003/29dec_magneticfield.htm

The North pole of a small magnet (compass) points towards geographic Northbecause Earth’s magnetic South pole is presently up there!!

PHY2049: Chapter 28 7

Law of MagnetismAnalogy with Coulomb’s Law does not work

So far no one has found magnetic monopoles (=magnetic charges)Force between two small magnets is complicated; turns out to be not fundamental (should be deduced from a law that governs more fundamental phenomena)

What does this last statement mean?Magnetic field produced by magnet is not fundamentalMagnetic force on magnet is not fundamental either

Two phenomena turn out to be fundamentalElectric current produces magnetic fieldMagnetic field exerts force on moving charge

The law consists of two parts, two equationsMagnetic field produced by electric current (Chapter 29)Force due to magnetic field on moving charge (Chapter 28)

PHY2049: Chapter 28 8

Reading Quiz

The magnetic force on a moving charged particle is:

(a) Perpendicular to the velocity(b) Parallel to the velocity(c) Parallel to the B field(d) Independent of the velocity(e) None of the above

PHY2049: Chapter 28 9

Magnetic Field BPostponing until Chapter 29 the questions of how magnetic field is

produced and how its strength is varied, large number of experiments show

Choose the unit (tesla) such that

Force magnitude depends on direction of v relative to Bv is parallel to B ⇒ sinφ = 0v is perpendicular to B ⇒ sinφ = 1v is at angle 45° to B ⇒ sinφ = 0.71

Force direction is perpendicular to both B and vRight hand rule (next slide)

For given direction of v, force magnitude is proportional to v and B

sinF qv BF qvB φ= ×=

r rr

F qvB=0F =

sin 45F qvB=

B

+q

v

F (into page)

BvqFrrr

×∝m

PHY2049: Chapter 28 10

Right Hand RuleFirst point fingers in direction of velocity

Curl fingers toward B field⇒ Thumb points toward forceThis is for positive charge q

F

v

B

PHY2049: Chapter 28 11

Consider +q moving relative to a B field as shown

(a) Force is parallel to v(b) Force is parallel to B(c) Force is into the page(d) Force is out of the page

Right Hand Rule

B

+q

PHY2049: Chapter 28 12

A charged particle moves in a straight line through some region of space. Can you conclude that B = 0 there?

(a) Yes(b) No B field can exist since if v || B

there is no magnetic force

Bq v

Right Hand Rule

PHY2049: Chapter 28 13

A particle with negative charge enters a magnetic field region. What path will it follow?

ABCDE

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

Magnetic Force

AB

C

D

E(1) RHR says it bends down (− charge)(2) But force cannot instantaneously change v(3) So the answer is D, not E

PHY2049: Chapter 28 14

Magnetic Field UnitsFrom the expression for force:

B = Fmax / q vUnits: newton/(coulomb⋅meter/s) = N/(A⋅m) ≡ tesla (SI unit)Another unit: gauss = 10-4 tesla

Examples:Earth: B = 0.5 gauss = 5 x 10-5 TGalaxy: B ∼ 10-6 gauss = 10-10 TBar magnet: B ∼ 100 gauss = 10-2 TStrong electromagnet: B = 2 T (35 T in Tallahassee)Superconducting magnet: B = 5 – 10 T (20 T in Tallahassee,

also coming soon to UF)Pulsed magnet: B ∼ 100 TNeutron star: B ∼ 108 – 109 TMagnetar: B ∼ 1011 T

PHY2049: Chapter 28 15

Magnetic fields — large and small

45 T magnet during assembly, National High Magnetic Field Laboratory, Tallahassee~Million times the earth field.Requires 30 MW (300 thousand light bulbs!)

Magnetoencephalography (MEG)detects magnetic fields produced by brain activity (electric currents): ~10-12 TSensors require a 4 K temperature

PHY2049: Chapter 28 16

PulsarsRapidly Rotating Neutron Stars

Enormous Magnetic Fields

Beam off Beam on

Crab PulsarR = 10 kmM = 1.4 solar massB ≈ 108 TPeriod = 1/30 sec

Lighthouse effect

PHY2049: Chapter 28 17

Particle with m = 1.5 g, q = −2 µC moves with velocity 2 km/s through a magnetic field of 2.5 T at an angle of 30° to the field.

Magnitude of force

Direction of forceUp out of the page, from RHR

Note the negative charge!

Example

( )( )( )( )6sin 2 10 2.5 2000 0.5 0.005NF qBv φ −= = × =

B

−q

v

F (out of page)

PHY2049: Chapter 28 18

Magnetic Field and Work

Magnetic force is always perpendicular to velocityTherefore B field does no work!Why? Because

ConsequencesKinetic energy does not changeSpeed does not changeOnly direction changesParticle moves in a circle (if )

( ) 0K F x F v t∆ = ⋅∆ = ⋅ ∆ =r rr r

v B⊥rr

PHY2049: Chapter 28 19

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

Trajectory in a Constant Magnetic FieldParticle with charge q enters B field with velocity v perpendicular to B. What path will the particle follow?

Force is always ⊥ velocity and ⊥ BPath will be a circle. F is the centripetal force needed to keep the charge in its circular orbit. Let’s calculate radius R

Fv

R

v

B

qF

v

PHY2049: Chapter 28 20

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

(continued)

BqF

v

2mv qvBR

=mvRqB

=

PHY2049: Chapter 28 21

Magnetic Force Two particles of the same charge enter a magnetic field with the same speed. Which one has the bigger mass?

(a) A(b) B(c) Both masses are equal(d) Cannot tell without more info

x x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x xA B

mvRqB

=

Bigger mass means larger inertia, less acceleration, thus bigger radius

PHY2049: Chapter 28 22

Work and Energy

A charged particle enters a uniform magnetic field. What happens to the kinetic energy of the particle?

(a) it increases (b) it decreases(c) it stays the same(d) it depends on the direction of the velocity(e) it depends on the direction of the magnetic field

Magnetic field does no work, so kinetic energy does not change.