chapter 28: magnetic fields - university of · pdf filephy2049: chapter 28 1 chapter 28:...

PHY2049: Chapter 28 1

Chapter 28: Magnetic Fields


Magnetic Fields Magnetic field (units, field lines)

Magnetic field of the earth and other astronomical objects

Effects of magnetic fields on charges and currentsForce on a moving chargeForce on a currentTorque on a current loopPath followed by particle in magnetic field

Generating magnetic fieldsLong wireCurrent loopSolenoid

InstrumentsMass spectrometersCyclotrons and synchrotrons


Reading QuizThe magnetic force on a moving charged particle is:

(1) Perpendicular to the velocity(2) Parallel to the velocity(3) Parallel to the B field(4) Independent of the velocity(5) None of the above


Reading QuizConsider +q moving relative to a B field as shown

Force is parallel to vForce is parallel to BForce is into the pageForce is out of the page

B

+q


Reading QuizWhen I cut a magnet into two pieces I get:

An isolated north and south magnetic poleTwo smaller magnetsThe two pieces are no longer magnets


Bar MagnetsTwo poles: “north” and “south”

Like poles repel

Unlike poles attract

Magnetic poles cannot be isolated

NS

Similar to dipole field from electrostatics


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!


Searches for Magnetic Monopoles


Earth is a big magnet!!

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

Particles moving along field lines cause Aurora Borealis.http://science.nasa.gov/spaceweather/aurora/gallery_01oct03.html


What Causes Magnetism?What is the origin of magnetic fields?

Electric charge in motion!For example, a current in a wire loop produces a field very similar to that of a bar magnet (as we shall see).

Understanding the source of bar magnet field lies in understanding currents at the atomic level within matter

Orbits of electrons about nuclei

Intrinsic “spin” of electrons (more important effect)


Magnetic Field UnitsFrom the expression for force on a current-carrying wire:

B = Fmax / I LUnits: Newtons/A⋅m ≡ Tesla (SI unit)Another unit: gauss = 10-4 Tesla

Some sample magnetic field strengths:Earth: B = 0.5 gauss = 0.5 x 10-4 TGalaxy: B ∼ 10-6 gauss = 10-10 TBar magnet: B ∼ 100 – 200 gaussStrong electromagnet: B = 2 TSuperconducting magnet: B = 5 – 10 TPulse magnet: B ∼ 100 TNeutron star: B ∼ 108 – 109 TMagnetar: B ∼ 1011 T


PulsarsRapidly Rotating Neutron Stars

Enormous Magnetic Fields

Beam off Beam on

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


Magnetic Field BMagnetic field defined by magnetic force on a test charge

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)

sinF qv BF qvB φ= ×=

F qvB=0F =

sin 45F qvB=

B

+q

v

F (into page)


Right Hand RuleFirst point fingers in direction of velocity

Curl fingers toward B field⇒ Thumb points toward force

F

v

B


ExampleParticle with m = 1.5 g, q = −2μC moves with velocity 2,000 m/s through a magnetic field of 2.5 T at an angle of 30° to the field.

Magnitude of force

Direction of force: up out of the page, from RHR

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

B

−q

v

F (up)


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

1. Yes2. No

A B field can exist since if v || Bthere is no magnetic force

Bq v


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

(1) A(2) B(3) C(4) D(5) E

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


Magnetic Force on Current-Carrying WireMagnitude of force

Easy to derive from charge, number density & drift velocity of individual charge carriers

Direction of force: RHR

sinF iBL φ=


ExampleA 4 m long wire carries current of 500A in NE direction

Magnitude of force (B = 0.5 gauss = 5 × 10-5 T, pointing N)

Direction of force: Upwards, from RHR

Can adjust current in wire to balance against gravity

Calculate mass from density, length and cross-sectional area

Good exam problem!

siniBL mgφ =

( )( )( )( )5sin 500 5 10 4 0.71 0.071NF iBL φ −= = × =

m LAρ=


Magnetic Force A vertical wire carries a current in a vertical magnetic field. What is the direction of the force on the wire?

(a) left (b) right (c) zero (d) into the page(e) out of the page

I

B

I is parallel to B, sono magnetic force


Magnetic Field and WorkMagnetic force is always perpendicular to velocity

Therefore 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 tΔ = ⋅Δ = ⋅ Δ =

v 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 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 FieldA charge q enters B field with velocity v perpendicular to B. What path will q 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

FFv

R

v

B

qF

v


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

Circular Motion of Positive Particle

BqF

v

2mv qvBR

=mvRqB

=


Cosmic Ray ExampleProtons with energy 1 MeV move ⊥ earth B field of 0.5 Gauss or B = 5 × 10-5 T. Find radius & frequency of orbit.

212

2KK mv vm

= ⇒ =

2mv mKReB eB

= =

( )( )6 19 13

27

K 10 1.6 10 =1.6 10 J

1.67 10 kgm

− −

−

= × ×

= ×

( )1

2 2 / 2v v eBf

T R mv eB mπ π π= = = = 760Hzf =

2900mR =

Frequency is independent of v!


Helical Motion in B FieldVelocity of particle has 2 components

(parallel to B and perp. to B)Only v⊥ = v sinφ contributes to circular motionv|| = v cosφ is unchanged

So the particle moves in a helical pathv|| is the constant velocity along the B fieldv⊥ is the velocity around the circle

v v v⊥= +

mvRqB

⊥=

Bv

φ

v⊥

v||


Helical Motion in Earth’s B Field


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

ABBoth masses are equalCannot 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 meansbigger radius


Mass SpectrometerPositive ions first enter a “velocity selector” where E ⊥ B and values are adjusted to allow only undeflected particles to enter mass spectrometer.

Magnetic force is down, electric force is upBalance forces

Spectrometer determines mass usingmeasured radius r and velocity v

/qE qvBv E B

==

qBrmv

=


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

(1) it increases (2) it decreases(3) it stays the same(4) it depends on the direction of the velocity(5) it depends on the direction of the magnetic field

Magnetic field does no work, so K is constant


Mass Spectrometer A beam of electrons travels right at v = 5 x 105 m/s

What value of magnetic field would make electrons go undeflectedthrough a region where E = 100,000 V/m pointing up vertically?

What is the frequency of the circular orbit of the electrons if the electric field is turned off?

5 5/ 10 /5 10 0.2T

eE evB

B E v

=

= = × =

( )( )5

95

1 5 10 6.8 10 Hz2 6.28 1.4 10

vfT Rπ −

×= = = = ×

×

( )( )( )( )

31 525

19

9.1 10 5 101.4 10 m

1.6 10 0.2mv mvevB RR eB

−−

−

× ×= ⇒ = = = ×

×


Torque on Current LoopRectangular current loop in uniform magnetic field (lengths a & b)

Forces in left & right branches are 0 Force in top branch is into planeForce in bottom branch is out of plane

Equal forces give net torque!Bottom side up, top side down (RHR)Rotates around horizontal axis

μ = NiA ⇒ “magnetic moment”Assuming N turnsτ = μB, true for any shape!!

If plane tilted angle θ to B fieldτ = μBsinθθ is angle between normal and B

B

a

b

a

b

( )Fd iBa b iBab iBAτ = = = =Plane normal is ⊥ B(θ = 90°)


Torque ExampleA 3-turn circular loop of radius 3 cm carries 5A current in a B field of 2.5 T. Loop is tilted 30° to B field.

Rotation is always in direction to align μ with B field

30°

( )22 23 3 5 3.14 0.03 0.0339A mi rμ π= = × × × = ⋅

sin30 0.0339 2.5 0.5 0.042 N mBτ μ= = × × = ⋅


Magnetic Force A rectangular current loop is in a uniform magnetic field. What direction is the net force on the loop?

(a) + x (b) + y (c) zero (d) – x(e) – y

B

x

z

y

Forces cancel onopposite sides of loop


Electromagnetic Flowmeter

Moving ions in the blood are deflected by magnetic forcePositive ions deflected down, negative ions deflected upThis separation of charge creates an electric field E pointing upE field creates potential difference V = Ed between the electrodesThe velocity of blood flow is measured by v = E/B

E


Hall Effect: Do + or – Charges Carry Current?

+ charges moving counter-clockwise experience upward force

Upper plate at higher potential

– charges moving clockwise experience upward force

Upper plate at lower potential

Equilibrium between electrostatic & magnetic forces:

This type of experiment led to the discovery (E. Hall, 1879) that current in conductors is carried by negative charges

up driftF qv B= Hdown induced w

VF qE q= = H drift "Hall Voltage"V v Bw= =


Partial Loops (cont.)Note on problems when you have to evaluate a B field at a point from several partial loops

Only loop parts contribute, proportional to angle (previous slide)Straight sections aimed at point contribute exactly 0Be careful about signs, e.g.in (b) fields partially cancel, whereas in (a) and (c) they add

chapter 28: magnetic fields - university of · pdf filephy2049: chapter 28 1 chapter 28:...

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