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Chapter 28: Magnetic Field and Magnetic Forces

Iron ore found near Magnesia

Compass needles align N-S: magnetic Poles

North (South) Poles attracted to geographic North (South)

Like Poles repel, Opposites Attract

No Magnetic Monopoles

Magnetic Field Lines = direction of compass deflection.

Electric Currents produce deflections in compass direction.

=>Unification of Electricity and Magnetism in Maxwell’s Equations.

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Magnetic Fields in analogy with Electric Fields

Electric Field:

– Distribution of charge creates an electric field E(r) in the surrounding space.

– Field exerts a force F=q E(r) on a charge q at r

Magnetic Field:

– Moving charge or current creates a magnetic field B(r) in the surrounding space.

– Field exerts a force F on a charge moving q at r

– (emphasis this chapter is on force law)

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Magnetic Fields and Magnetic Forces

Magnetic Force on a moving charge

– proportional to electric charge

– perpendicular to velocity v

– proportional to speed v (for a given geometry)

– perpendicular to Magnetic Field B

– proportional to field strength B (for a given geometry)

F = q v B

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F = q v B

F = |q| v B sin = |q| v B (v B) v

B

F

+

F = q v B

F = |q| v Bv

B

F

+

v

F = q v B

F = |q| v B

vB

F

+ B

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Units of Magnetic Field Strength:

[B] = [F]/([q][v])

= N/(C m s-1)

= Tesla

Defined in terms of force on standard current

CGS Unit 1 Gauss = 10-4 Tesla

Earth's field strength ~ 1 Gauss

Direction = direction of velocity which generates no force

Electromagnetic Force:

F = q ( E + v B )

= Lorentz Force Law

Magnetic Fields

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Magnetic Field Lines and Magnetic Flux

Magnetic Field Lines

Mapped out with compass

Are not lines of force (F is not parallel to B)

Field Lines never intersect

Magnetic Flux

dB = B . dA

d B dA

B dA

B dA

B

B

0 no magnetic charge! (no monopoles)

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• SI Unit of Flux:

– 1Weber = 1Tesla x 1 m2

– for a small area B = dB /dA

– B = “Magnetic Flux Density”

Flux through an open surface will play an important role

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Motion of Charged Particles in a Magnetic Field

Charged Particle moving perpendicular to the Magnetic Field

– Circular Motion!

– (simulations)

+

+

vF

F

v

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Charged Particle moving perpendicular to a uniform Magnetic Field

+

+

v

vF q vBmv

R

Rmv

q B

v

R

q B

m

| |

| |

| |

2

cyclotron frequency

R

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In a non-uniform field: Magnetic Mirror

Net component of force away from concentration of

field lines.

B

v

F

Magnetic Bottle Van Allen Radiation Belts

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Work done by the Magnetic Field on a free particle:

=> no change in Kinetic Energy!

Motion of a free charged particle in any magnetic field has constant speed.

dW F dx

qv B vdt

0!

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Recall:

Charged Particle moving perpendicular to a uniform Magnetic Field

F q vBmv

R

Rmv

q B

| |

| |

2

+

+

v

vR

Applications of Charged Particle Motion in a Magnetic Field

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Velocity Selector

makes use of crossed E and B to provide opposing forces

No net deflection => forces exactly cancel:

|q| v B=|q| E

v = E/B

E

upwards

F = q v B

downwards

F = qE

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J. J. Thomson’s Measurement of e/m

Electron Gun

and velocity selector:

E

V

2

2

2

2

21

VBE

me

eVmv

BE

v

e/m = 1.76x1011 C/kg

with Millikan’s measurement of e

=> mass of electron

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Example: Using an accelerating Potential of 150 V and a transverse Electric Field of 6x106 N/C. Determine

a) the speed of the electrons,

b) the magnetic field magnitude required for no net deflection

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Mass Spectrometer

E

vE

B

Rmv

qB

mRqB

v

RqB

E

2

One method

velocity selector + circular trajectory

R1

R2

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Example: Vacuum System Leak Detector uses Helium atoms. Ionized helium atoms (He +) are detected with a mass spectrometer with a magnetic field strength of .1 T. With a velocity selector tuned to 1x105 m/s, where must the detector be placed to detect 4He +

ions?

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Magnetic Force on a Current Carrying Wire

vd

I

dl

B

Fi

A

F F q v B

Nqv B n volume qv B

nAdlqv B JAdl B

Idl B RHR

i i i

d d

d

( )

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F ILB

A m T

N

sin

. sin

.

50 1 12 45

42 4

Example: A 1-m bar carries 50 A from west to east in a 1.2 T field directed 45° North of East. What is the magnetic force on the bar?

Force will be directed upwards (out of the plane of the page)

I dl

B

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Torque on a Current Loop (from F = I l x B )Rectangular loop in a magnetic field (directed along z axis) short side length a, long side length b, tilted with short sides at an angle with respect to B, long sides still perpendicular to B.

B

Fa

Fa

Fb

Fb

Forces on short sides cancel: no net force or torque.

Forces on long sides cancel for no net force but there is a net torque.

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Torque calculation: Side view

Fb = IBb

Fb

moment arm

a/2 sin

= Fb a/2 sin Fb a/2 sin

Iab B sin = I A B sin Magnetic Dipole ~ Electric Dipole

U =

Switch current direction every 1/2 rotation => DC motor

magnetic moment

B

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Hall Effect

Conductor in a uniform magnetic fieldx

y

z

Bz

Jx

Magnetic force on charge carriers F = q vd B

Fz = qvdB Charge accumulates on edges

Jx

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Equilibrium: Magnetic Force = Electric Force on bulk charge carriers

Bz

Charge accumulates on edges Fz = 0 = qvdBy + q Ez

Ey

vE

B

J nqv nqE

B

nqJ B

E

dz

y

x dz

y

x y

z

Hall EMF V E w

I J tw

nqIB

V t

H z

x

y

H

w

t Jx

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Negative Charge carriers:

velocity in negative x direction

magnetic force in positive z direction

=> resulting electric field has reversed polarity

Bz

Jx

Ey

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Example: A ribbon of copper 2.0 mm thick and 1.5 cm wide carries a 75 A current in a .40 T magnetic Field. The resulting Hall emf is .81 V. What is the density of charge carrying electrons?

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