8 磁場

8 磁場

How is an aurora so thin yet so tall and wide?

Introduction

How does magnetic resonance imaging (MRI) allow us to see details in soft nonmagnetic tissue?

How can magnetic forces, which act only on moving charges,

explain the behavior of a compass needle?

The magnetic field

The electric field and the magnetic field

Electromagnets and permanent magnets

8-1 The definition of B

B v

q

F 

 





0



 q E F

 

The tracks in a bubble

chamber

Figure 27.21

The SI unit for B

1 tesla = 1T =1 N/A‧m=10

gauss

10⁸ T 1.5 T 10^-2 T 10^-4 T 10^-10 T 10^-14 T

Magnetic Field Lines

 Magnetic vs. electric dipoles

A horseshoe and a C-shaped

magnets

例 1 A 5.3 MeV proton

7

15

12 2

2 / 3.2 10 m/s sin 6.1 10 N / 3.7 10 m/s

B

B

v K m

F qvB a F m



  

  

  

B = 1.2 mT

8-2 Crossed Fields:

Discovery of the Electron

 A cathode ray tube

 Thomson’s procedure:

設定E = 0 、 B = 0 ，並記錄光點 位置

開啟電場、測光點偏移量

開啟磁場，並調整其值，直至光

點回復未偏移位置

2 2

2 2

2 ,

/ ,

2

y qEL qE qvB mv

m B L v E B

q yE

 

 

Calculation

2 2 2

2 2 , 1

x

x y

y

mv y QEL

t v L

t a y

m QE m

a F











the charge-to-mass ratio of the electron ：

1.758819610¹¹ C·kg^-1

8-3 Crossed Fields: The Hall Effect

 By the conduction electrons in copper:

) (

d

l A Vle

n Bi

neA i ne

v J

B ev

eE

Ed V

d

d













例 2 A cube generator

mV 0

.

 3







V

dvB V

Ed V

evB eE

d = 1.5 cm, v = 4.0m/s, B = 0.05T

8-4 A Circulating Charged Particle

qB m

v r

T

qB mv

r

r mv

qvB

r mv

ma F

/ 2

/ 2

/

/

/

2

2

















Figure 27.17a

Figure 27.17b

The frequency and angular frequency

2 2

1

m f qB

m qB

f  T     



頻率與軌跡

The magnetic bottle machine

Helical Paths

cos 2 m p v T v

qB

 

 

The pitch (螺距) of the helical path cos sin

v  v





V_∥ and V_⊥

極光橢圓圈

例 3 The Mass Spectrometer (質 譜儀)

2

2 1

/ 2

2

1 ₂

r x

q mV B

qB r mv

m qV

v

qV mv











u 93 .

8 203 2 2 2

2

2 









V qx m B

q mV r B

x

Isotope Separation

Centrifuge and diffusion chamber

質譜儀

8-5 Cyclotrons and Synchrotrons

Fermilab: 6.3km ring

(迴旋加速器與同步加速器)

Tevatron Collider

Main Injector

 The resonance condition:

 When proton energy > 50Mev:

Out of resonance (relativistic effect)

A huge magnet (4×10⁶ m²) is needed for high energy (500Gev) protons

 The proton sychrotron at Fermilab can produces 1Tev proton

fosc

f 

Synchrotrons

CERN LHC

The LHC is 27km long and sits 100m below the surface.

Large Hadron Collider

• Baryon fermion

• Meson boson

8-6 Magnetic Force on a Current-

Carrying Wire

B L

i F

iLB B

qv F

v iL

it q

B

d B

d





  











sin /

For a wire segment:

B L

id F

d 

 





Magnetic Force

例 4 A length of wire with a

semicircular arc

) (

2 2

cos

sin sin

) (

3 2

1

0

0 0

2

3 1

R L

iB F

F F

F

iBR iBR

d iBR

dF F

Rd iB

iBdL dF

iLB F

F





























 



 











Calculation

線圈

8-7 Torque on A Current Loop



F

and F

cancel



F

and F

form a force couple



















sin )

( sin

) 2 sin

( )

2 sin (

cos )

90

2 sin(

B NiA

N

iabB

iaB b iaB b

ibB ibB

F

 







 





 ^

N turns coil

例 5 A galvanometer for analog meters

-6 -4 2

8

sin sin

[(250)(100 10 A)(2.52 10 m ) (0.23T)(sin 90 )] / 28

5.2 10 N m / degree NiAB

NiAB

  

 





 



  



  

The direct-current motor

Figure 27.40

8-8 The Magnetic Dipole



The magnetic dipole moments



The magnetic potential energy

) :

cf (

sin

E p

B

B

NiA    

    























) (

)

( p E U B

U







B

B B

U





 2

) (

) (













磁能