Lectures 12-13 (Ch. 27)
Magnetism, Magnetic Force
1.
Introduction
(permanent magnets,
magnetic field, B)
2. Magnetic force.
3. Motion of q in B (Uniform B, Nonuniform B)
4. Motion of q in crossed B and E
5. Force on a segment of a current
6. Torque on a loop of a current
7. Applications
Permanent magnets
Similar to electric attraction and/or repulsion.
But each magnet has two opposite poles like
an electric dipole has two opposite charges.
The poles can not be separated.
Magnets produce magnetic field, B
The structure of B of a magnet is similar
to the structure of E of a dipole. But B
lines are always closed lines (contrary to
E lines which start on + and end on - q)
Again: magnetic monopoles do not exist
Electric currents also produce B
Amper’s equivalence hypothesis
Magnet consists of the loops of the current.
They are due to circulation of electrons around nucleus.
I
Spin axis
The core is very hot so
that iron is not
magnetic.
B is attributed to
dynamo effect of the
circulating electric
current in the core of
the earth.
The rotation of the Earth
plays a part in generating
the currents.
Evidence for 171
magnetic field reversals
during the past 71 million
years has been reported.
Magnetic force on a moving charge

 
F  qv  B
 
F  B,
 
F v 
1.Work done by magnetic force=0
2. Force does not change the
magnitude of v, just the direction
Right hand rule (RHR)

 
F  qv  B
Let v vector enter a palm of your right hand. Then curl the fingers to grab B vector. Your
thumb points in direction of F vector for positive charge and opposite to F vector for
negative charge.
Units of B
[B]=[F]/[q][v]=Ns/Cm=N/Am=1T (tesla)
1T=104 G (gauss)
Range of B:
Magnetically shielded room 10-10 G
Interstellar space 10-6 G
Earth's magnetic field 0.5 G
Small bar magnet 100 G
Strong lab magnet 10 Tesla
Big electromagnet 10 Tesla
Surface of neutron star 108 T
Magstar 1011 T
The largest superconducting magnet used in LHC
Motion
of
the
charged
particles
in
B

 
1. v  B
Circular motion (in plane  B )
mv 2
F  ma  qvB 
R
mv
qB
R
,   v/R 
qB
m
2R 2m
or T 

,
v
qB
1
qB
(hertz)
f  
T 2m
qB (radian per sec)
  2f 
m
Cyclotron frequency
If B is out of page +q makes clockwise rotation
Cyclotron
is a device to accelerate the charged particles
B
F
v
E(t)
Top view: “Dees” (two metal halfcylindrical empty cans).
E
qB

m
Cyclotron frequency does
not depend on v !
Charged particles are accelerated by E (t) in the
interval between the “Dees”. E(t) is a periodic
function with the frequency = cyclotron frequency.
When v approaches c, relativistic effects should
be taken into account:
m
m0
  (v )
v
1  ( )2
c
One has to tune the frequency of the applied E(t) to
provide the further acceleration of the particles.
Synchrophasotron.
Example.
You need to built a cyclotron to accelerate the protons from K0 =1MeV to Kf =29MeV. You may
provide B=1T and maximum voltage, V=10kV.
1)What should be the frequency of the applied voltage?
2)What should be the minimum radius of “Dees”?
3) How much time does it require?
qB 1.6 10 19 C 1T
8 rad
1. 


0
.
96

10
m
1.67 10  27 kg
s
2K f m
2  29 1061.6 10 19 J 1.67 10  27 kg
mv
2. R 


 0.78m
19
qB
qB
1.6 10 C 1T
3.t  Tn
2
6.28s
8
T


6
.
3

10
s
7
 9.6 10
K f  K 0 28 106 J
3
n


1
.
4

10
2qV
2 10 4 J
t  90 s
Discovery of the positron in a bubble chamber,
in the strong B
Paul Dirac
1902–1984
Prediction of a positron:
Paul Dirac, 1928
Discovery of a positron:
Carl Anderson, 1932
Uniform B, V has a component in B direction
Circular motion in a plane perpendicular
to B with the radius R  mv
qB
is combined with a motion with constant
velocity,
v
along B, resulting in a helical path.
Pitch (the distance between two neighboring
turns):
2m
d  v  T , T 
qB
Example1. At t=0 proton with v=(1.5x105 i+2x105 j)m/s enters at the origin of
the coordinate system the region with B=0.5iT. Describe and plot the pass.
Find the coordinates of the proton at t=T/2 where T is the period of the circle.
2

 
F  qv  B
R
y
mvy
qB
, T
F  ma  qv y B 
mvy
R
2R 2m

, y (t  T / 2)  0,
vy
qB
z (t  T / 2)  2 R, x(t  T / 2)  vx  T / 2
Example2. A proton enters the region of the uniform
magnetic field, B=BxiT, experiencing the magnetic
force F=FzkN. What could you tell about initial
velocity of the proton?
F
z







Fz k  q(vx i  v y j  vz k )  Bx i  q(v y Bx k  vz Bx j )
y
Vy
z
B
R
F
vz  0, v y  
Fz
, nothing can be sad about vx
qBx
Nonuniform B
Magnetic bottle: the force is toward the center, the particles are trapped. Radius
is the largest in the center where B is the weakest (R~1/B). Most of the particles
reaching the ends slow down and turn back under the action of force, oscillating
between the ends. The most energetic particles may escape only on the most
ends where B is uniform.
The Van-Allen Radiation Belts
protect life on the earth from the cosmyc radiation
The principle of magnetic bottle: charged particles are trapped by the
nonuniform B of the Earth. They may escape only near the poles regions
resulting in Northern and Southern lights.
Aurora borealis (Northern lights)
Tokamak (toroidal magnetic coil )
Plasma confinement for nuclear fusion
Positively charged ions and negatively charged
electrons in a fusion plasma are at very high
temperatures (~106K), and have correspondingly
large velocities. In order to maintain the fusion
process, particles from the hot plasma must be
confined in the central region, or the plasma will
rapidly cool. Magnetic confinement fusion
devices exploit the fact that charged particles in
a magnetic field feel a magnetic force and follow
helical paths along the field lines.
Magnetic deflection
 
vB
charged particle starts to orbit around B lines.
If
If the size of the region with B is smaller then the radius of the orbit, the particle will be
deflected at the exit of the region from original direction by a distance (see figure):
x1  R  R 2  d 2
Example. A particle with q=2.15 μC and m=3.2x10-11kg enters the region of B as shown on Fig. Find
its displacement at the exit of the region and a total displacement on the wall.
x2
mv (3.20  1011 kg)(1.45  105 m/s)
R

 5.14 m.
6
qB
(2.15  10 C)(0.420 T)
x1  R  R 2  d 2

tan  
=0.42T
d
R2  d 2
x2  (D  d ) tan 

R d
2
2
 1.45 105 m / s
x  x1  x2
Motion of q in B┴E.
Velocity selector
E
qvB  qE  v 
B
Independent on the mass and q
Particles with this velocity will be undeflected. Particles with larger velocity will
be deflected by B. Particles with smaller velocity will be deflected by E.
Mass spectrometer
(discovery of the isotopes by J.J. Tomson,1913)
It consists of two regions (1) and (2).
In the region (1) there is crossed E and B.
It is a velocity selector.
Particles propagating with v=E/B are undeflected
and enter the region (2).
In the region (2) only B’ is present ( it may be the same as in
region (1) or different from it) . Particles start to move in
circles and after making a half of circle they strike the
photographic plate. Measuring the diameter of the circle
allows to calculate the mass of the particle if q is known.
(1)
(2)


mv 2
Fm  ma  qvB' 
R
qB' R
m
V
E
qB ' BR
v  ,m 
B
E
Using this device
isotopes were
discovered.
Neon 20 and 22
Example.
In the mass spectrometer, a distance between P and P’ plates is 0.5cm, B=B’=0.7T. The particles
propagate undeflected at v=2.68x104 m/s. 1. Find voltage between the plates. (2) Find the distance,
d, between the lines on the photographic plate for singly charged positive ions of 24Mg and 26Mg.
1amu=1.66x10-27kg.
E V
(1)v  
 V  vdB  93.8V
B dB
mv 2
mv
(2)
 qvB  R 
R
qB
R24  9.53 10 3 m
R26  10.32 10 3 m
d  2( R26  R24 )  1.58mm
The cathod ray tube
J.J. Thomson e/m experiment, 1897
discovery of e
Sir J.J.Tomson
(1856 -1940)
Nobel prize ,1906
V
mv 2
2eV E
eV 
v

2
m
B
e
E 1
e
11 C
( )
  1.76 10
m
B 2V
m
kg
The Hall effect (Edwin Hall, 1879)
When current flows along the
strip conductor in the presence
of B perpendicular to the strip
the deflection of carriers results
in inducing of voltage across
the strip (Hall voltage).
L
Separation of charges
occurs till
qE= qvB→ VH =EL=vBL
1. Sign of VH allows one to define the sign of careers (see figure)
2. Measuring of VH allows to find out drift velocity : vd 
3.
E VH

B BL
Measuring of VH and I allows to find out n (density of carriers):
I
IBL
I  qnAvd  n 

qAvd qAVH
The Hall resistance:
VH
EL vd B
B
RH 



I
JLd
Jd nqd
J  nqvd
B  1T , low temperature
h
RH  2 p, p  1,2...
e
h  6.63  10 34 Js
B  10T
hp
RH  2 , k  1,2...
ek
RH
1T
10T
B
Quantum Hall effect, Nobel Prize 1982, von Klitzing
Fractional Hall effect, Nobel Prize 1998 ,Stromer,Tsui,Laughlin
Force on a segment of a current

 
Fq  qv  B


 
F   Fi  qi vd  B
i
i

 
F  qnAlvd  B
  
F  I  Bl
Example
.
z

Fm
y
Fm  50 A 1m 1.2T  60 N
Fg  mg  60 N  m  6.12kg
x

Fg
Example. Electromagnetic rail gun (a simple motor)
Data: I=1000A, B=1T,m=50kg,L=1m
Find the distance the bar must travel along the rails if it
is to reach the escape speed for the earth (11.2km.s)
F
x
at 2
F IBL
x
,a  
2
m
m
v
v  at  t 
a
v2
v 2m
x


2a 2 IBL
(1.12 10 4 m / s ) 2 50kg
 3.14km
2000 A 1m  T
x
Example
Force on a half of a circle
  
dF  I  Bdl
dFy  dF sin   IBdl sin 
dl  Rd 
 /2
Fy  2 RIB  sin d  2 RIB
0
Ftotal  IBL  2IBR  IB( L  2R)
A quarter of a circle



F  Fx i  Fy j
F  ( IBR ) 2  ( IBR ) 2  2 IBR
-
=
F=0
F  2IBR
I
Loudspeaker
Magnetic dipole moment
  IA
(A is the area of the loop)
Torque on a current loop
 
  2r  F
  
2r  b, F  I  Ba
  
  B

  IA  Iab
Back to Amper’s equivalence
=
μ
S
μ
N
Analogy between electric and dipole moments


μ
S
B
μ tends to be parallel to B

x

-

  B
 
U    B

N
x
 
  p E
 
U  pE

+
p
E
p tends to be parallel to E

S
μ
N
Stable equilibrium
-S
p
+
E
Stable equilibrium
B
Solenoid (or magnetic coil)
many loops of current with the same direction of current

  B
  NIA
  NIAB sin 


DC current motor

  B


Force on a current loop in a nonuniform B
Replace a loop with equivalent magnet
Replace a loop with equivalent magnet
N
S
N
repulsion
attraction
S
Attraction of an unmagnetized iron to the magnet
Unmagnetized iron:μ=0
1. Alignment (randomly oriented μ become parallel to B,
i.e. iron become magnetized )
S
B
N
2. Attraction(opposite poles attract each other)
S
N
S
N
B