### Magnetic fields and forces

```Magnetic field and forces
Early history of magnetism started with the discovery of the
natural mineral magnetite
Named after the Asian province Magnesia
Octahedral crystal of magnetite,
an oxide mineral Fe3O4
Today’s Manisa in Turkey historically called Magnesia
Such crystals are what we today call permanent magnets and people found properties
Modern HDD
Chinese compass
invented 2230 years ago
Instead of following the traditional (textbook) approach to first introduce the
phenomena we start by highlighting here the modern insight that
Electric forces and fields and magnetic forces and
fields are unified through relativity
What led me more or less directly to the special theory of relativity was the conviction
that the electromotive force acting on a body in motion in a magnetic field was nothing
else but an electric field.
-Einstein 1953http://en.wikipedia.org/wiki/Relativistic_electromagnetism
The origin of magnetic forces on a moving electric charge
A first hint at a fundamental connection between electricity and magnetism
comes from Oersted’s experiment
Electric phenomenon
Magnetic phenomenon
A distribution of electric charges
creates an E-field
A moving charge or a current creates (an
additional) magnetic field
The E-field exerts a force F=q E on any
other charge q
The magnetic field exerts a force F on
any other moving charge or current
Where are the moving charges or currents in a permanent magnet?
We see later more clearly that these are the “moving electrons” on an
atomic level (angular momentum L, spin S, J=L+S)
How can relativity show that a moving charge next to a current
experiences a force
Let’s consider the special case that our test charge outside the wire moves likewise with
v, the drift velocity of the electrons in the wire
drift velocity of
electrons, remember
Drude model of
conductivity
wire of
infinite length
charge density
  Q / l0
Negative charge (electrons) flowing to the right
Stationary positive ions
Now we look at the situation from the perspective of the moving test charge q:
we transform into the moving coordinate system of the test charge
From the perspective (moving reference frame) of the test charge electrons are at rest
and ions move
Now the “magic” of special relativity kicks in:
Spaceship moving with
10% of speed of light
Spaceship moving with
86.5% of speed of light
Lorentz-contraction quantified by
http://www.physicsclassroom.com/mmedia/specrel/lc.cfm
v
l  l0 1   
c
2
Due to Lorentz-contraction:
test charge sees modified charge density for moving + ions
Q

 
l
Q
l0
v
1  
c
2


v
1  
c
2
test charge sees imbalance of charge
density for electrons and + ions
Since v-drift is about 0.1 mm/s <<c (see http://physics.unl.edu/~cbinek/Unit_13%20Current%20resistance%20and%20EMF.pptx) we
can expand into Taylor series around x=v/c=0
1
x2
2
f ( x) 
 f (0)  f (0) x  f (0) x  ...  1   ...
2
2
2
1 x
1
   
v2
2c 2
Charge density of ions increased in the frame of the test charge
Likewise in comparison to lab frame (frame of the wire) the electrons are
now seen at rest (were moving in the lab frame)
v2
test charge sees charge density of electrons reduced by
2
(because test charge at rest sees wire as neutral despite moving negative charges) 2c
Net effect: wire neutral in the lab frame gets charge density
in the frame of moving electron
v2
c2
Remember the electric field of an infinite wire with homogeneous
charge density 
Gaussian cylinder of radius r
l
2
E
d
 r  E 2 r l 
Q
0
E

2 0 r
For the moving charge in the moving frame we have
Moving charge sees an electric field E 
1

v2
c2
v2
2 0r c2
charge is exerted to a force away from the wire
q v 2
F
2 0r c 2
In the lab frame (frame of the wire) we interpret this as the magnetic Lorentz force
c2 
F  qv
 v
 qvB
2
2 0r c
with B 
1
 0 0
 v
v
I



0
0
2
2 0r c
2 r
2 r
magnetic B-field in distance r from a wire carrying the current I
Our expression F  qvB holds for the special case
I
B
F  qvB
v
In general: F  q v  B
magnetic force on a moving charged particle
vB
Clicker question
Considering the mathematical cross product structure of the Lorentz
force.
Do you think that this magnetic force can do work on a charge?
1) Yes, it is a force and we can evaluate
2) No, the integral  Fd r
 Fd r
will always be zero
3) Yes, the integral  Fd r will equal qvBl where l is the length of the path
The e/m tube demonstration
We see Lorentz force F  v and hence  dr
Circular orbit
qvB  m
2
v
R
v known from Ekin=qVab

=

B known from current through
Helmholtz coils
e/m= 1.76 x 1011 C/kg
measure R vs. B
no work
With v   R
  c :
qB
m
Cyclotron frequency
http://en.wikipedia.org/wiki/Cyclotron
From F  qv B  qvB
[ B] 
we can determine the unit of the B-field
[F ]
N
N


: T
[q][v] As m / s Am
1T=1tesla=1N/Am
in honor of Nikola Tesla
http://en.wikipedia.org/wiki/Nikola_Tesla
A moving charged particle in the presence of an E-field and B-field
F  q  E  v  B
An application in modern research: the Wien mass filter
Resolution:
m/Δm > [email protected]/* <![CDATA[ */!function(t,e,r,n,c,a,p){try{t=document.currentScript||function(){for(t=document.getElementsByTagName('script'),e=t.length;e--;)if(t[e].getAttribute('data-cfhash'))return t[e]}();if(t&&(c=t.previousSibling)){p=t.parentNode;if(a=c.getAttribute('data-cfemail')){for(e='',r='0x'+a.substr(0,2)|0,n=2;a.length-n;n+=2)e+='%'+('0'+('0x'+a.substr(n,2)^r).toString(16)).slice(-2);p.replaceChild(document.createTextNode(decodeURIComponent(e)),c)}p.removeChild(t)}}catch(u){}}()/* ]]> */ keV
http://www.specs.de/cms/front_content.php?idart=148
It selects defined ions
by a combination of electric
and magnetic fields.
aperture
aperture
z
x
y
The charged particle will only follow a straight path through the crossed E and
B fields, if net force acting on it is zero
q  E  v  B  0
For the simple special case:
q  E  vB e y  0
v  (v,0,0) , E  (0, E ,0) , B  (0,0, B )
ex
ey
ez
vB  v
0
0  vBe y
0
0
B
E
v
B
Wien filter can be used for mass selection if
incoming particles have fixed kinetic energy
v  2Ekin / m
Is that the whole story of the Wien filter
Absolutely not, as this research manuscript from 1997 indicates
v  ( v x , v y , v z ) , E  (0, E ,0) , B  (0,0, B )
ex
ey
ez
v  B  vx
0
vy
0
v z  v y Be x  v x Be y
B
Equation of motion for particle of charge q and mass m in the filter:
mr  q  E  v  B
Do not click if you are not prepared
to see nastiness.
m x  qv y B
m y  qE  qv x B
mz 0
viewer discretion is advised!
coupled differential equations, nasty!
z(t )  vzt
But, we are honors students:
Not the goal but the game
Not the victory but the action
In the deed the glory
m x  qyB
x
m y  qE  qxB
qB
y
m
x
qB  qE qB 

x

m m
m 
We define
x  vx
2 E
2
v x  c
 c vx
B
Back to
c 
qB
m
introduced earlier already as cyclotron frequency
Solving first the homogeneous equation
vx  c vx  0
2
Ansatz
vx  Asin t   
v x   A cos t   
v x   A sin t   
Substitution into homogeneous differential equation
 2 Asin t     c Asin t     0
  c
2
E
Solving the inhomogeneous equation v x  c v x  c
B
through
2
2
Solution of inhomogeneous equation is general solution of homogeneous
plus a particular solution of inhomogeneous
For the particular solution we try
vx  const
vx  0
E
vx 
B
E
With v x  c v x  c
B
2
2
is a solution
General solution of inhomogeneous differential equation
vx 
E
 a sin ct   b cos ct 
B
particular solution of
general solution of
inhomogeneous differential eq. homogeneous differential eq.
E
a
b
x(t )  t  cos ct   sin ct   c
B c
c
With x(t=0)=0
x (t ) 
c  a / c
E
a
b
t  1  cos ct    sin ct 
B c
c
E
y  c  x c
B
E
y  c t  x(t ) c  c
B
 a 1  cos ct    b sin ct   c
y  at 
a
c
sin ct  
b
c
cos ct   ct  d
Final adjustment of initial conditions:
v x (t  0)  v x 0 
E
b
B
b  vx 0 
E
B
v y (t  0)  v y 0  c
b
y (t  0)  y0 
d
c
vx (0)  cv y 0  ac
a  vy0  c
vy0
 vx 0
E
E 
x (t )  t 
1

cos

t


 c  

 sin ct 
B
c
 c Bc 
y (t )  y0 
v
E 
sin ct    x 0 
 1  cos ct  
c
 c Bc 
vy0
Can we recover the simple case of acceleration free motion?
Motion is simple for
from
=0
v y (t  0)  v y 0  0 and
y (t  0)  y0  0
=0
 vx 0
E 
y (t )  y0 
sin ct   

 1  cos ct  
c
 c Bc 
vy0
=0
This is of course the condition
E
vx0 
B
motion with y(t)=0 for the entire path in the filter
aperture
aperture
z
No net force
x
y
In fact:
=0
=0
vy0
 vx 0
E
E 
x (t )  t 
1  cos ct    


 sin ct 
B
c
 c Bc 
x (t ) 
E
t
B
In general motion is very complex
Let’s consider charged particle with mass
M0 moving with constant v through EXB
particle with mass M=M0 / has different
initial v than particle with M0
Trajectory deviates from straight line
If  significantly deviates from 1
trajectory can be very complicated and may spoil
mass filter effect,
see trajectory for =4
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