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Chapter 21
Magnetic Force
Clicker
What is the potential difference across the clams when I take bulb
out of circuit?
A) 1.7 V
B) 3.4 V
C) 5.1 V
D) 6.8 V
E) 12 V
Magnetic Field of a Moving Charge
The Biot-Savart law for a
moving charge
 0 qv  rˆ
B
4 r 2
The Biot-Savart law for a
short piece of wire:

 0 Il  rˆ
B 
4 r 2
How magnetic field affects other charges?
Magnetic Force on a Moving Charge

 
Fmagnetic  qv  B
q – charge of the particle
v – speed of the particle
B – magnetic field
N
N
T

C  m/s A  m
Direction of the magnetic force depends on:
the direction of B
the direction of v of the moving charge
the sign of the moving charge
A negative charge is placed at rest in a magnetic field as
shown below. What is the direction of the magnetic force
on the charge?
B
A.
B.
C.
D.
E.
Up
Down
Into the page
Out of the page
No force at all.
A negatively charged particle is moving horizontally to the right
in a uniform magnetic field that is pointing in the same direction
as the velocity. What is the direction of the magnetic force on
the charge?
V
A.
B.
C.
D.
E.
Up
Down
Into the page
Out of the page
No force at all.
B
Now, another negatively charged particle is moving upward and
to the right in a uniform magnetic field that points in the
horizontal direction. What is the direction of the magnetic force
on the charge?
V
A.
B.
C.
D.
E.
Left
Up
Down
Into the page
Out of the page
B
Magnetic Force on a Current-carrying Wire

 
Fm  qv  B
Current: many charges are moving
Superposition: add up forces on individual charges
Number of moving charges in short wire:
nAl
Total force:
I
 

Force of a short wire: Fm  Il  B
In metals: charges q are negative. Will this equation still work?
Forces Between Parallel Wires
Definition of 1 Ampere:
Ampere is defined as a current at which two very long
parallel wires 1 m apart create a force on each other of
2.10-7 N per meter length.
From this also follows that 0/(4) = 10-7 T.m/A
Forces Between Parallel Wires
For long wire:
m0 2 I1
B1 =
4p d
Magnetic force on lower wire:
Fm = IDl ´ B
F21 = I 2 LB1 sin 90 o
m0 2I1
F21 = I 2 L
4p d
Magnetic force on upper wire:
m0 2 I 2
B2 =
4p d
F12 = I1LB2 sin 90 o
m0 2I 2
F12 = I1 L
4p d
What if current runs in opposite directions?
Electric forces: “likes repel, unlikes attract”
Magnetic forces: “likes attract, unlikes repel”
Effect of B on the Speed of the Charge

 
Fmagnetic  qv  B
What is the effect on the magnitude of speed?
 
F  dl  0
Kinetic energy does not change
Magnetic field cannot change a particle’s energy!
Magnetic field cannot change a particle’s speed!
Magnetic force can only change the direction of velocity but
not its magnitude
Magnitude of the Magnetic Force

 
Fmagnetic  qv  B
Single electron in television tube:

dp
 qvB sin 
dt

dv
q
 vB sin 
dt m
(v<<c)
e/me = 1.78.1011 C/kg
Circular Motion at any Speed

 
Fmagnetic  qv  B
Any rotating vector:


dX
  X …angular speed
dt


dp


d
p
 p
 qv  B  q vB sin 90o
dt
dt

mv
1 v / c
2
2
 q vB
qB

1  v2 / c2
m
Cyclotron Frequency
Circular Motion at Low Speed
qB

1  v2 / c2
m
qB
if v<<c:  
m
independent of v!
Alternative derivation:
F  ma
2
v
q vBsin 90o  m
R
q B  m
Period T:
2

T
Circular motion:
v2
v
a

R
R
qB

m
m
T  2
qB
Non-Relativistic
A Cyclotron
qB
v<<c :  
m
V  V0 cos t
Vequivalent  2 NV0
Vequivalent ~ 108 V  v~c :
need to adjust B or 
Exercise

v||
What if v is not perpendicular to B?

 
Fmagnetic  qv  B
Direction?

v
Magnitude?
Fm  qv B
Trajectory: helix
Determining e/m of an Electron
p  q Br
q
v

m Br
mv2
 q V
2
2
q
v2
   2 2
Br
m
q
v  2V
m
2
2
q 2 V
q
  
2 2
m
m
B
r
 
e 2V
 2 2
m Br
Joseph John Thomson (1856-1940)
1897: m/e >1000 times smaller than H atom

 
Fmagnetic  qv  B


Fe  qE


 
F  qE  qv  B
Magnetic force can only change the direction of velocity but
not its magnitude
2
v
F  ma a 
2
R
v
o
q vBsin 90  m
R
(high q – small R, large m large R)
Applications: find p, e/m, separate particles
(including 238U and 235U (natural abundance 0.7%) isotopes
Proton
energy
Velocity,
m/sec
R, m
1MeV
1.3*107
0.15 m
7000 GeV
0.9999995c
4.2 km
e = 1.6*10-19 C
mproton = 1.7*10-27 kg
Giga 109
What is the radius of circle of proton accelerated over 7000GeV
in B =5.5 T?
Proton speed up to 0.9999995c
mv
1
R
B q 1  v 2 / c2
Particle Accelerator
Fermilab: Tevatron
CERN: Large Hadron Collider
Applications:
study nucleus – accelerate particles
(protons) and shoot at nucleus to observe
nuclear reactions
create new particles by colliding particles
Synchrotron – source of synchrotron
radiation
To keep R constant B is increased
up to 3 T (superconducting coils)
Argonne National Laboratory