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Fall 2008: ICE 0116
General Physics II
Ch. 28 Magnetic field
Instructor: Prof. Kondekar Pravin N
Information and Communications University
Text: Fundamentals of Physics (8th
ed.) by Halliday, Resnick, Walker
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Topics
The magnetic field (B )
The Hall effect
A circulating charged particle
Magnetic force on a current-carrying wire
Torque on a current loop
The magnetic dipole moment
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The magnetic field
Magnets and electromagnets We know about magnetic forces produced by
Static charges produce and feel electric forces.
Moving charges produce and feel magnetic forces too.
In magnets, the moving charges are the electrons in the atoms that make thematerials.
In electromagnets, they are the charges that make up the current in the wire.
magnets
electromagnets
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The magnetic field
No magnetic monopoles The situation is very similar to electrostatics, if we substitute poles
where we used to say charge : like poles repel, opposite poles attract.
However : no isolated poles occur in nature. They all occur in pairs. Cut amagnet in half, you still have two magnets with two poles each!
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The magnetic field
Magnetic fields We define magnetic fields and magnetic field lines in the same way we did for
electric fields. The magnetic field lines of a magnet are very much like that ofelectric dipole :
Magnetic field lines go from north pole to south pole(on the outside). However,
magnetic field lines are always closed: there are no monopoles where the linescan end! So, that go from south to north in the inside.
The earth has a magnetic field too: this is the magnetoshere.
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The definition of B
Units of magnetic field Tesla, (Gauss=10-4T)
Magnetic field in nature
Well-shielded lab : ~10-14T
Earths surface : ~100T(1G)
Fridge magnet : ~10mT(100G)
Electromagnet : ~1T(10kG)
Superconducting NMR magnet : ~2T
Superconducting lab magnet : 10T-20T
High magnet field lab(e.g. Los Alamos) : 30T-60T
Neutron star : 100MT
mA
N
meterondcoulomb
newton
ondmetercoulomb
newtonTtesla
!
!
!!
1
))(sec/(1
)sec/)((111
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The definition of B
Force exerted by magnetic field Observations of MOVING charges show the presence of a velocity-dependent
FORCE that is given by :
We attribute such a force to a MAGNETIC FIELD = B
Units of B : Tesla =(N.s)/(C.m)=N/(A.m)
Compare with for electric fields : definition of fields!EqF!
BvqF v!
Newtons Coulombs Meter/second
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Magnetic and electric forces
Right hand rule
The force acting on a charges paticle moving with velocity though amagnetic field is always perpendicular to and .
BF v
v BB
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Magnetic and electric forces
Magnetic and electric force
Charges thatdo not move, do not feel magnetic force (what about magnets?)
Magnetic forces are perpendicular to both the velocity of charge and to themagnetic field (electric forces are parallel to the field). Since magnetic forcesare perpendicular to the velocity they do not work
Speed of particles moving in a magnetic field remains constant in magnitude,the direction changes. Kinetic energy is constant (no work)
EqF !For electrostatic forces :
For magnetic forces : BvqF v!
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Magnetic and electric forces
Example An electron is traveling with a constant velocity v.
It enters a box in which there is a uniformmagnetic field B.
Which of the following is TRUE?
(a) The electron speeds up.
(b) The electron slows down.
(c) The electron speed is constant.
(d) It depends on the direction of themagnetic field.
Force is qv x B
Direction of force is ALWAYSnormal to velocity! Speed CANNOT CHANGE! Direction of velocity DOESCHANGE i.e. acceleration is NOT 0!
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Magnetic and electric forces
Sample problem Uniform magnetic field : , Kinetic energy : 5.3 MeV
Magnetic deflecting force ?
mTB 2.1!
s
kg
MeVJMeVKv
mvK
/102.3
1067.1
)/1060.1)(3.5)(2(2
2
1
7
27
13
2
v!
v
v!!
!
N
vBqFo
B
15101.6
90sin
v!
!
212
27
15
/107.31067.1
101.6sm
kg
N
m
Fa B v!
v
v!!
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Crossed fields : discovery of the electron
Experiment Discovery of the electron in 1897 by J. J. Thomson
Ions are injected in the region of crossed E and B fields, which fixedtheir velocity
Electric field (perpendicular) electric force: (perpendicualar)
Magnetic field(horizontal) magnetic force (perpendicular)
EeBev
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Crossed fields : discovery of the electron
e/m of electron E = 0, B = 0 : note the position of the spot on screen S due to the
undeflected beam
E 0, B = 0 : Measure the resulting beam deflection (perpendiculardirection)
E 0, b 0 and adjust its value until the beam returns to theundeflected position
2
2
2mv
eEL
y !
yE
LB
e
m
B
E
vevBeE 2
22
!!!
),1913,106.1( 19 MillikanInCe
v!
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Crossed fields : Hall effect
Hall effect In 1879, Edwin H. Hall
Copper strip of width d in crossed field
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Crossed fields : Hall effect
Hall effect Hall potential difference V
Different charge (negative or positive) the opposite direction
Number density
The electric and magnetic forces are in balance
Drift speed
Number density
Ed!
BeveE d!
neA
i
ne
Jvd !!
le
Bin ! )/( dAl!
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Crossed fields
Example In the boxed region:
Uniform Balong -y
Uniform Ealong +z
An electron (me, -e) enters atleft with velocity v along +x
Can the electron travel throughthe box undisturbed, as shown?
(a) No, this is impossible!
(b) Yes, if v = (eB)/E
(c) Yes, if v = E/BForce on electron due to E is along -z (intopage)Force on electron due to B is along +z(out of page) Balance these out: eE = evB [sin(900) = 1] v = E/B
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A circulating charged particle
Circular motion Since magnetic force is transverse
to motion, the natural movementof charges is circular.
The radius of the circular path
Period T
qB
mvr
r
mvqvB
rvmF
!!
!
2
2
m
qBf
m
qB
Tf
qB
m
v
r
T
!!
!!
!!
T[
T
TT
2
2
1
22
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A circulating charged particle
Helical paths If the velocity of a charged particle has a component parallel to the (uniform)
magnetic field, the particle will move in a helical path about the direction ofthe field vector.
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A circulating charged particle
Sample problem (mass spectrometer) B = 80.000mT, V = 1000.0V , q = +1.6022 X 10-19C, x =1.6254 m
q
mV
Brx
q
mV
Bm
qV
qB
m
qB
mvr
mqVv
qVmv
UK
222
212
2
02
1
0
2
!!
!!!
!
!
!((
ukgqxB
m 93.203103863.38
25
22
!v!! )106605.11( 27kgu v!
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Cyclotrons and synchrotrons
The cyclotron Suppose you wish to accelerate
charged particles as fast as you can
The proton synchrotron
Linear accelerator (long)
)( conditionresonanceffosc
!
oscfmqB T2!
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Cyclotrons and synchrotrons
Sample problem Cyclotron, oscillator frequency : 12MHz, a dee radius R=53cm
(a) Magnetic field? m=3.34 X10-27kg
(b) Kinetic energy?TT
C
skg
q
mfB osc
6.157.1
1060.1
)1012)(271034.3)(2(219
16
}!
v
vv!!
TT
J
smkg
mvK
sm
kg
Tm
m
RqBv
12
2727
2
7
27
107.2
)/1099.3)(1034.3(2
1
2
1
/1039.3
1034.3
)57.1)(191060.1)(53.0(
v!
vv!
!
v!
v
v!!
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Magnetic force on a current-carrying wire
Magnetic force on a wire
Note : if wire is not straight, compute force ondifferential elements and integrate
BvqF
v
Liitq
d
d
v!
!!
BiLBq
iLqF v!v!
B
LiF
v!
BLid
Fd v!
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Magnetic force on a current-carrying wire
Sample problem i=28A, linear density of the wire : 46.6 g/m
What are the magnitude and direction of theminimum magnetic field needed to suspend thewire-that is, to balance the gravitational force on it
B
T
A
smmkgB
i
gLm
iL
mgB
mgiLB
2
23
106.1
28
)/8.9)(/106.46(
)/(
sin
sin
v!
v!
!!
!
J
J
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Torque on a current loop
Rectangular coil : a x b, current = i
Net force on current loop = 0
But : Net torque is NOT zero!
For a coil with N turns
iaBFF !!31
UX sin)( BNiA! (WhereA is the area of coil)
UUX sinsinTorque 11 iabBbFbF !!v!!
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The magnetic dipole moment
Magnetic moment We just showed : Define : magnetic dipole moment
The coil behaves like a bar magnet placedin the magnetic field
Current carrying coil: magnetic dipole
As in the case of electric dipoles magneticdipoles tend to align with the magnetic field
UX sinNi B!
^
)( nNi!Q
Bv! QX
Right hand rule : curl fingers in
direction of current : thumb
points along
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The magnetic dipole moment
Electric and magnetic dipoles
Epv!X Bv!X
EpU !)(UBU ! QU )(
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The magnetic dipole moment
Example