Transcript Magnetism

Magnetism
Chapter 28
Magnetism
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Chapter 28
Magnetic Fields
In this chapter we will cover the following topics:
Magnetic field vector B
Magnetic force on a moving charge FB
Magnetic field lines
Motion of a moving charge particle in a uniform magnetic field
Magnetic force on a current carrying wire
Magnetic torque on a wire loop
Magnetic dipole, magnetic dipole moment 
Hall effect
Cyclotron particle accelerator
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(28 – 1)
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Magnetism was known
long ago.
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New Concept
The Magnetic Field
– We give it the symbol B.
– A compass will line up
with it.
– It has Magnitude and
direction so it is a
VECTOR.
• There are some
similarities with the
Electric Field but also
some significant
differences.
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What produces a magnetic field
One can generate a magnetic field using one of the
following methods:
Pass a current through a wire and thus form what is knows
as an "electromagnet".
Use a "permanent" magnet
Empirically we know that both types of magnets attract
small pieces of iron. Also if supended so that they can
rotate freely they align themselves along the north-south
direction. We can thus say that these magnets create in
the surrounding space a "magnetic field" B which
manifests itself by exerting a magnetic force FB .
We will use the magnetic force to define precicely
Magnetism
(28 – 2)
the magnetic field vector B.
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Magnetism
• Refrigerators are attracted to magnets!
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Where is Magnetism Used??
• Motors
• Navigation – Compass
• Magnetic Tapes
– Music, Data
• Television
– Beam deflection Coil
• Magnetic Resonance Imaging
• High Energy Physics Research
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FE
(28 – 8)
FE  qE
Cathode
FB  qv  B
Anode
FB
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Magnet Demo – Compare to
Electrostatics
N
Magnet
What Happens??
S
Pivot
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Results - Magnets
S N
Shaded End is NORTH Pole
Shaded End of a compass points
to the NORTH.
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• Like Poles Repel
• Opposite Poles
Attract
• Magnetic Poles are
only found in pairs.
– No magnetic
monopoles have
ever been
observed.
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Magnets
Cutting a bar magnet in half produces TWO bar
magnets, each with N and S poles.
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Consider a Permanent Magnet

B
N
S
The magnetic Field B goes from North to South.
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Introduce Another Permanent Magnet

B
N
N
S
pivot
S
The bar magnet (a magnetic dipole) wants to align with the B-field.
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Field of a Permanent Magnet

B
N
N
S
S
The south pole of the small bar magnet is attracted towards the north pole of the big
magnet.
The North pole of the small magnet is repelled by the north pole of the large magnet.
The South pole pf the large magnet creates a smaller force on the small magnet than
does the North pole. DISTANCE effect.
The field attracts and exerts a torque on the small magnet.
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Field of a Permanent Magnet

B
N
N
S
S
The bar magnet (a magnetic dipole) wants to align with the B-field.
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Convention For Magnetic Fields
X
Field INTO Paper
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B

Field OUT of Paper
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Typical Representation
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Experiments with Magnets Show
• Current carrying wire produces a
circular magnetic field around it.
• Force (actually torque) on a Compass
Needle (or magnet) increases with
current.
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Current Carrying Wire
Current into
the page.
B
Right hand RuleThumb in direction of the current
Fingers curl in the direction of B
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Current Carrying Wire
• B field is created at ALL POINTS in space
surrounding the wire.
• The B field has magnitude and direction.
• Force on a magnet increases with the
current.
• Force is found to vary as ~(1/d) from the
wire.
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Compass and B Field
• Observations
– North Pole of magnets
tend to move toward
the direction of B while
S pole goes the other
way.
– Field exerts a
TORQUE on a
compass needle.
– Compass needle is a
magnetic dipole.
– North Pole of
compass points
toward the NORTH.
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Planet Earth
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A Look at the Physics

B
q

v

q B
There is NO force on
a charge placed into a
magnetic field if the
charge is NOT moving.
There is no force if the charge
moves parallel to the field.
• If the charge is moving, there
is a force on the charge,
perpendicular to both v and B.
F=qvxB
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The Lorentz Force
This can be summarized as:

 
F  qv  B
F
or:
F  qvBsin 
v
B
mq
 is the angle between B and V
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Nicer Picture
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Another Picture
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VECTOR CALCULATIONS
i
a  b  ax
bx
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j
ay
by
k
az
bz
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Practice
B and v are parallel.
Crossproduct is zero.
So is the force.
Which way is the Force???
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Units
F  Bqv Sin(θ )
Units :

F
N
N
B


qv Cm / s Amp  m
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1 tesla  1 T  1 N/(A - m)
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teslas are
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The Magnetic Force is Different
From the Electric Force.
Whereas the electric force
acts in the same direction as
the field:
The magnetic force acts in a
direction orthogonal to the
field:


F  qE



F  qv  B
(Use “Right-Hand” Rule to
determine direction of F)
And
--the
charge
must
be
moving
!!
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v
F
C
.B
electron
.
r
r
mv
qB

qB
m
Motion of a charged particle in a uniform magnetic field
(also known as cyclotron motion)
A particle of mass m and charge q when injected with a speed
v at right angles to a uniform magnetic field B, follows a
circular orbit, with uniform speed. The centripetal force
required for such motion is provided by the magnetic force
FB  qv  B
The circular orbit of radius r for an electron is shown in the figure. The magnetic force
v2
mv
2 r 2 mv 2 m
FB  q vB  ma  m  r 
. The period T 


r
qB
v
q Bv
qB
qB
qB
1

. The angular frequency   2 f 
T 2 m
m
Note 1 : The cyclotron period does not depend on the speed v. All particles of the
same mass complete their circular orbit during the same time T regardless of speed
Note 2 : Fast particles move on larger radius circular orbits, while slower particles move
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on smaller radius orbits. All orbits have the same period T
(28 – 9)
The corresponding frequency f 
r
mv
qB
T
2 m
qB
Helical paths
We now consider the motion of
a charged in a uniform magnetic
field B when its initial velocity
v forms and angle  with B.
We decompose v into two
components.
One component  v
v  v cos 
 parallel to B and the other  v  perpendicular to B (see fig.a)
v  v sin 

The particle executes two independent motions.
One is the cyclotron motion is in the plane perpendicular to B we have
mv
2 m
analyzed in the previous page. Its radius r   . Its period T 
qB
qB
The second motion is along the direction of B and it is linear motion with constant
speed v . The combination of the two motions results in a helical path (see fig.b)
2 mv cos 
The
pitch p of the helix is given by: p  Tv 
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qB
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(28 – 10)
Wires
• A wire with a current
contains moving charges.
• A magnetic field will
apply a force to those
moving charges.
• This results in a force
on the wire itself.
– The electron’s sort of
PUSH on the side of the
wire.
F
Remember: Electrons go the “other way”.
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Magnetic force on a current carrying wire.
Consider a wire of length L which carries a current i as shown in
the figure. A uniform magnetic field B is present in the vicinity
FB
of the wire. Experimentaly it was found that a force FB is
exerted by B on the wire, and that FB is perpendicular
to the wire. The magnetic force on the wire is the vector sum
of all the magnetic forces exerted by B on the electrons that
constitute i. The total charge q that flows through the wire
in time t is given by:
L
q  it  i
Here vd is the drift velocity of the electrons
vd
in the wire.
The magnetic force FB  qvd B sin 90  i
L
vd B  iLB
vd
FB  iLB
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(28 – 11)
The Wire in More Detail
Assume all electrons are moving
with the same velocity vd.
L
L
q  it  i
vd
F  qvd B  i
L
vd B  iLB
vd
vector :
F  iL  B
B out of plane of the paper
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Vector L in the direction of the
motion of POSITIVE charge (i).
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(28 – 12)
Magnetic force on a straight wire in a uniform
magnetic field.
If we assume the more general case for which the
magnetic field B froms and angle  with the wire
the magnetic force equation can be written in vector
form as: FB  iL  B
FB  iL  B
B
i
dF
.
dFB = idL  B
FB  i  dL  B
Magnetism
Here L is a vector whose
magnitude is equal to the wire length L and
has a direction that coincides with that of the current.
The magnetic force magnitude FB  iLB sin 

dL
Magnetic force on a wire of arbitrary shape
placed in a non - uniform magnetic field.
In this case we divide the wire into elements of
length dL which can be considered as straight.
The magnetic force on each element is:
dFB = idL  B The net magnetic force on the
wire is given by the integral: FB  i  dL  B
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Current Loop
What is force
on the ends??
Loop will tend to rotate due to the torque the field applies to the loop.
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Top view
 net  iAB sin 
Side view
(28 – 13)
CFnet  0
C
Magnetic torque on a current loop
Consider the rectangular loop in fig.a with sides of lengths a and b which carries
a current i. The loop is placed in a magnetic field so that the normal nˆ to the loop
forms an angle  with B. The magnitude of the magnetic force on sides 1 and 3 is:
F1  F3  iaB sin 90  iaB. The magnetic force on sides 2 and 4 is:
F2  F4  ibB sin(90   )  ibB cos  . These forces cancel in pairs and thus Fnet  0
The torque about the loop center C of F2 and F4 is zero because both forces pass
through point C. The moment arm for F1 and F3 is equal to (b / 2) sin  . The two
torques tend to rotare the loop in the same (clockwise) direction and thus add up.
The net torque    1 + 3 =(iabB / 2) sin   (iabB / 2) sin   iabB sin   iAB sin 
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Magnetic dipole moment :
  B
The torque of a coil that has N loops exerted
by a uniform magnetic field B and carrries a
current i is given by the equation:   NiAB
U    B
We define a new vector  associated with the coil
which is known as the magnetic dipole moment of
U  B
U   B
the coil.
The magnitude of the magnetic dipole moment   NiA
Its direction is perpendicular to the plane of the coil
The sense of  is defined by the right hand rule. We curl the fingers of the right hand
in the direction of the current. The thumb gives us the sense. The torque can
expressed in the form:    B sin  where  is the angle between  and B.
In vector form:     B
The potential energy of the coil is: U    B cos      B
U has a minimum value of   B for   0 (position of stable equilibrium)
U has a maximum value of  B for   180 (position of unstable equilibrium)
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Note
: For both positions the net torque   0
(28 – 14)
The Hall effect
In 1879 Edwin Hall carried out an experiment in which
R
he was able to determine that conduction in metals is due
to the motion of negative charges (electrons). He was also
able to determine the concentration n of the electrons.
He used a strip of copper of width d and thickness . He passed
a current i along the length of the strip and applied a magnetic
field B perpendicular to the strip as shown in the figure. In the
R L
L
presence of B the electrons experience a magnetic force FB that
L
R
pushes them to the right (labeled "R") side of the strip. This
accumulates negative charge on the R-side and leaves the left
side (labeled "L") of the strip positively charged. As a result
of the accumulated charge, an electric field E is generated as
shown in the figure so that the electric force balances the magnetic
force on the moving charges. FE  FB  eE  evd B 
E  vd B (eqs.1). From chapter 26 we have: J  nevd 
Magnetism
vd 
J
i
i


ne Ane
dne
(eqs.2)
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(28 – 15)
The other sides
1=F1 (b/2)Sin()
=(B i a) x (b/2)Sin()
total torque on
the loop is: 21
Total torque:
=(iaB) bSin()
=iABSin()
(A=Area)
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Dipole Moment Definition
Define the magnetic
dipole moment of
the coil  as:
=NiA
= X B
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We can convert this
to a vector with A
as defined as being
normal to the area as
in the previous slide.
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A Coil
For a COIL of N turns, the net
torque on the coil is therefore :
τ  NiABSin(θ )
Normal to the
coil
RIGHT HAND RULE TO FIND NORMAL
TO THE COIL:
“Point or curl you’re the fingers of your right
hand in the direction of the current and your
thumb will point in the direction of the normal
to the coil.
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An Application
The Galvanometer
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A 40.0-cm length of wire carries a current of
20.0 A. It is bent into a loop and placed with
its normal perpendicular to a magnetic field
with a magnitude of 0.520 T. What is the
torque on the loop if it is bent into
(a)an equilateral triangle?
(b)What is the torque if the loop is
(c) a square or
(d) a circle?
(e) Which torque is greatest?
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