Download Magnetic Fields

Magnetic Fields
Magnetism on the Atomic Level
Magnetic Domains
Magnetic Poles
Magnetic Fields and the Definition of B
The Field. We can define a magnetic field B to be a vector quantity that exists when it
exerts a force FB on a charge moving with velocity v. We can next measure the
magnitude of FB when v is directed perpendicular to that force and then define the
magnitude of B in terms of that force magnitude:
FE
E
q
B=
FB
qv
1N
1N
1T

m
1C 1 s 1 A 1 m
where q is the charge of the particle. We can summarize all these results with the
following vector equation:
F B  qv  B
that is, the force FB on the particle by the field B is equal to the charge q times the
cross product of its velocity v and the field B (all measured in the same reference
frame). We can write the magnitude of FB as
FB  q vB sin 
where ϕ is the angle between the directions of velocity v and magnetic field B.
Force on a Moving Charge
+ charge
FB  q vB sin 
v
+
B
FM
- charge
FM
v
–
B
Magnetic Fields and the Definition of B
F B  qv  B
This equation tells us the direction of F. We know the cross product of v and B is a
vector that is perpendicular to these two vectors. The right-hand rule (Figs. a-c) tells
us that the thumb of the right hand points in the direction of v × B when the fingers
sweep v into B. If q is positive, then (by the above Eq.) the force FB has the same
sign as v × B and thus must be in the same direction; that is, for positive q, FB is
directed along the thumb (Fig. d). If q is negative, then the force FB and cross
product v × B have opposite signs and thus must be in opposite directions. For
negative q, F is directed opposite the thumb (Fig. e).
Magnetic Fields and the Definition of B
F B  qv  B
Answer:
(a) towards the positive z-axis
(b) towards the negative x-axis
(c) none (cross product is zero)
Magnetic Fields and the Definition of B
𝐹𝐵 never has a component parallel to 𝑣. This means that 𝐹𝐵 cannot
change the particle’s speed 𝑣 (and thus cannot change the particle’s
KE). The force can only change the direction of 𝑣. Only in this sense
does a magnetic field accelerate a particle.
Concept Check – Magnetic Force (2)
A positive charge enters a uniform magnetic field as shown. What is the
direction of the magnetic force?
1.
2.
3.
4.
5.
out of the page
into the page
downward
upward
to the left
x x x x x x
x x x x vx x
q
x x x x x x
Concept Check – Magnetic Force (2)
A positive charge enters a uniform magnetic field as shown. What is the
direction of the magnetic force?
1.
2.
3.
4.
5.
out of the page
into the page
downward
upward
to the left
x x x
F
x x x
q
x x x
x x x
x vx x
x x x
Using the right-hand rule, you can see that the magnetic force is
directed upward. Remember that the magnetic force must be
perpendicular to BOTH the B field and the velocity.
Concept Check – Magnetic Force (3)
A positive charge enters a uniform magnetic field as shown. What is the
direction of the magnetic force?
1.
2.
3.
4.
5.
out of the page
into the page
downward
upward
to the left


v

q

Concept Check – Magnetic Force (3)
A positive charge enters a uniform magnetic field as shown. What is the
direction of the magnetic force?
1.
2.
3.
4.
5.
out of the page
into the page
downward
upward
to the left


v

F 
q

Using the right-hand rule, you can see that the magnetic force is
directed into the page. Remember that the magnetic force must
be perpendicular to BOTH the B field and the velocity.
Concept Check – Magnetic Force (4)
A positive charge enters a uniform magnetic field as shown. What is the
direction of the magnetic force?
1.
2.
3.
4.
5.
out of the page
into the page
zero
to the right
to the left








v

q





Concept Check – Magnetic Force (4)
A positive charge enters a uniform magnetic field as shown. What is the
direction of the magnetic force?
1.
2.
3.
4.
5.
out of the page
into the page
zero
to the right
to the left




 v
 
 q
 
F=0




The charge is moving parallel to the magnetic field, so it does not
experience any magnetic force. Remember that the magnetic
force is given by: F = q v B sin(q). What is q for this scenario?
Concept Check – Atom Beam
A beam of atoms enters a magnetic field region. What path will the
atoms follow?
x x x x x x x x x 1x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
2
x x x x x x x x x x x x
x x x x x x x x x x x x
4
x x x x x x x x x x x x
3
Concept Check – Atom Beam
A beam of atoms enters a magnetic field region. What path will the
atoms follow?
x x x x x x x x x 1x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
2
x x x x x x x x x x x x
x x x x x x x x x x x x
4
3
x x x x x x x x x x x x
Atoms are neutral objects whose net charge is zero. Thus they do
not experience a magnetic force.
Follow-up: What charge would follow path 3? What about path 1?
Concept Check – Force on Charge in Magnetic Field
A proton beam enters into a magnetic field region as shown below.
What is the direction of the magnetic field B?
1.
2.
3.
4.
5.
+y
–y
+x
+ z (out of page)
– z (into page)
y
x
Concept Check – Force on Charge in Magnetic Field
A proton beam enters into a magnetic field region as shown below.
What is the direction of the magnetic field B?
1.
2.
3.
4.
5.
+y
–y
+x
+ z (out of page)
– z (into page)
y
x
The picture shows the force acting in the +y direction. Applying
the right-hand rule leads to a B field that points into the page.
The B field must be into the plane because B  v and B  F.
Magnetic Fields and the Definition of B
Magnetic Field Lines
We can represent magnetic fields with field lines, as we did for electric fields. Similar
rules apply:
(1) the direction of the tangent to a magnetic field line at any point gives the direction
of B at that point
(2) the spacing of the lines represents the magnitude of B —the magnetic field is
stronger where the lines are closer together, and conversely.
Two Poles. The (closed) field lines enter one end of a magnet and exit
the other end. The end of a magnet from which the field lines emerge is
called the north pole of the magnet; the other end, where field lines enter
the magnet, is called the south pole. Because a magnet has two poles, it
is said to be a magnetic dipole.
Magnetic Fields and the Definition of B
(a) The magnetic field lines for a bar magnet.
(b) A “cow magnet” — a bar magnet that is intended to be slipped down into the rumen of a cow
to recover accidentally ingested bits of scrap iron and to prevent them from reaching the
cow’s intestines. The iron filings at its ends reveal the magnetic field lines.
Sample 28.01
Magnetic Fields
Magnetic Fields
Bar Magnet
The direction of a magnetic field can be seen by placing a compass near a
bar magnet. The North pole of the compass points in the direction of the field,
away from North and toward South. The magnetic field can also be seen by
placing the magnet under a piece of paper and sprinkling small iron filings on
top. The filings line up in the shape of the field, as shown at right.
Magnetic Fields
Magnetic Field of Earth
Spin
axis
The geographic North Pole is a
magnetic south pole.
The North pole of a compass points to the Earth's geographic North Pole, which
means that there must be a magnetic South Pole up there.
Magnetic Field of Earth
Van Allen Belts
Van Allen Belts
Aurora Over Alaska
Aurora Over Prelude Lake, Canada
Aurora and Meteor Over Norway
Above Aurora Australis
Northern Lights Over Canada
Aurora Over Alaska
Aurora in West Texas Skies
Aurora Over Missouri, Oct. 24, 2011
Aurora Over Arkansas, Oct. 24, 2011
Concept Check – Moving in a Magnetic Field
In what direction would a B field have to point for a beam of electrons
moving to the right to go undeflected through a region where there is a
uniform electric field pointing vertically upward?
1.
2.
3.
4.
5.
E
up (parallel to E )
down (antiparallel to E )
into the page
out of the page
impossible to accomplish
B=?
electrons
v
Concept Check – Moving in a Magnetic Field
In what direction would a B field have to point for a beam of electrons
moving to the right to go undeflected through a region where there is a
uniform electric field pointing vertically upward?
1.
2.
3.
4.
5.
E
up (parallel to E )
down (antiparallel to E )
into the page
out of the page
impossible to accomplish
B=?
electrons
v
Without a B field, the electrons feel an electric force downward. In
order to compensate, the magnetic force has to point upward.
Using the right-hand rule and the fact that the electrons are
negatively charged leads to a B field pointing out of the page.
Crossed Fields: Discovery of The Electron
A modern version of J.J.
Thomson’s apparatus for
measuring the ratio of mass to
charge for the electron. An
electric field E is established by
connecting a battery across the
deflecting-plate terminals. The
magnetic field B is set up by
means of a current in a system of
coils (not shown). The magnetic
field shown is into the plane of the
figure, as represented by the
array of Xs (which resemble the
feathered ends of arrows).
If a charged particle moves through a region containing both an electric field and a
magnetic field, it can be affected by both an electric force and a magnetic force.
When the two fields are perpendicular to each other, they are said to be crossed
fields.
If the forces are in opposite directions, one particular speed will result in no deflection
of the particle.
Crossed Fields: Discovery of The Electron
y  12 a y t 2
FE q E
ay 

m
m
L
L
v
 t
t
v
2
y
1
2
q EL
q EL2
  
m v
2mv 2
E
If FE  FB , then q E  q vB sin 90  v 
B
q EL2
2 2
m
B
L
y
2

2
E
q
2
yE
2m  
B
Crossed Fields: The Hall Effect
As we just discussed, a beam of electrons in a vacuum can be
deflected by ad magnetic field. Can the drifting conduction
electrons in a copper wire also be deflected by a magnetic
field?
In 1879, Edwin H. Hall, then a 24-year-old graduate student at
the Johns Hopkins University, showed that they can.
This Hall effect allows us to find out whether the charge
carriers in a conductor are positively or negatively charged.
Beyond that, we can measure the number of such carriers per
unit volume of the conductor.
Figure (a) shows a copper strip of width d, carrying a current i
whose conventional direction is from the top of the figure to the
bottom. The charge carriers are electrons and, as we know,
they drift (with drift speed vd) in the opposite direction, from
bottom to top.
As time goes on, electrons move to the right, mostly piling up
on the right edge of the strip, leaving uncompensated positive
charges in fixed positions at the left edge as shown in figure (b).
Crossed Fields: The Hall Effect
When a uniform magnetic field B is applied to a conducting strip
carrying current i, with the field perpendicular to the direction of
the current, a Hall-effect potential difference V is set up across
the strip.
The electric force FE on the charge carriers is then balanced by
the magnetic force FB on them.
The number density n of the charge carriers can then be
determined from
Bi
n=
Vle
in which l (= A/d) is the thickness of the strip. With this equation
we can find n from measurable quantities.
Crossed Fields: The Hall Effect
Moving Conductor
When a conductor moves through a uniform magnetic field B
at speed v, the electrons in the conductor also move as a
result. The conduction electrons will experience a force
causing them to move and set up an E-field and potential
difference, V. Equilibrium will be established and the Halleffect potential difference V across it is
FE  FB
eE  evB
V  Ed
V = vBd
Where d is the width perpendicular to both velocity v and
field B.
Sample 28.02
Sample 28.02
A Circulating Charged Particle
A beam of electrons is projected into a
chamber by an electron gun G. The
electrons enter in the plane of the page
with speed v and then move in a region of
uniform magnetic field B directed out of
that plane. As a result, a magnetic force
FB= q (v×B) continuously deflects the
electrons, and because v and B are always
perpendicular to each other, this deflection
causes the electrons to follow a circular
path. The path is visible in the photo
because atoms of gas in the chamber emit
light when some of the circulating electrons
collide with them.
A Circulating Charged Particle
FB  mac
mv 2
q vB =
r
mv
r=
qB
A Circulating Charged Particle
mv
r=
qB
T
2 r 2 mv 2 m


v
v qB qB
f 
qB
1

T 2 m
qB
  2 f 
m
The quantities T, f, and  do not depend on the speed of the particle. Fast particles
move in large circles and slow ones in small circles, but particles with the same
charge to mass ratio take the same time T to complete one round trip. You can also
see, according to the right hand rule, that the direction of rotation for a positive
particle is ccw when looking in the direction of B.
Concept Check – Mass Spectrometer
Two particles of the same speed and charge enter a magnetic field and
follow the paths shown. Which particle has the bigger mass?
3. both masses are equal
4. impossible to tell from the picture
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
1
2
Concept Check – Mass Spectrometer
Two particles of the same speed and charge enter a magnetic field and
follow the paths shown. Which particle has the bigger mass?
3. both masses are equal
4. impossible to tell from the picture
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
The relevant equation for us is:
mv
R
qB
x x x x x x x x x x x x
x x x x x x x x x x x x
1
2
According to this equation, the
bigger the mass, the larger the radius.
Follow-up: This is one way to separate isotopes.
Concept Check – Mass Spectrometer
Two particles of the same mass and speed enter a magnetic field and
follow the paths shown. Which particle has the bigger charge?
3. both charges are equal
4. impossible to tell from the picture
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
1
2
Concept Check – Mass Spectrometer
Two particles of the same mass and speed enter a magnetic field and
follow the paths shown. Which particle has the bigger charge?
3. both charges are equal
4. impossible to tell from the picture
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
The relevant equation for us is:
mv 2
FC 
 q vB
R
mv
R 
qB
x x x x x x x x x x x x
x x x x x x x x x x x x
1
2
According to this equation, the
bigger the charge, the smaller the radius.
Follow-up: What is the sign of the charges in the picture?
Concept Check – Mass Spectrometer (2)
A proton enters a uniform magnetic field that is perpendicular to the
proton’s velocity. What happens to the kinetic energy of the proton?
1.
2.
3.
4.
5.
it increases
it decreases
it stays the same
depends on the velocity direction
depends on the B field direction
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
Concept Check – Mass Spectrometer (2)
A proton enters a uniform magnetic field that is perpendicular to the
proton’s velocity. What happens to the kinetic energy of the proton?
1.
2.
3.
4.
5.
it increases
it decreases
it stays the same
depends on the velocity direction
depends on the B field direction
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
x x x x x x x x x x x x
The velocity of the proton changes direction but the magnitude
(speed) doesn’t change. Thus the kinetic energy stays the same.
Helical Paths
Helical Paths
v  v cos 
v  v sin 
pitch p  Tv  Tv cos 
Sample 28.03
Sample 28.04
Cyclotrons and Synchrotrons
The Cyclotron: The figure is a top view of the region of a cyclotron
in which the particles (protons, say) circulate. The two hollow Dshaped objects (each open on its straight edge) are made of sheet
copper. These dees, as they are called, are part of an electrical
oscillator that alternates the electric potential difference across the
gap between the dees. The electrical signs of the dees are
alternated so that the electric field in the gap alternates in direction,
first toward one dee and then toward the other dee, back and forth.
The dees are immersed in a large magnetic field directed out of the
plane of the page. The magnitude B of this field is set via a control
on the electromagnet producing the field.
The key to the operation of the cyclotron is that the frequency f at which the proton
circulates in the magnetic field (and that does not depend on its speed) must be equal
to the fixed frequency fosc of the electrical oscillator, or
f  f osc
(resonance condition)
This resonance condition says that, if the energy of the circulating proton is to
increase, energy must be fed to it at a frequency fosc that is equal to the natural
frequency f at which the proton circulates in the magnetic field.
Cyclotron
Cyclotron
Cyclotrons and Synchrotrons
Proton Synchrotron: The magnetic field B and the oscillator frequency fosc,
instead of having fixed values as in the conventional cyclotron, are made to vary
with time during the accelerating cycle. When this is done properly, (1) the
frequency of the circulating protons remains in step with the oscillator at all
times, and (2) the protons follow a circular — not a spiral — path. Thus, the
magnet need extend only along that circular path, not over some 4 ✕106 m2. The
circular path, however, still must be large if high energies are to be achieved.
Fermi National Accelerator Laboratory
Photo Tour of Fermilab
Magnetic Force on Current-Carrying Wire
A straight wire carrying a current i in a uniform
magnetic field experiences a sideways force
F B  iL  B
(force on a current)
Here L is a length vector that has magnitude L
and is directed along the wire segment in the
direction of the (conventional) current.
Crooked Wire. If a wire is not straight or the field
is not uniform, we can imagine the wire broken
up into small straight segments. The force on the
wire as a whole is then the vector sum of all the
forces on the segments that make it up. In the
differential limit, we can write
d F B  id L  B
and the direction of length vector L or dL is in the
direction of i.
A flexible wire passes between the
pole faces of a magnet (only the
farther pole face is shown). (a)
Without current in the wire, the wire
is straight. (b) With upward current,
the wire is deflected rightward. (c)
With downward current, the
deflection is leftward.
Forces on Wires
 t  qvtB sin q q
FM  qvB sin q  qvB sin q   

 vt  B sin q

t
t
 t 
FM  IlB sin q
Concept Check – Magnetic Force on a Wire
A vertical wire carries a current and is in a vertical magnetic field. What
is the direction of the force on the wire?
I
1.
2.
3.
4.
5.
left
right
zero
into the page
out of the page
B
Concept Check – Magnetic Force on a Wire
A vertical wire carries a current and is in a vertical magnetic field. What
is the direction of the force on the wire?
I
1.
2.
3.
4.
5.
left
right
zero
into the page
out of the page
B
When the current is parallel to the magnetic field lines, the force
on the wire is zero.
Concept Check – Magnetic Force on a Wire (2)
A horizontal wire carries a current and is in a vertical magnetic field.
What is the direction of the force on the wire?
1.
2.
3.
4.
5.
left
right
zero
into the page
out of the page
I
B
Concept Check – Magnetic Force on a Wire (2)
A horizontal wire carries a current and is in a vertical magnetic field.
What is the direction of the force on the wire?
1.
2.
3.
4.
5.
left
right
zero
into the page
out of the page
I
B
Using the right-hand rule for charge moving in a magnetic field,
we see that the magnetic force must point out of the page. Since F
must be perpendicular to both I and B, you should realize that F
cannot be in the plane of the page at all.
Sample 28.06
Torque on a Current Loop
As shown in the figure
(right) the net force on the
loop is the vector sum of
the forces acting on its four
sides and comes out to be
zero. The net torque acting
on the coil has a
magnitude given by
t = NiABsinq
where N is the number of turns in the coil, A
is the area of each turn, i is the current, B is
the field magnitude, and θ is the angle
between the magnetic field B and the normal
vector to the coil n.
The elements of an electric motor: A
rectangular loop of wire, carrying a
current and free to rotate about a fixed
axis, is placed in a magnetic field.
Magnetic forces on the wire produce a
torque that rotates it. A commutator (not
shown) reverses the direction of the
current every half-revolution so that the
torque always acts in the same
direction.
Concept Check – Magnetic Force on a Loop
A rectangular current loop is in a uniform magnetic field. What is the
direction of the net force on the loop?
1.
2.
3.
4.
5.
+x
+y
zero
-x
-y
B
z
y
x
Concept Check – Magnetic Force on a Loop
A rectangular current loop is in a uniform magnetic field. What is the
direction of the net force on the loop?
1.
2.
3.
4.
5.
+x
+y
zero
-x
-y
B
z
y
x
Using the right-hand rule, we find that each of the four wire
segments will experience a force outward from the center of the
loop. Thus, the forces of the opposing segments cancel, so the
net force is zero.
Concept Check – Magnetic Force on a Loop (2)
If there is a current in the loop in the direction shown, the loop will:
1.
2.
3.
4.
5.
move up
move down
rotate clockwise
rotate counterclockwise
both rotate and move
N
S
Concept Check – Magnetic Force on a Loop (2)
If there is a current in the loop in the direction shown, the loop will:
1.
2.
3.
4.
5.
move up
move down
rotate clockwise
rotate counterclockwise
both rotate and move
N
S
Look at the North Pole: here the magnetic field points to the right
and the current points out of the page. The right-hand rule says
that the force must point up. At the south pole, the same logic
leads to a downward force. Thus the loop rotates clockwise.
Torque on a Current Loop
Torque on a Current Loop
FM  NIlv B
  2  FM   12 lh sin  
  Force LeverArm
  NIlvlh B sin 
   FM   12 lh sin  
  NIAB sin 
Summary
The Magnetic Field B
• Defined in terms of the force FB
acting on a test particle with charge
q moving through the field with
velocity v
Eq. 28-2
A Charge Particle Circulating in
a Magnetic Field
• Applying Newton’s second law to
the circular motion yields
Eq. 28-15
• from which we find the radius r of
the orbit circle to be
Eq. 28-16
Magnetic Force on a Current
Carrying wire
• A straight wire carrying a current i in
a uniform magnetic field
experiences a sideways force
Eq. 28-26
• The force acting on a current
element i dL in a magnetic field is
Eq. 28-28
Torque on a Current Carrying
Coil
• A coil (of area A and N turns,
carrying current i) in a uniform
magnetic field B will experience a
torque τ given by
Eq. 28-37
Summary
The Hall Effect
• When a conducting strip carrying a
current i is placed in a uniform
magnetic field B, some charge
carriers (with charge e) build up on
one side of the conductor, creating
a potential difference V across the
strip. The polarities of the sides
indicate the sign of the charge
carriers.
Orientation Energy of a
Magnetic Dipole
• The orientation energy of a
magnetic dipole in a magnetic field
is
Eq. 28-38
• If an external agent rotates a
magnetic dipole from an initial
orientation θi to some other
orientation θf and the dipole is
stationary both initially and finally,
the work Wa done on the dipole by
the agent is
Eq. 28-39