Download Magnetic Fields and Magnetic Forces Chapter 27

Chapter 27
Lecture 10
Magnetic Fields and Magnetic Forces
Announcements
— Stay tuned for update on TA for 11am and 12pm Discussion Sections
— Exam next Thursday (will cover through Ch 26)
— Ch 26 HW assigned today (due next week)
Brief History of Magnetism
̣
̣
̣
̣
̣
̣
13th century BC: Chinese used compasses
• Possibly an invention of Arabic or Indian origin
800 BC: Greeks
• Discovered magnetite (Fe3O4) attracts pieces of iron
1600: William Gilbert
• Expanded experiments with magnetism to a variety of materials and
suggested Earth was a magnet
1819: Hans Christian Oersted
• An electric current in a wire deflected a nearby compass needle:
connects Electricity and Magnetism
1820’s: Faraday and Henry
• A changing magnetic field creates an electric field
1820’s: Maxwell and his equations
• A changing electric field produces a magnetic field.
Magnetic Poles
̣
Every magnet, regardless of its
shape, has two poles
➡ Called north and south poles
̣ Like poles repel
• N-N or S-S
̣ Unlike poles attract
• N-S
̣
The force between two poles varies as the inverse
square of the distance between them
Magnetic Poles
̣
A single magnetic pole has never
been isolated.
̣ No matter how many times a
permanent magnetic is cut in
two, each piece always has a N
& S pole
A magnet can attract items that are not themselves
magnets
• Analogous to electrostatic attraction of charge
objects to neutral objects via polarization
• “Ferromagnets” (like Iron) can get induced to
become magnets in the presence of a strong
magnetic field.
Magnetic Poles, cont.
●
●
The name “pole” comes from the
way a magnet behaves in the
Earth’s magnetic field
If a bar magnet is suspended so
that it can move freely, it will
rotate
● The magnetic north pole
points toward the Earth’s north
geographic pole. This implies
● Earth’s north geographic
pole is a magnetic south
pole
● Earth’s south geographic
pole is a magnetic north
pole
Magnetism &
Electricity
First evidence of relationship between magnetism and moving charges
discovered in 1820 by Hans Oersted. Current => Magnetic field.
Similar experiments by Ampere, Faraday and Henry discovered that a
moving magnet near a conducting loop can cause a current in the loop.
Ultimately, Maxwell showed that electricity and magnetism are different
manifestations of the same physical laws
•“Maxwell’s Equations” for electromagnetism
Magnetism &
Electricity
First evidence of relationship between magnetism and moving charges
discovered in 1820 by Hans Oersted. Current => Magnetic field.
Similar experiments by Ampere, Faraday and Henry discovered that a
moving magnet near a conducting loop can cause a current in the loop.
Ultimately, Maxwell showed that electricity and magnetism are different
manifestations of the same physical laws
•“Maxwell’s Equations” for electromagnetism
Magnetic Fields
✦
✦
✦
Reminder: an electric field surrounds any
electric charge
The region of space surrounding any moving
electric charge also contains a magnetic
field
A magnetic field also surrounds a permanent
magnet
Example Magnetic Fields for Common Sources
Magnetic Field
!
̣ Magnetic field is a vector: B
Direction is given by the direction
a north pole of a compass needle
points in that location
̣ Magnetic field lines can be used to
show how the field lines, as traced
out by a compass, would look
̣
The compass can be used to trace
the field lines
̣ The lines outside the magnet point
from the North pole to the South pole
̣
●
●
Iron filings in the figures show
the pattern of the magnetic field
lines
The direction of the field is the
direction the north pole of a
compass would point
●
Remember N of compass would be
rotated towards S of magnet
Force on a Charge Moving in a Magnetic Field
!
! !
FB = qv × B
Direction: Force is perpendicular to the plane
defined by the velocity and the magnetic field.
Force on a Charge Moving in a Magnetic Field
!
! !
FB = qv × B
Magnetic
force
charge
Magnetic
field
Velocity of
charge
Direction: Force is perpendicular to the plane
defined by the velocity and the magnetic field.
Direction of Force: Right-Hand Rule
!
! !
FB = qv × B
cross products review
Quiz
At a certain instant, a proton
is moving in the direction i
through a magnetic field in the
–k direction. What is the
direction of the magnetic force
exerted on the proton?
!
! !
FB = qv × B
A.
B.
C.
D.
E.
i
–k
j
–j
The force is zero.
Quiz
At a certain instant, a proton
is moving in the direction i
through a magnetic field in the
–k direction. What is the
direction of the magnetic force
exerted on the proton?
!
! !
FB = qv × B
A.
B.
C.
D.
E.
i
–k
j
–j
The force is zero.
Quiz: Magnetic Force I
A positive charge enters a uniform magnetic field as shown. what is
the direction of the magnetic force ?
A.Left
B.Right
v
C.Zero
D.Into the page
q
E.Out of the page
Quiz: Magnetic Force I
A positive charge enters a uniform magnetic field as shown. what is
the direction of the magnetic force ?
A.Left
B.Right
v
C.Zero
D.Into the page
q
E.Out of the page
The charge is moving parallel to the magnetic field, so it does not
experience any magnetic force.
Recall:
Example
Electrons are traveling at
in a magnetic field:
What is the magnetic force exerted on the electrons (magnitude and direction)?
!"
" !"
F = qv × B
Example
Electrons are traveling at
in a magnetic field:
What is the magnetic force exerted on the electrons (magnitude and direction)?
!"
" !"
F = qv × B
Do we expect the force to be non-zero?
Example
Electrons are traveling at
in a magnetic field:
What is the magnetic force exerted on the electrons (magnitude and direction)?
!"
" !"
F = qv × B
Do we expect the force to be non-zero?
Which direction?
Example
Electrons are traveling at
in a magnetic field:
What is the magnetic force exerted on the electrons (magnitude and direction)?
!"
" !"
F = qv × B
Do we expect the force to be non-zero?
Which direction?
x-comp. of velocity “feels” y-comp. of B, force in z direction
y-comp. of velocity “feels” (-) x-comp. of B, force in z direction
both of those forces point in -z direction
Example
Electrons are traveling at
in a magnetic field:
What is the magnetic force exerted on the electrons (magnitude and direction)?
!"
" !"
F = qv × B
⎛
î
⎜
F = qe ⎜ 2.0 ×10 6
⎜
0.03
⎜⎝
ĵ
3.0 ×10 6
−0.15
(
k̂ ⎞
⎟
0 ⎟
0 ⎟⎟
⎠
) (
)
= qe ⎡ 0 î + 0 ĵ + ⎡⎣ −0.15 × 2.0 ×10 6 − 0.03 × 3.0 ×10 6 ⎤⎦ k̂ ⎤
⎣
⎦
= qe ⎡⎣ −0.3 ×10 6 − 0.09 ×10 6 ⎤⎦ k̂
=-qe ⎡⎣ 0.39 ×10 6 ⎤⎦ k̂
Differences Between E and B Fields
̣ Direction
●
of force
The electric force acts along the direction of the E
(electric) field ~
~
F = qE
●
The magnetic force acts perpendicular to the B
(magnetic) field
̣ Motion
●
●
~
F~ = q~v ⇥ B
The electric force acts on a charged particle regardless
of whether the particle is moving
The magnetic force acts on a charged particle only
when the particle is in motion
Quiz
A charged particle moves an infinitesimal distance, ds, under the
influence of a magnetic field (and no other force). What is the
work done by the magnetic field (B)? (FB below is the magnetic
force.)
A: q B v ds
B: - FB ds
C: |q| B v ds
D: 0
E: None of the above because it depends on the angle
between the velocity and the magnetic field.
!
! !
FB = qv × B
Quiz
A charged particle moves an infinitesimal distance, ds, under the
influence of a magnetic field (and no other force). What is the
work done by the magnetic field (B)? (FB below is the magnetic
force.)
A: q B v ds
d~s = ~v dt
dW = F~ · d~s
B: - FB ds
C: |q| B v ds
F~B ? ~v
D: 0
E: None of the above because it depends on the angle
between the velocity and the magnetic field.
!
! !
FB = qv × B
Work done by Magnetic Fields
!
!
W = ∫ a FB • d s =
b
∫
b
F
cos(90°
)ds
=
0
B
a
!
! !
FB = qv × B
€While electric force does work in displacing a charged
particle, the magnetic force associated with a steady
magnetic field does no work when a particle is
displaced
This is because the force is perpendicular to
the displacement
Work done by Magnetic Fields: KE
̣ The
kinetic energy of a charged particle moving
through a magnetic field cannot be altered by the
magnetic field alone.
̣ When a charged particle moves with a given
velocity through a magnetic field, the field can
alter the direction of the velocity, but not the
speed or the kinetic energy
KE = ½ m v2
v2 depends only on speed,
not direction!
Units of Magnetic Field
●
The SI unit of magnetic field is the tesla (T)
!
! !
FB = qv × B
m
N =C T
s
●
N s
N
Vs
T=
=
= 2
m C mA m
A non-SI commonly used unit is a gauss (G)
1 T = 104 G
€● Earth’s field is about 0.5 G (5e-5 T) but varies over the
surface.
●
Notation
●
When vectors are
perpendicular to the page,
dots and crosses are used
●
●
The dots represent the arrows
coming out of the page
The crosses represent the
arrows going into the page
Quiz: Magnetic Force II
The figure shows a negative charge moving to the right, and a uniform magnetic
field pointing into the screen.
Which path does the charge follow?
A.Path 1
B.Path 2
C.Path 3
D.Not enough information to determine
!
! !
FB = qv × B
Quiz: Magnetic Force II
The figure shows a negative charge moving to the right, and a uniform magnetic
field pointing into the screen.
Which path does the charge follow?
A.Path 1
B.Path 2
C.Path 3
D.Not enough information to determine
!
! !
FB = qv × B
q < 0, so “qV” points left
Now… what path will the particle travel if we keep watching….?
Charged particle in uniform magnetic field
●
●
●
Consider a particle moving in an
external uniform magnetic field
If its initial velocity is
perpendicular to the field then
the force is always directed
toward the center of a circular
path
The magnetic force causes a
centripetal acceleration,
changing the direction of the
velocity of the particle
Circular Motion of a
Charged Particle in B field
●
Equating the magnetic and
centripetal forces:
mv
FB = qvB =
r
2
●
mv radius
p depends on velocity
= x mass (momentum),
Solving for radius r =
charge, and B-field strength
qB qB
●
Period T = 2πr = 2π mv = 2π m
v
●
Angular €
speed
€
v qB
qB
v qB
ω= =
r m