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Ch. 21 - Magnetic Forces and Fields
3 D- visualization of vectors and
current
21.1 Magnetic Fields
The needle of a compass is permanent magnet that has a north
magnetic pole (N) at one end and a south magnetic pole (S) at
the other.
21.1 Magnetic Fields
The behavior of magnetic
poles is similar to that of
like and unlike electric charges.
21.1 Magnetic Fields
Surrounding a magnet there is a magnetic field, an “alteration of
space” caused by the magnet. The direction of the magnetic field at
any point in space is the direction indicated by the north pole of a
small compass needle placed at that point. The variable we use for
the magnetic field is B.
21.1 Magnetic Fields
The magnetic field lines and pattern of iron filings in the vicinity of a
bar magnet and the magnetic field lines in the gap of a horseshoe
magnet.
21.1 Magnetic Fields
The iron core of the earth acts as a magnet and the earth has a magnetic
field as shown below
21.2 The Force That a Magnetic Field Exerts on a Charge


F  qE
Recall that when a charge is placed in an electric field, it experiences a
force, according to
The direction of the force is always parallel to or 180° from the
direction of the field.
Magnetic Force on a moving charge
due to an external magneticB field
• A moving charge is a
magnet.
• In the presence of an
external magnetic field,
the charge will
experience a force and
change its speed and
direction.
• The magnetic field (B) is
coming out of the page in
this picture.
21.2 The Force That a Magnetic Field Exerts on a Charge
Right Hand Rule No. 1. Extend the right hand so the fingers point
along the direction of the magnetic field and the thumb points along
the velocity of the charge. The palm of the hand then faces in the
direction of the magnetic force that acts on a positive charge.
If the moving charge is negative,
the direction of the force is opposite
to that predicted by RHR-1.
Magnetic Force on a charge moving in a magnetic
field (B)
•
The magnitude,|Fm|= qvB(sinθ) where θ is the angle
between the direction of motion and the direction of B
• The direction of Fm is given by RHR1
• The unit of B is the Tesla (Newton/Amp-meter)
F=0
1 gauss  104 tesla
F<Fmax
and is often used when
studying the magnetic field of
the Earth
F = Fmax
Magnetic Force on a moving charge
due to an external B field
Fm = qvBsinθ
• Only a moving charge
experiences a force.
• There must be a component
of velocity perpendicular to
the external field, or F = 0.
Magnetic Force on a moving charge
due to an external B field
Fm = qvBsinθ
• The force is mutually
perpendicular to v
and B.
• The force on a
negative moving
charge is shown by
the left hand rule!
Conceptual Problem
A positive charge is
traveling in the direction
shown by the arrow. It
enters the external B field
(dots mean out of the
page). The particle
experiences a force:
a. to the right
b. to the left
c. into the page
d. out of the page
Conceptual Problem
A positive charge is
traveling in the direction
shown by the arrow. It
enters the external B field
(dots mean out of the
page). The particle
experiences a force:
a. to the right
b. to the left
c. into the page
d. out of the page
Numerical problem
A particle with a charge of
+8.4 μC and a speed of 45
m/s in the direction shown
(purple) enters an external
magnetic field (red) of 0.30 T.
Find the magnitude and the
direction of the magnetic force
on the particle.
Numerical problem
A particle with a charge of
+8.4 μC and a speed of 45
m/s in the direction
shown(purple) enters an
external magnetic field (red) of
0.30 T. Find the magnitude
and the direction of the
magnetic force on the particle.
Note that vy = v sin 150° is the
component perpendicular to
the magnetic field.
Answer: 5.67 x 10-5 Newtons,
into the page
vy
21.3 The Motion of a Charged Particle in a Magnetic Field
•The magnetic force always remains
perpendicular to the velocity and is directed
toward the center of the circular path.
•The magnetic force results in a centripetal
acceleration. According to Newton’s Law:
v2
Fm  m
r
v2
qvB  m
r
21.4 The Mass Spectrometer
Chemists and biologists use the “mass spec” to identify the the
mass and/or charge of molecules.
2
v
qvB  m
r
21.4 The Mass Spectrometer
The mass spectrum of naturally
occurring neon, showing three
isotopes.
How much Deuterium in my sample?
Deuterium, (“heavy hydrogen”) has an extra
neutron in its nucleus. Researchers want
to know the percentage of deuterium in a
sample of hydrogen. The hydrogen is
ionized and then is accelerated through
the metal plates with a potential
difference of 2100 V It then enters a
uniform B field 0.1T, into the page. Ignore
effects due to gravity.
At what radius should the detector be
placed?
mp = mn = 1.67 x 10-27 kg
q = e = 1.60 x 10-19 C
We need to use both Conservation of
Energy and Newton’s Law for uniform
circular motion to solve this problem.
How much Deuterium in my sample?
1. find the speed of the particle
upon exiting the capacitor, using
conservation of energy:
EPE0 = KEf
q ∆V = mv2/2, v = 4.49 x 105 m/s.
2. Use Newton’s Law to find an
expression for r:
2
v
Fm  qvB  m
r
r = 9.37 x 10-4 m
Ranking Task
3 particles move
perpendicular to a
uniform B field and
travel in circular paths
as shown. They have
the same mass and
speed. Rank the
particles in order of
charge magnitude,
greatest to least:
v2
Fm  qvB  m
r
a. 3,2,1
b. 3,1,2
c. 2,3,1
d.1,3,2
e. 1,2,3
Ranking Task
3 particles move
perpendicular to a
uniform B field and
travel in circular paths
as shown. They have
the same mass and
speed. Rank the
particles in order of
charge magnitude,
greatest to least:
v2
Fm  qvB  m
r
a. 3,2,1
b. 3,1,2
c. 2,3,1
d.1,3,2
e. 1,2,3
charge and radius are inversely
proportional
21.5 The Force on a Current in a Magnetic Field
The magnetic force on the
moving charges pushes the
wire to the right.
21.5 The Force on a Current in a Magnetic Field
F  qvB sin 
 q 
F   
vt B sin 

t L

I
F  ILB sin 
21.6 The Torque on a Current-Carrying Coil
The two forces on the loop have equal magnitude but an application
of RHR-1 shows that they are opposite in direction.
21.6 The Torque on a Current-Carrying Coil
The loop tends to rotate such that its
normal becomes aligned with the magnetic
field.
21.6 The Torque on a Current-Carrying Coil
Net torque    ILB 12 w sin    ILB 12 w sin    IAB sin 
magnetic
moment

  NIA B sin 
number of turns of wire
What happens to the loop?
The magnetic field
exerts:
a. a net force and a net
torque
b. a net force but no net
torque
c. a net torque, but no
net force
d. neither a net force,
nor a net torque
What happens to the loop?
The magnetic field
exerts:
a. a net force and a net
torque
b. a net force but no net
torque
c. a net torque, but no
net force
d. neither a net force,
nor a net torque
What happens to this loop?
B = 0.25T, I = 12 A
Single turn square with
sides of 0.32m length
Part 1: Is there a torque?
a. yes, top out of the page,
bottom into the page
b. yes, left side out of the
page, right side into the
page
c. no torque
What happens to this loop?
B = 0.25T, I = 12 A
Single turn square with
sides of 0.32m length
Part 1: Is there a torque?
a. yes, top out of the page,
bottom into the page
b. yes, left side out of the
page, right side into the
page
c. no torque
What happens to this loop?
B = 0.25T, I = 12 A
Single turn square with
sides of 0.32m length
Part 2: What is the
magnitude of the torque?
What happens to this loop?
B = 0.25T, I = 12 A
Single turn square with sides
of 0.32m length
Part 2: What is the magnitude
of the torque?
 = F(L/2)
 = NIAB
top
+ F(L/2)bottom Or
= 30.7 N-m
21.6 The Torque on a Current-Carrying Coil
Example 6 The Torque Exerted on a Current-Carrying Coil
A coil of wire has an area of 2.0x10-4m2, consists of 100 loops or turns,
and contains a current of 0.045 A. The coil is placed in a uniform magnetic
field of magnitude 0.15 T. (a) Determine the magnetic moment of the coil.
(b) Find the maximum torque that the magnetic field can exert on the
coil.
magnetic
moment
(a)

NIA  100 0.045 A  2.0 10  4 m 2  9.0 10  4 A  m 2
magnetic
moment



4
2

4
(b)   NIA B sin   9.0  10 A  m 0.15 T sin 90  1.4  10 N  m


21.6 The Torque on a Current-Carrying Coil
The basic components of
a dc motor.
21.6 The Torque on a Current-Carrying Coil
21.7 Magnetic Fields Produced by Currents
Right-Hand Rule No. 2. Curl the fingers of the
right hand into the shape of a half-circle. Point
the thumb in the direction of the conventional
current, and the tips of the fingers will point
in the direction of the magnetic field.
21.7 Magnetic Fields Produced by Currents
A LONG, STRAIGHT WIRE
o I
B
2 r
o  4 10 7 T  m A
permeability of
free space
21.7 Magnetic Fields Produced by Currents
Example 7 A Current Exerts a Magnetic Force on a Moving Charge
The long straight wire carries a current
of 3.0 A. A particle has a charge of
+6.5x10-6 C and is moving parallel to
the wire at a distance of 0.050 m. The
speed of the particle is 280 m/s.
Determine the magnitude and direction
of the magnetic force on the particle.
21.7 Magnetic Fields Produced by Currents
 o I 
 sin 
F  qvB sin   qv
 2 r 
o I
B
2 r
21.7 Magnetic Fields Produced by Currents
Current carrying wires can exert forces on each other.
21.7 Magnetic Fields Produced by Currents
Conceptual Example 9 The Net Force That a Current-Carrying Wire
Exerts on a Current Carrying Coil
Is the coil attracted to, or repelled by the wire?
21.7 Magnetic Fields Produced by Currents
A LOOP OF WIRE
B
o I
2R
center of circular loop
21.7 Magnetic Fields Produced by Currents
Example 10 Finding the Net Magnetic Field
A long straight wire carries a current of 8.0 A and a circular loop of
wire carries a current of 2.0 A and has a radius of 0.030 m. Find the
magnitude and direction of the magnetic field at the center of the loop.
21.7 Magnetic Fields Produced by Currents
o I1 o I 2 o  I1 I 2 
B

 
 
2 r 2R
2 r R 

4 10
B

T  m A  8.0 A
2.0 A 

  1.1105 T

2
  0.030 m 0.030 m 
7
21.7 Magnetic Fields Produced by Currents
The field lines around the bar magnet resemble those around the loop.
21.7 Magnetic Fields Produced by Currents
21.7 Magnetic Fields Produced by Currents
A SOLENOID
number of turns per
unit length
Interior of a solenoid
B  o nI
21.7 Magnetic Fields Produced by Currents
A cathode ray tube.
21.8 Ampere’s Law
AMPERE’S LAW FOR STATIC MAGNETIC FIELDS
For any current geometry that produces a
magnetic field that does not change in time,
 B    I
||
o
net current
passing through
surface bounded
by path
21.8 Ampere’s Law
Example 11 An Infinitely Long, Straight, Current-Carrying Wire
Use Ampere’s law to obtain the magnetic field.
 B    I
||
o
B   o I
B2 r  o I
o I
B
2 r
21.9 Magnetic Materials
The intrinsic “spin” and orbital motion of electrons gives rise to the magnetic
properties of materials.
In ferromagnetic materials groups of neighboring atoms, forming
magnetic domains, the spins of electrons are naturally aligned with
each other.
21.9 Magnetic Materials
21.9 Magnetic Materials