# Chapter 21

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```					Chapter 21

Magnetic Forces and Fields
Chapter 21: Magnetic Forces and Fields
Magnetism
Magnetic  Force and Field
Trajectory of a Charge in a Magnetic Field
Magnetic Forces and Torques on Electric Currents
Magnetic Fields Produced by Currents
Ampere’s Law
Magnetic Materials
Magnetism
The unity of physics:
   heat and thermodynamics          kinetic energy, momentum,
conservation of
energy

   electricity      force and energy

   magnetism:
 magnetic fields exert forces on moving charges
 moving charges produce magnetic fields
Magnetism
Magnesia: now called “Manisa,” in western Turkey

Here, around 500 BC, people first noticed that a particular sort of iron
ore, suspended from a thread, sought a constant orientation relative to
the earth.
Magnetism
No isolated poles (instead, opposed pairs: dipoles)
N & S (“north” and “south”)

Unlike poles attract each other
Like poles repel each other

Pole names result from tendency toward geographic
alignment
Magnetism
The “north” pole is the
end of a magnetic
dipole that seeks the
geographic north
direction on Earth.
Magnetic Fields -- Qualitative
The magnetic field is a
vector that points
from a magnetic
north pole, toward a
magnetic south pole.

The field points in the
same direction as the
north pole of a
compass needle.
Magnetic Fields -- Qualitative
Magnetic field lines
point toward a south
magnetic pole.

The magnetic field
vector is tangent to
the field line at each
point.
Magnetic Fields -- Quantitative
The magnetic field vector is mathematically defined
from its property of exerting a force on a moving
charge.
Magnetic Fields -- Quantitative
The magnetic force on a moving charge:
     
F  qv  B
magnetic field
magnetic force
velocity
charge

Magnitude:         F  qvBsin 
Direction: given by right-hand rule #1
Magnetic Fields -- Quantitative
Right-hand rule #1, two ways:
v
F
F

B

B
v       F

B                                         v
right hand, palm up
Magnetic Fields -- Quantitative
The magnetic force equation serves as a defining
equation for the magnetic field, B: F  qvBsin 

F
B
qv sin 
Ns
SI units of magnetic field, B:        T (tesla)
Cm

1
Other common unit of B: the gauss ( =        T     )
10 000
Nikola Tesla

Serbian-American inventor and electrical engineer
1856 - 1943
Johann Carl Friedrich Gauss

German mathematician and physicist
1777 - 1855
Magnetic Fields and Forces
Properties of the magnetic force on moving charges

 The magnetic force acts only on a moving charge. No
force on a stationary charge.
 The magnetic force acts only on a charge whose velocity
has a nonzero component perpendicular to B. No magnetic
force is exerted if the motion of the charge is parallel to B.
 The magnetic force is perpendicular to the magnetic field.
 The magnetic force does no work on the moving charge.
Magnetic Fields and Forces
The magnetic force does no work on the moving
charge?
q                         B

θ
By right-hand rule #1,
v sin(θ)

F points out of the
plane of the drawing
θ     v

v cos(θ)

W  Fd cos  0 because  90
Trajectory of a Charge in a Magnetic Field
Consider a charged object moving at a velocity v in a plane
perpendicular to a uniform magnetic field B:

B

q
v
Trajectory of a Charge in a Magnetic Field
The magnetic force on the charge is always perpendicular to its
velocity.

B

F

v
q
Trajectory of a Charge in a Magnetic Field
At a later time, the force is still perpendicular to the velocity.

v

B
F       q
Trajectory of a Charge in a Magnetic Field
A force that always acts on an object perpendicular to its velocity
is a centripetal force.

This force produces circular motion. (Recall Chapter 5.)

The force, velocity, and path radius are related by:

v2
FC  ma C  m
r
Trajectory of a Charge in a Magnetic Field
v2
FC  ma C  m
r

But FC is also the magnetic force:

FC  qvBsin   qvB
2
mv                   mv
qvB                   r
r                   qB
Mass Spectrometry
If ions have a mass m and a
charge q:

1 2
mv  qV
2
2qV
v2 
m
2qV
m
mv2                     mv           m
FC  qvB                  qB       
r                      r          r
qB 2 r 2
solve for m : m 
2V
Magnetic Force on a Current-Carrying Wire
Recall the magnetic force on a charge moving perpendicular to a
magnetic field:
F  qvBsin 

Now, if a current I flows through a straight wire whose length is
L:
F  qvBsin 
q
F  qvBsin    vtB sin    ILBsin 
t
t           t
Magnetic Force on a Current-Carrying Wire

F  ILBsin 
 is now the angle between the
field vector B and the wire
The direction of F is still given
by right-hand rule #1
(the thumb is now the
conventional current)
Torque on a Current-Carrying Loop
A rectangular loop makes an angle f with a magnetic field:

B

A     I                 A                      FM
B                                           f
/2
L      W/2 sin f                          W
f
FM
N                      S                                 W
/2

B

W                                           VIEW A-A
FM  ILB
W
  2 FM  sin f  ILBW sin f  IAB sin f
2
Torque on a Current-Carrying Loop
A rectangular loop makes an angle f with a magnetic field:

B

A     I                 A                      FM
B                                           f
/2
L      W/2 sin f                          W
f
FM
N                      S                                 W
/2

B

W                                           VIEW A-A

  IAB sin f (single loop)
N turns :   NIAB sin f
Torque on a Current-Carrying Coil

  NIABsin f

Maximum torque occurs when f = 90° (plane of
coil is parallel to B):  m ax  NIAB

Minimum torque occurs when f = 0° (plane of coil
is perpendicular to B):   0
m in
Magnetic Field Produced by a Current
Experimental observation (due to Oersted): a current-
carrying wire produces a magnetic field, directly
proportional to the magnitude of the current, and
inversely proportional to the distance from the wire:
I
B
r
Convert to an equation, using a constant of proportionality:
m0 I
B   
2 r
m0 is called the permeability of free space: m0  4  10 7 T m / A
Hans Christian Oersted: 1777 - 1851
Danish physicist; discovered
the field due to a current-
carrying wire by accident,
one evening in April 1820,
while preparing for a
lecture at the University of
Copenhagen
Magnetic Field
Produced by a Current

m0 I
Magnitude:    B
2r

Direction: right-hand rule #2
Magnetic Forces Between Two Current-Carrying Wires
Each wire produces a magnetic field at the location of the
other wire

Each wire experiences a magnetic force due to the field
produced by the other
B2

F2                 F1

I1                                 I2

r
B1
Magnetic Field Produced by a Current Loop
RHR #2 shows that
the magnetic field
points out of the
plane of the loop
R
on the inside of the
loop, and into the
plane on the
outside of the loop.          B

The field magnitude
is not uniform.
I
Magnetic Field Produced by a Current Loop
The field magnitude
at the center is
given by:
m0 I
B                    R
2R
If the loop becomes a
flat coil, with N           B
turns:
m0 I
BN
2R
I
Magnetic Field Produced by a Solenoid
Solenoid: a coil whose length is large compared to its radius.

Field magnitude inside the solenoid:     B  m0 nI

# of turns per unit length
Ampere’s Law
For an arbitrary closed path about an arbitrary current
distribution I:
 
 B  dl  m0 I
We can also write this as a sum of “sufficiently” small
length elements around the closed path:

B    parallel   l  m0 I
Ampere’s Law
Ampere’s Law can be used to calculate the magnetic fields produced
by simple current geometries. Example: the long straight wire:

B    parallel   l  m0 I
Right-hand rule #2 says that B
is everywhere parallel to l. So:

 B l  B l
parallel

B  l  B  2r  m I       0

m0 I
B
2r
Magnetic Materials
Electrons have orbital and spin rotations in atoms.

These circular motions of charge act as current loops, and
produce magnetic dipoles on an atomic scale.

These magnetic dipoles mostly cancel each other. The
remaining dipoles tend to be oriented randomly, so bulk
materials do not show magnetic dipole behavior.
Magnetic Materials
In ferromagnetic materials (iron, cobalt, nickel, and some
alloys), there is a spin coupling of some electrons over
small portions of the material called domains.

The domains do behave as magnetic dipoles.

In an “ordinary” object made of a ferromagnetic material,
the domains are themselves randomly oriented, and the
object is not a magnet.
Magnetic Materials
The ferromagnetic domains may be brought into a common
alignment by an external magnetic field (induced
magnetism).

In some materials, the common domain alignment is
persistent, even when the external field is removed. The
object is then a permanent magnet.

The ferromagnetic field can be several orders of magnitude
higher than the external field that causes the domain
alignment.

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