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Magnetic Fields and Forces
Getting to the heart of the matter:
Charges in motion create magnetic fields. (Stationary charges do not.)
Magnetic fields can cause forces on charges in motion. (But not on
stationary charges.)
Since one or more charges in motion are called “current”, we can restate
these more clearly as:
(1) Magnetic fields are caused by currents.
(2) Currents in magnetic fields can experience magnetic forces.
We are intentionally using the word “can” here since, for certain
alignments, magnetic fields may not cause a force on a given current.
Magnetism is caused by “electricity in motion”. As you know, magnets have
“North and South poles”, that lead to attraction and repulsion. But, there
are no free “magnetic charges (N, S)” associated with these poles. We’ll
see that these poles are caused by electric current distributions, and that
electricity and magnetism are unified interactions. This was one of the
great physics discoveries of the 19th century.
The connection between electricity and magnetism
Hans Christian Ørsted 1777-1851
Danish physicist and chemist, He is best known for
discovering the relationship between electricity and
magnetism known as electromagnetism.
While preparing for an evening lecture on April 21st 1820,
Ørsted developed an experiment which provided evidence
that surprised him. As he was setting up his materials, he
noticed a compass needle deflected from magnetic north
when the electric current from the battery he was using was
switched on and off. This deflection convinced him that
magnetic fields radiate from all sides of a live wire just as
light and heat do, and that it confirmed a direct relationship
between electricity and magnetism.
Three months later he began more intensive investigations,
and soon thereafter published his findings, proving that an
electric current produces a magnetic field as it flows through
a wire.
But magnetism was known and applied from ancient times…
(From Wikipedia) Lodestone refers to either:
(1) Magnetite, a magnetic mineral form of Fe3O4, one of
several iron oxides.
(2) A piece of intensely magnetic magnetite that was used
as an early form of magnetic compass.
Iron, steel and ordinary magnetite are attracted to a
magnetic field, including the Earth's magnetic field.
Only magnetite with a particular crystalline structure,
lodestone, can act as a natural magnet and attract and
magnetize iron.
In China, the earliest literary reference to magnetism lies in
a 4th century BC book called Book of the Devil Valley
Master (鬼谷子): "The lodestone makes iron come or it
attracts it." The earliest mention of the attraction of a
needle appears in a work composed between 20 and
100 AD (Louen-heng): "A lodestone attracts a needle."
By the 12th century the Chinese were known to use
the lodestone compass for navigation.
Fun with magnets: the rules of attraction and repulsion
Conclusion:
Opposite poles attract,
like poles repel.
(Same rule as for
electric charges.)
Magnets can “magnetize” some materials,
creating “induced magnetization”
What does this tell us about
the induced poles?
How does this happen?
There are no free magnetic poles
If you try to separate the N and S poles of a magnet by cutting it
in half, new N and S poles will appear so that each new magnet
always has a NS pair!
This will make sense once we define the magnetic poles and
see how they are created.
How will we measure magnetic fields?
Q
We can use a small “test
magnet” to find the direction
of the magnetic field, since
the test magnet will align
parallel to the field lines.
We cannot use a test charge since
stationary charges experience no
force from a magnetic field.
The compass is an example of a “test magnet”
The action of the test magnet.
Similar to an electric dipole in an
electric field.
Historically, mapping Earth’s
magnetic field has been of great
interest and utility:
The result: Earth’s magnetic field is a “magnetic dipole”
Electric currents circulating in
the Earth’s interior create a
magnetic South pole near the
geographic North pole and a
magnetic North pole near the
geographic South.
Notice that, outside the Earth,
the shape of the magnetic field
is very similar to the shape of
an electric dipole field with a
negative charge at “S” and a
positive charge at “N”, where
the field lines emerge. The
shape is similar but the physics
is different!
The Earth’s geomagnetic field: measurement + simulation
The pattern of magnetic field
lines inside the Earth is
much more complicated than
those inside the magnets
we’ll study.
This is because our
“geodynamo” consists of a
complex network of currents,
mostly in the outer core,
driven by the Earth’s
rotation.
Fortunately, the magnetic
field pattern near and
beyond the surface is an
almost perfect dipole.
(Out further, it is distorted by
the “solar wind” of charged
particles.)
Now, to the basics:
The magnetic field
is a vector field
(like E), and this is
its symbol:

B
Units:
 N 
1 T  1 tesla  

A

m


(Nikola Tesla,1856-1943. We meet him later. )
In this chapter, we will be focusing on magnetic forces on charges and
currents moving in a given magnetic field.
How magnetic fields are produced will be considered in detail in the
next chapter, “Magnetic sources”.
But, in order know what magnetic field configurations to expect, we
will look at a few pictures of basic field configurations, and the current
distributions that cause them. 
Magnetic field of a line of current, I.
 Demo. Also magnetic field lines
Magnetic field of a line of current.
Right-hand rule, and 2D view of test magnets.
Magnetic field of a line of current.
2D end view and 3D view.
This B field has cylindrical
symmetry. Compare and
contrast to E field of line
charge.
Magnetic dipole field due to a current loop, or a magnet.
This B field has
dipole form at long
distances.
Magnetic field due to a solenoid (series of loops)
This B field has dipole
form at long distances.
Inside, this B field is almost uniform
throughout the solenoid interior !
Next chapter !
Drawing conventions for 2D pictures of B fields
Now we return to the central topic of this chapter,
magnetic forces.
Magnetic force on a charged particle of velocity v in field B:
First, consider the force magnitude:
F  q v B  q vBsin(  )
1. The force is proportional to the charge.
2. The force is proportional to the speed.
3. The force is proportional to the magnetic field.
4. The force depends on the angle between v and B.
The full story is contained in the cross product:

 
F  q (v  B )
So the force vector is also
perpendicular to both v and B.
(We will look at evaluation of the
cross product in more detail.)

F

v


B
Lorenz force equation: combining the electric and magnetic
forces acting on a charged particle.
If a charged particle is traveling in a
region that has both electric and
magnetic fields, the forces due to both
fields may be added vectorially
(superposed):
 

F  FE  FB
The resulting equation is known as the
Lorenz force equation:



F

FB

FE

E

v


B
 
F  q( E  v  B)
(Ludvig Valentin Lorenz, 1829 - 1891: Danish mathematician and physicist.)
We’ll begin by considering problems where only a magnetic field is
present. But later, we’ll look at the more general case.
Magnetic force on a charged particle of velocity v in field B:
Again, the magnetic force equation:

 
F  q (v  B )
magnitude
F  q v B  q vBsin(  )
The sin() factor tells us that the
particle will experience the
maximum force when the vectors
are lined up as shown, with the
velocity perpendicular to the field.
But the other thing to notice is that the
magnetic force is always perpendicular to
the velocity. This means that the force
never acts along the particle’s direction of
motion. So, F does no work on the particle!
F can change the particle’s direction, but
not its speed (kinetic energy).

F

v


B
In a region with uniform magnetic field B:
As we said, alignment matters.
Negative charge flips the force.
We can use “right hand rules” to figure out
directions associated with the cross product.
 Cross product demonstrator
Charged particle motion in a uniform B field
From the discussion above we see
that, in 2D, this must be circular
motion at constant speed. Why??
The magnetic force provides the
radial acceleration needed to
maintain circular motion:
v2
F  qv B  qvB  mar  m
r
One power of v cancels, giving:
mv p
qB 

r
r
mv p
 r

qB qB
And the orbital frequency, f, is:
v qB
 
r m


qB
f 

2 2 m
f is also known as “cyclotron frequency”
Discuss the equations for r and f.
Magnets at the LHC proton collider, CERN,
Geneva, Switzerland
~8000 magnets of various types. Magnetic fields at full strength: 8.4 T.
Proton energies: 7 TeV per particle. Radius of main ring: 4.3 km.
Discovery of the positron (anti-electron)
These particles are in a uniform
magnetic field. A high energy
photon (gamma ray) has
collided with an electron in a
hydrogen atom. The electron
proceeds forward. There was
also enough energy in the
collision to produce another pair
of particles (electron and its antimatter partner, the positron) via
E = mc2.
For each particle, from the
direction in which it turns we can
find the charge sign, and from
the radius of curvature right after
the collision we can calculate
the momentum:
p  qBr
(The particles spiral in to smaller radii as
they lose energy colliding with atoms.)
Charged particle motion in a uniform B field in 3D
If vparallel = 0, then the
motion is circular, parallel
to the y-z plane.
Otherwise, the particle
coasts at constant
velocity in the +x or –x
direction, and the
resulting trajectory is a
helix. How would we
calculate its pitch angle?
Charged particle motion in a “magnetic bottle”
Since the magnetic field is strongest at
these locations, the cyclotron frequency
is highest here.
The Van Allen Belts: nature’s magnetic bottle
The dipole field of the
Earth is strongest near
the poles, creating a
magnetic bottle effect
between the poles. As
the particles stream into
the poles (and back out
again), their cyclotron
frequency rises, then
falls. They also radiate
photons, observable as:
(1) Aurorae at the poles.
(2) Radio “whistlers”.
D
Magnetic force on a conductor carrying a current I in
a uniform magnetic field, B
We’ve considered magnetic forces on individual
charges. Now we find the total force on a
segment of conductor of length L carrying a
current I in a uniform field B. Because the
charge carriers have drift velocity vd , each
carrier feels a magnetic force. But, the electric
fields in the conductor constrain them to motion
in the direction of the current.
The force acting on all charges in this volume is:
F  Qtotvd B  (qnAL)vd B  (nqvd ) ALB  ( JA) LB  ILB
Giving us the simple result:
F  ILB
This can be used for any problems where the current is perpendicular to the
magnetic field, and hence to the drift velocity. But we need to generalize this
formula to account for the cross product relationship between vd and B.
Generalizing the force on a conductor to any orientation
For a single particle in a uniform B field:
Cross product:
Magnitude with tilt:
Magnitude at 90o:

 
F  qv  B
F  qvB sin(  )
F  qvB
Our derivation showed that we should
replace qv by IL:
Magnitude at 90o:
Magnitude with tilt:
Cross product:
F  ILB
F  ILB sin(  )

 
F  IL  B
This last equation is general, and it makes sense, since the direction of
v is now the same as the direction of the conductor, and I contains q.
Evaluating the cross product
Two equations we’re using have cross products:

 
F  qv  B

 
F  IL  B
The cross product is calculated by (1) filling a 3x3
matrix with x-y-z unit vectors in the first row, and
putting the two vectors in order into the second and
third rows, then (2) finding the determinant of this
matrix. In the first equation above, we would have:
iˆ ˆj kˆ 


 
v  B  det v x v y v z 
B B B 
 x y z
  ˆ
v  B  i (v y Bz  vz By )  ˆj (vx Bz  vz Bx )  kˆ(vx By  v y Bx )
Then, since F = q(v x B), if q is negative, it will flip the direction of F.
 Example with uniform B field in z direction
Define the “magnetic moment”:
useful in torque and energy equations
The “magnetic moment”, m , also called the
“magnetic dipole moment”, tells us the
strength of a magnetic dipole field, in the
same way that p, the electric dipole
moment tells us the strength of an electric
dipole field. We’ll define it here, then use it
in upcoming derivations:
m  IA
The top picture shows us the direction
of the magnetic moment for a current
loop; and the bottom picture, for a
hydrogen atom—caused by the current
of the electron traveling in its orbit.
S
N
Magnetic moment for a coil of many turns
(Ignore the external B field for now.) Find the
magnetic moment if the coil above has (1) a single
turn, (2) 500 turns. The general expression? Also,
note the units.
Torque on a current loop in a uniform magnetic field, B
Maximum torque
F  ILB
The long sides of the loop,
each of length L, experience
forces F = ILB, perpendicular
to the wire and to the
magnetic field. There is a
torque on the loop about the
y axis, similar to the torque
felt by an electric dipole in a
uniform electric field. Note
the direction of m when the
torque is minimum and
maximum.
D
Zero torque
Torque on a current loop in a uniform magnetic field, B
Discuss forces on ends,
of length b. Then…
Total force on each side:
F  IaB
Torque due to each side:
F  ILB
b
IabB sin(  )
 side  F sin(  ) 
2
2
Total torque on loop:
  2 side  IabB sin(  )  IAB sin(  )  mB sin(  )
F  IaB
But , m, and B are all vectors. We can see from the picture, and from the
sin() factor that they are related by the cross product:



  mB
Application of loops in a magnetic field: the DC electric motor
Materials that can be magnetized
These are called “ferromagnetic materials”, and they include iron (Fe),
nickel (Ni), cobalt (Co), and gadolinium (Gd).
You can see that if this magnetization is induced by an external magnetic field, the
situation resembles the induced polarization of a dielectric by an electric field.
A closer look at magnetization
Note: Some materials, such as “soft iron” are easy to magnetize, and when
their magnetic domains align, the resulting field can be much larger than the
external field causing the magnetization.
In some materials, the magnetization persists
“Soft iron” and “soft steel” are examples of
alloys that magnetize easily, with internal
magnetic fields up to 2T. But when the
external field is removed, their magnetic
domains readily return to random
orientation, and the magnetization
disappears. These materials are useful for
transformers and inductors (more later).
“Hard steel” and other special ferromagnetic
alloys, are harder to magnetize. But after
the external field is removed, their magnetic
domains retain alignment. Permanent
magnets can be made from these materials,
and from specialized ceramic materials.
D
Magnetic flux
 
 E   E  dA   E cos( )dA
Recall electric flux: General expression
A
Uniform field, flat surface
 
 B   B  dA   B cos( )dA
 Magnetic flux is: General expression
A
Uniform field, flat surface
General
A
 
 E  E  A  EA  E ( A cos  )
A
 
 B  B  A  BA  B( A cos  )
Uniform field, flat surface
Gauss’s Law for magnetic fields
What can we calculate with magnetic flux? Not much yet, until we have seen
Faraday’s Law of induced emf. But for now, we can jump directly to Gauss’s
Law for magnetic fields. Start with Gauss’s Law for electric fields, the one we
have seen already:
  Q
 E   E  dA 
A
enclosed
0
As we have seen, magnetic forces come from electric charges in motion.
There are no free magnetic charges. Magnetic field lines diverge from N
poles and converge into S poles, but they do not begin or end at either pole.
Then Qmagnetic = 0, so that there cannot be enclosed charge. Gauss’s Law for
magnetism is then:
 
 B   B  dA  0
A
 Sketch a bar magnet and look at magnetic Gaussian surfaces
“containing” the poles, and containing the whole magnet.
Contrast to case of the electric dipole.
Crossed E and B fields: a velocity selector
 Find v in terms of E and B
D
The mass spectrometer: a
velocity selector followed by a
uniform B field to give momentum
information. Why do different
beam spots correspond to different
masses?
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