Concepts: Magnetic Field
Magnet
Magnetic Force
Cyclotron Motion
Magnetic Dipole Moment
Chapter 32.1–2, 5, 7, 10

• A magnet is a material that produces a magnetic field
• A permanent magnet stays magnetized for a long time, without external aid
• Magnets have two poles: north and south

Opposite poles attract and like poles repel

Works down to the microscopic level!

• Unlike electric charges, which can be +ve or –ve, magnetic poles appear to come only in pairs Æ magnetic dipoles
• There is no such thing as a “magnetic monopole” in the present universe, as far as we can tell, but maybe one could exist

• Magnets pick up some objects, but not others
– pick up: paperclips, iron filings
– won’t pick up: plastic, glass, copper
• If one pole of a magnet picks up an object, so will the other

• Materials which are attracted to a magnet or which a magnet sticks to are called magnetic materials
• All electrically neutral objects are attracted to an electrically charged object, but only a few materials are attracted to magnets
• Magnets have magnetic fields , just as electrically charged objects have electric fields

• Recall: electric dipoles in an external electric field will align with the field because the net torque on a non-aligned dipole is not zero
• Similarly: magnets and magnetic materials will align with external magnetic fields
– this is how compasses work!

North end of each compass points in the direction of the magnetic field.

What kind of magnetic pole is at the geographic north pole of the Earth, in the Arctic?
A. North
B. East
C. South
D. West
E. Other

• Note that, for historical reasons, we use the letter “B” for magnetic fields r
E r
B

• Magnetic field lines are closed curves
• They point into the south pole and out the north pole of a magnet
• They are more concentrated where the magnetic field is strongest
• They do not cross
N
S

• The magnetic field of a bar magnet looks a lot like the electric field of an electric dipole
• One big difference:
– electric field lines start on positive charges and end on negative ones
– magnetic fields lines are closed curves (no start or end)
• This is because there are no magnetic monopoles for fields to start or end on

N
S

• Place compasses around a conducting wire with no electric charges moving through it (i.e. no current)
– result: the compass needles point north to align with B field of Earth

• Now run a current through the wire
– current means that charges are moving along the wire
– compasses now register a magnetic field circling the wire

• The direction of the field is given by the right-hand rule :

• First you say magnetic fields come from permanent magnets
• Now you say that magnetic fields come from electric currents
• Which is it?
• In fact, both types of magnetism have the same cause
• Moreover, when we look at the atomic scale, we shall see that electricity and magnetism are two aspects of the same force —the electromagnetic force

An electric current flowing in a straight wire produces a cylindrical magnetic field
(circular field lines).
Note that the convention is to point the current in the direction opposite to the actual flow of electrons.

• What if we bend the wire?
• The B field still wants to make circles around the wire, but now the circles get pushed together inside the bend and spread apart outside the bend r
I

• Bend the wire all the way around into a loop: current flows out of page current flows into page

looking this way gives

• Second use for right hand rule : wrap fingers of right hand in direction of current and thumb points in direction of magnetic field (i.e. north)

• Where is the north magnetic pole of this current loop?
A
D B
C

• Stack many loops of current together and you get a solenoid (in reality, a tightlywound spiral of wire)

• To build up the field of a solenoid, start with a single current loop:

• Now consider three loops. The field outside the middle one starts to drop.

Many loops:

• If you add up enough loops, the field inside the solenoid approaches uniformity
(like the electric field of a parallel-plate capacitor)
• Fifteen current loops:

• In what direction is the current flowing through this solenoid at the position of the red dot?
A. clockwise
B. counterclockwise
C. into the page
D. out of the page

• If a current-carrying wire generates a magnetic field, than two such wires should exert magnetic forces on each other
• Ampere’s experiment:
– set up two parallel wires
– run currents through them in either direction
– see which configurations result in attraction and repulsion

• Can’t explain these results merely by electric repulsion of the wires
– like currents attract but like charges repel
– opposite currents repel but opposite charges attract
• Ampere’s experiment shows that a magnetic field exerts a force on a current
• Therefore, write magnetic force on a moving charge, q

• Magnetic force on charge q moving with v r r
F
B
q v r
r
B [ N ]
• CAUTION : DO NOT accidentally reverse the sense of the cross product here!
• Magnitude of this force: v r
φ
F
B
= q v B sin
q r
B

r
B
r
B
r
A r
A
r
A r
B
r
A r
B

• Electric current has units of Amperes, A:
1 Ampere = 1 “Amp” = 1 A = 1 C/s
• Magnetic field has units of Tesla, T:
1 Tesla = 1 N/(Am)

• An electron moves with speed 2.0
× 10 5 m/s in a
1.2 Tesla uniform magnetic field. The electron is moving due west and experiences an upward magnetic force of 3.2
× 10 -14 N. What is the direction of the magnetic field?

B y
F
B
= q v B sin
B
θ θ sin
=
F
B q B v v (west)
F (up) x sin
=
0 .
8323
= v r
But where do we measure this angle from?
× r
F r
B
B
=
= −
− r
F e v r
/
× e r
B points into page.
B
Two different directions work… sin( 56 o
)
=
0 .
823 and sin( 180 o −
56 o
)
56 o
=
0 .
823

An electron moves perpendicular to a magnetic force, as shown. What is the direction of the magnetic field?
A. right
B. left
C. into the page
D. up
E. out of the page

• Only a moving charge experiences a magnetic force.
antiparallel to a magnetic field.
• Charges moving perpendicular to the magnetic field experiences a maximum force, which is: F
B
,
Max
= q v B

• The magnetic force is always v r r
B
• The magnetic force on a positive charge is in the direction of v
×
B
• The magnetic force on a negative charge
− v
×
B

A
In which case does the force on the particle point downward?
C
D
B

Beam 1 e e e e e e -
Beam 2 e e e e e e v v

• Examine the forces on a +ve charge in a uniform external B field:
⊗ ⊗ ⊗
⊗ ⊗ ⊗
⊗
⊗
⊗
⊗
⊗
⊗
B
⊗ F ⊗
⊗ ⊗
⊗
+q
⊗
⊗ ⊗
⊗ ⊗
+q
⊗ ⊗
⊗
⊗
⊗
⊗
⊗ ⊗

• Charged particles experience centripetal forces in uniform magnetic fields
• A charged particle moving perpendicular to a uniform magnetic field undergoes uniform circular motion

• We call the circular motion of charged particles in magnetic fields cyclotron motion

• Apply Newton’s 2 nd law:
F
= m a r q v B
= m v
2 r cyc
= r mv qB
B
This is the cyclotron radius for this configuration
F r cyc
+q v

• Cyclotron motion is put to good use in the device the motion is named after: the cyclotron
• The goal of a cyclotron is to accelerate a charged particle (usually an ion) to very high speeds
• Take advantage of particle’s curved path to build a compact device (instead of a linear configuration)

• Cyclotrons are particle accelerators
• Medicine:
– collisions between particles at high speeds can produce radioactive isotopes
– put these radioactive isotopes into metabolically active molecules and inject them into patient
– radioactive molecules concentrate in organ of interest
– radioactive isotopes decay inside the body and emit decay products
– doctors scan the body for the decay products to learn about the diseased or malfunctioning organ
– example: positron emission tomography (PET) scans

Ion
Source x x x Uniform B x x x x x x radius depends on mass x x x
E.g.
12 C and 13 C; same charge:
Find the ratio of diameters if the ions have the same speeds

q v
Σ
F
= r ma r
B
m v
2
( d / 2 ) d
=
2 mv q B d
13
= d
12
2 m
13 v q B
2 m
12 v q B
= m
13 m
12

• We have seen that charged particles with velocities perpendicular to uniform magnetic fields move in circles with uniform angular speed

• We have also seen that charged particles moving parallel to magnetic field lines experience no acceleration r
F
B
= q v r
× r
B
= qvB sin 0
=
0

• If the B field is uniform, but the particle’s velocity is neither parallel nor perpendicular to the field, what happens?
v
||
|| q r
B v
⊥ v r
= v r
⊥
+
⊥ v r
|| r
F r
F
B r
F
B
B
q v r
=
q q ( v r
⊥ v r
⊥
×
r
B r
B v r
||
+
) q
v r
||
B
× r
B
F
B
= q v
⊥
B

• Since the force on the particle is F
B it is accelerated only in the direction
= q v perpendicular to the B field (as expected)
⊥
B
┴
r
B

Ion spiraling in such a non-uniform B:
- Will experience force component slowing down its sideways motion towards denser part of the field
- Reverses path!
Read about
“magnetic bottle”
Earth’s magnetic field interacts with ions from Solar wind: source of aurorae near the poles

• The northern lights (Aurora Borealis) are caused by collisions between energetic particles emitted by the Sun (the “solar wind”) and particles in Earth’s atmosphere
• The aurora usually only appears near
Earth’s magnetic poles because the charged particles from the Sun are channeled there by Earth’s magnetic field

• The electric force on a proton in an external electric field is directed:
A. parallel to the electric field
B. antiparallel to the electric field
C. perpendicular to the electric field
D. it depends on the velocity of the proton

• The magnetic force on a proton moving in an external magnetic field is directed:
A. parallel to the magnetic field
B. antiparallel to the magnetic field
C. perpendicular to the magnetic field

• Proton is accelerated parallel to external E field
• Proton is accelerated perpendicular to external
B field

• A particle with charge q traveling at velocity v enters a region with both E and
B fields
• What is the net force on this particle?
r
F
= r
F e
+ r
F
B
= q r
E
+ q v r
× r
B

• In mass spectrometer, we want to know what v of each particle is.
• Easiest way to do this is to select out only particles with a particular velocity.
• We do this using a velocity selector, which uses crossed E and B fields

y
FBD for a positively charged particle moving through these fields:
F
B
Σ
Σ r
F y r
F y
= r
F e
+ r
F
B
= qE
− qvB
+q
F e
To select particles with velocities directed straight through the selector, set Σ F y
0
= qE
−
=0: qvB v
=
E / B
Only particles with exactly this v will pass through undeflected.

• Recall: all magnets are dipoles dipole moment,
μ
μ r

• For a current loop with surface area A and current I, the magnetic dipole moment is: direction of determined by right-hand rule
μ =
A I [ m 2 A ]

The magnetic moment of a bar magnet is around 1 m 2 A
The magnetic moment of the Earth is about 8 X 10 22 m 2 A
The magnetic dipole moment of a magnetotactic bacterium is about
10 –16 –10 –15 m 2 A

• When an object with magnetic moment is placed in an external magnetic field, r
B
μ r it experiences a torque:
r
=
r
× r
B
• Very similar in form to the torque on an electric dipole caused by an electric field.

μ μ r
|| r
B
• Thus, just as electric dipoles try to align with electric fields, magnetic dipoles try to align with magnetic fields

• As we have seen, moving charges produce magnetic fields
• Consider an electron orbiting an atomic nucleus:
• The motion of the electron gives the atom a magnetic moment!

• Most atoms have many electrons, and their magnetic moments are arranged to mostly or entirely cancel each other out
• If you add up all the little magnetic moments of the atoms in a substance, they typically cancel out

• However , individual particles, such as electrons, have their own magnetic moments
• Why? Because they are charged and rotating Æ moving charge *
μ r
*This explanation is a bit of a cheat. More on this when we talk about quantum mechanics.

• As you may recall from chemistry, atoms gather electrons in pairs with opposite “spins”
• The net magnetic moment of each pair is zero
• But not all atoms have even numbers of electrons
• Add up all the unpaired electron magnetic moments, and usually you get an overall magnetic moment of roughly zero, because they all point in random directions
• This is why most materials are non-magnetic and can’t be picked up with a magnet

• In some materials, notably iron, the magnetic moments of unpaired electrons interact in such a way that they all line up
• We call such materials ferromagnets (“ironlike magnets”)

• In ferromagnets, the magnetic moments are usually arranged over short distances into magnetic domains
• The overall magnetic moment still isn’t huge, which is why ordinary iron isn’t itself a good magnet

• If we apply an external magnetic field to a ferromagnetic substance, we can align up all of the magnetic domains and create a much stronger magnet
• If the external magnetic field is withdrawn, the domains can remain aligned, creating a permanent magnet
• You can’t make a permanent magnet out of just any ferromagnetic material, because in some such materials, the domains will tend to come unaligned again

• A single magnet is brought near a ferromagnetic substance, producing the magnetic moment as shown (orange arrows). Which magnet was it?
A. 1 or 2
B. 3 or 4
C. 1 or 3
D. 2 or 4
1 2
3 4