Videos
FerroFluid Sculptures Demo
http://youtu.be/uDzfkg8QSkQ
MagField Demo
http://youtu.be/f7t-PSHxZSc
MagField Demo
https://youtu.be/8llkHQtaOlg
10 Minute Compilation
https://youtu.be/V-M07N4a6-Y?list=PLzQYvo_
Tb2BmD-jIOvpTaJag3Lmth0if0
Wolfram Demonstrations
http://demonstrations.wolfram.com/search.
html?query=Magnetic_Field
SP212
Ch. 28 - Magnetic Fields
Maj Jeremy Best USMC
Physics Department, U.S. Naval Academy
February 23, 2016
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SP212
February 23, 2016
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Find the Physics
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Magnetic Tapes
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Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
~
The Magnetic Field (B)
Magnetic Field production
The electric field:
A magnetic field can be produced two different ways.
1
The first way is to use a current in a wire to
produce an electromagnet . These will be covered
in more detail in the next chapter (29).
2
The second way is to to use intrinsic properties of
charged particles (electrons) that are in certain
materials. These can add together to produce a
NET magnetic field in magnetic materials resulting
in a Permanent magnet.
~
~E = F
q0
We can define magnetic fields in a similar (though not
quite as simple) manner:
~FB = q~v × B
~
FB = |q|vB sin φ
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Permanent Magnets
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Magnetic Field Lines
Magnetic fields, like electric fields, can be drawn using
magnetic field lines. Magnetic field lines always start on
north poles and end on south poles . All magnets have
these two poles. There are ONLY magnetic dipoles.
Similarly to electric charges, like magnetic poles repel,
opposite poles attract!
Figure:
Figure from ”Grain boundary engineering by magnetic field application ” by T. Watanabe and S. Tsurekawa
and X. Zhao and L. Zuo , Science Direct 2006,
(a) Microstructure observed after heating at T = 1153 K, for 33 min and cooling at 10 C/min with a magnetic eld of 14 T.
(b) Schematic illustration of nucleation of ferrite phase at austenite/grain boundary triple junctions along the magnetic eld
direction.
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Permanent Magnets
The Magnetic Field
The unit for the magnetic field is the tesla (T):
1T =
1N
Am
Sometimes you will still see magnetic field measured in
gauss (G):
1 G = 1 × 10−4 T
The earth’s magnetic field is about .5 G at the surface.
Figure: tutorvista.com
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Earth’s Magnetic Field
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Crossed Fields
Previously, we’ve done problems about the Millikan Oil
Drop experiment, which involved balancing the force of
gravity with the force due to an electric field. We can
also balance the force of an electric field using a
magnetic field. The fields must be orthogonal to one
another (do you see why?) and are therefore called
“crossed fields”.
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Crossed Fields
Crossed Fields
J.J. Thompson used the following setup in 1897 to
discover the electron:
Back in Chapter 22, we solved the problem of a particle
moving through a uniform electric field, and found the
deflection at the end of the plates was
Deflection in E Field
qEL2
y=
2mv 2
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The electrons are boiled off the filament and accelerated
by the potential difference V . When we turn on the B
field, we create a magnetic force pointing in the opposite
direction of the E field. When the particles pass through
undeflected, We know the forces are equal in magnitude
but opposite in direction
Equal Forces
FE
|q|E
|q|E
v
= FB
= |q|vB sin φ
= |q|vB
= E /B
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SP212
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Crossed Fields
We know the speed of the electrons from the
accelerating potential
K = qV
(1/2)mv 2 = qV
r
2qV
v=
m
Substituting, we find:
q
E2
=
m
2B 2 V
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Crossed Fields
The Hall Effect
I’ve stated that the moving charge carriers in circuits are actually
the negatively charged electrons, not positive particles in direct
contrast to Ben Franklin’s assumption. Who is right?
Substituting that result in for v in our deflection
equation gives the charge-to-mass ratio of the electron:
|q|
2yE
= 2 2
m
B L
VS
The Hall Effect provides a direct experimental demonstration who
wins this battle.
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Using the Hall Effect
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The Hall Effect
The buildup of charge carries creates a
potential difference across the width of
the conductor, which creates an electric
field
V = Ed
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This electric field builds up until it
cancels out the magnetic force on the
charge carriers:
Circulating Charges
Now we can re-arrange this to find all sorts of good stuff!
The Hall Effect
mv
|q|B
2πm
2πr
=
T =
v
|q|B
|q|B
f = 1/T =
2πm
|q|B
ω = 2πf =
m
r=
FE = F B
eE = evd B
i
e(V /d) = eB
neA
Bid
n=
VAe
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Helical Motion
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Helical Motion
The last discussion assumed the particle was only moving
in a plane perpendicular to the magnetic field. If the
particle has a component of velocity parallel to the field
however, this component is not affected by the magnetic
field. What results is helical motion, where the
perpendicular component creates the circling motion,
and the parallel component adds the “progress.” We call
the distance between successive cycles the pitch (p) of
the helix.
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v|| = v cos φ
v⊥ = v sin φ
Maj Jeremy Best USMC (Physics Department, U.S. Naval Academy)
SP212
Magnetic Fields and Currents
Consider a length L of wire carrying a
current i. A current is a bunch of
charged particles, moving with a
velocity. They must respond to
magnetic forces! We know that current
and charge are related: q = it . Also,
since we know that currents move at
constant speed, this is easy to find:
v = L/t . Put these into what we
know:
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Magnetic Fields and Currents
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Torque on a Current Loop
Cool, magnetic fields exert forces on current carrying
wires, but Ch. 27 kept talking about circuits, which were
always complete loops. It turns out, magnetic fields
exert torques on current carrying loops.
Magnetic Force
~FB = q~v × B
~
~
= (it)(~L/t) × B
~
= i ~L × B
Where ~L is a vector with magnitude
equal to the length of the wire, in the
direction of conventional current flow
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The system
Torque on a Current Loop
Let’s find the net force acting on this
loop.
~F2 = −~F4
Boring!
We define the normal vector (~n) for this loop using the
right hand rule. Point your fingers in the direction of i,
your thumb gives the direction of ~n.
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Torque on a Current Loop
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Torque on a Current Loop
But ~F1 and ~F3 do not cancel out. In
fact, they produce a torque on the loop
.
τ = |~r × ~F| = rF sin θ
τ 0 = τ1 + τ3
b
b
τ0 =
iaB sin θ +
iaB sin θ
2
2
τ 0 = ibaB sin θ
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SP212
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OK, magnetic fields produce torques on current carrying
loops. We can make the torque stronger if we add N
more loops:
τ = Nτ 0 = NiBA sin θ
Where A = ab is the area of the loop Current carrying
loops move to align their area vector ~n with an external
magnetic field
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The Magnetic Dipole Moment
So current carrying loops align themselves with external
magnetic fields, just like bar magnets. Thus, we we can
consider the loop itself to be a magnetic dipole, and we
~ , the magnetic dipole moment. The
can define µ
~ is the same as the normal vector ~n, and its
direction of µ
magntidue is µ = NiA.
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The Magnetic Dipole Moment
~ to describe the energy of a
Finally, we can also use µ
dipole in a magnetic field.
The Magnetic Dipole Moment
~ to write our last results more compactly
We can use µ
τ = NiAB sin θ
= µB sin θ
~
~ =µ
~ ×B
τ
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Wiley Plus Homework
Chapter 28: Questions 1, 2, 6, 7, 9. Problems: 1, 3, 5,
9, 11, 14, 17, 27, 41, 49, 55, 56, 66, 70, 77, 86.
~
U(θ) = −~
µ·B
Dipoles have their lowest energy −µB when they are
aligned with the external field (they “want” to be aligned
with the field). They have their highest energy +µB
when they are anti-parallel to the field.
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