Concept 24.1: A magnetic dipole in a magnetic
field has a potential energy of orientation.
24. Magnetic dipoles
It takes work to rotate a magnetic
dipole against the torque due to a
magnetic field. Since this work
depends only on the initial and
final orientation, it can be
represented as a potential energy.
• potential energy of a dipole
• force on a magnetic dipole
• polarization and permanent
magnets
• why does a magnet stick to the
fridge?
U (θ 0 ) = −W cons = − ∫ τ app d θ =
Serway and Beichner
Section 29.3
µ
θ
B
I
r
r
r
r
τ app = −τ B = − µ × B
∫ µ B sin θ
dθ
r r
= − µ B cos θ = − µ • B
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Physics 1E03 Lecture 24
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Concept 24.2: A magnetic dipole feels a force
in a non-constant magnetic field.
Quiz of concept 24.1
Recall that a force is related to potential energy by
Which of these current loops has the lowest potential
energy of orientation?
r
r
dU
U = −Wc = − ∫ Fc • dr so Fx = −
dx
(Similar for Fy and Fz)
For a magnetic dipole in a magnetic field,
a)
b)
d)
c)
Fx = −
I
I
I
Physics 1E03 Lecture 24
r d r
r r
d
( − µ • B ) = µ • B ( x, y , z )
dx
dx
A constant field gives no net force on the dipole. A field
which changes with position is required.
I
3
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Concept 24.3: Magnetic materials (like iron and
nickel) are easily polarized by a magnetic field.
Quiz of concept 24.2
The magnetic
polarizability of
materials is measured
by the “susceptibility”,
χ.
The force on the current loop due the bar magnet varies
with their separation, x, according to:
y
S
a) 1/x
µM
b) 1/x2
N
µL
c) 1/x3
I
x
d) 1/x4
Atomic dipoles aligned
by a field B0 create an
average dipole
moment µave which
then is a source of its
own dipole field,
r
r
µ ave = χB0
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A few materials have large atomic dipole moments
which are so easily aligned that χ is large and they
become permanent magnets.
One dipole in the material
Each individual dipole feels
the field B0 plus the field
from all the other dipoles,
B0
as a total field Beff. Adding
the response of all the
individual dipoles gives
r
r
field inside dipole
µ ave = χBeff
r
r
r r
r
r
but Beff = B0 + B( µ ave ) = B0 + cµ ave
r
χ r
µ ave
so µ ave =
B0
B0
1 − cχ
r
When χ=1/c, r µ ave exists
r r
B( µ ave )
even when B0 → 0.
This is a permanent magnet.
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µ ave = 0
Random atomic dipoles
in a material
µ ave ≠ 0
B0
Torques partially align the
dipoles in a magnetic field.
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Concept 24.4: Magnets are attracted to the
door of the fridge by the field from the
magnetically polarized dipole moment in the
metal.
air
steel
S
µM
N
µave
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Summary
• a magnetic dipole has a potential energy of orientation in
a magnetic field
• a non-uniform magnetic field can create a force on a
magnetic dipole.
• a permanent magnet has large susceptibility, χ, so that it
“polarizes itself” once the dipoles are aligned
• magnets attract iron by polarizing a dipole moment within
the metal
Practice problems: Chapter 30, #38, 65
Next lecture: read sections 31.2, 31.1 (reverse order!)
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