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8.022 Lecture Notes Class 28 - 11/2/2006
Pure Dipole
Physical Dipoles
Dipole in Spherical Polar Coords
θˆ
� = µ0 m·sin
A
� = mˆ
z
4π r2 φ Dipole such that m
� = µ0 m (2 cos θr̂ + sin θθ̂)
B
4πr3
�
m
� =I
dA = IAâ
2
Boundary Conditions
� ·B
� =0
Perpendicular : B⊥,a − B⊥,b b/c �
� arm� − B
� � = µ0 (K
� × n̂)
Parallel : B
b
�a − B
� b =)
(B
�
� · d�l =
A
=
=
�
� × Ad�
� a
�
�
� · d�a
B
�LB
� → 0 , just have Babove+below , so
� ·A
�=0
perp: A⊥,a − B⊥,b = 0 b/c �
� �a − A
�� = 0
parallel: A
b
Magnetization
• substance with little
magnetic dipoles inside
� )
Diamagnetic (m
� anti aligns with B
� )
Paramagnetic ( m
� aligns with B
Ferromagnetic ←− really hard, non linear (depends on entire history
of magnet)
Force
3
• Forces on sloping sides cancel out
• Forces on other side also cancel out No net force But there is a
torque.
Torque
� =
N
x̂ · (a sin θ) · IB · b
=
IabB sin θx̂
=
(I · A)B sin θx̂
=
mB sin θx̂
�
= m
� ×B
; paramagnetism (unpaired)
Diamagnetism
Examine using classical model- electron around proton; it moves fast
enough that we can consider it constant current.
eV
I = − Te = − 2πR
|m| =
IπR2
m
� = − 12 eV R℘ˆ
� S
�
quantum stuff: L,
e −→ −e
� =m
� weak effect
N
� ×B
1
e2
v2
·
= me
4π�0 R2
R
add magnetic force :
(mp >> me )
4
1
e2
v2
· 2 + ev � B = me
4π�0 R
R
ev � B =
ev � B ≈
=⇒
me �2
R (v
(mp >> me )
− v2) =
me
R (2v
me
R (v
�
�
+ v � )(v � − v)
)(v − v)
eRB
= v � − v = Δv
2me
What ? Magnetic field speeds up an electron? No, actually B generates
a E that does the work
1
−e2 R2 �
Δm
� = − eΔvRẑ =
B
2
4me
So, charge in magnetization is opposite of magnetic field. Change of
velocity is independent of orbit direction! (works for both paired and
unpaired , but its so weak that it’s noticeable only when no paramag­
netization) Diamagnetization!
� = magnetization = m
M
� per unit vol.