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Physics 202, Lecture 14
Today s Topics
  Sources of the Magnetic Field (Ch 27)
  Review: The Ampere s Law
  Applications And Exercises of ampere s Law
  Straight line, Solenoid, Toroid
  Magnetism in Matter
 Ampere
Review: Ampere s Law
! !
s Law:
"! B• dl = µ0 I
I
any
closed path
  It applies to any closed path
  It applies to any static B field
  It is practically useful
in symmetric cases
dl
  Ampere s Law can be derived from Biot-Savart Law
  Key point: Derivation relies on the fact that the
current has no divergence(is a continuous flow
does not increase or decrease at any point). This
works for current loops or infinite currents.
Solenoid
  The B field inside an ideal solenoid:
  Ideal: Infinitely long and tightly wound
"!
! !
B• dl = µ 0 I
any
closed path
Bl = µ 0 NI
B = µ 0 nI
n=N/L
B field independent of where
you are inside the solenoid ideal solenoid
segment 3
at ∞
Solenoid Ideal vs. Real
  The B field inside an ideal solenoid is:
B = µ0nI
n=N/L
ideal solenoid
Compare Solenoid and Bar Magnet
N
S
B
Loose Solenoid
Bar Magnet
Tight Solenoid
I
Toroid
 The B field inside a toroid
"!
! !
B• dl = µ 0 I
any
closed path
B2! r = µ 0 NI
µ0 NI
B=
2πr
b<r<c: B=0 outside the Torus
because the Ampere’s loop does
not enclose any net current
Review: Electric Dipole Moments
 Electric dipole moment p.
+
+q

τ = p×E
U = -p • E
-q
-
∑F = 0
p= qd
Dielectric material
contains electric dipoles In an external field E0 , the dipoles line up
at atomic level
Eind is always opposite to E0
 E=E0/κ<E0, C=κC0
(dielectric constant κ>1)
More on Magnetic Dipole Moments
 Magnetic dipole moment µ.
Note: B produced (at the center) is always in same direction as µ
definition of magnetic moment
Macroscopic
µ=IA
Microscopic
From orbiting electrons
(depends on their angular
momentum)
and quantum mechanical
spin (special type of
angular momentum)
I=q/T=qv/2πr
µ=IA =1/2 qvr = (q/2m)L
2
µ 0 IR
Bz=! =
3
2z
µ0µ
B=
3
2z
B
Magnetism in Matter
µ
∑F = 0


τ = µ ×B

U = -µ • B
µ in B Field
External B and
dipole B in the
same direction!
 Total magnetic dipole moment µ and
the magnetic field
  If all the individual dipoles line up
the field can be very strong. Induced
by applying an external field
  Configuration like a solenoid with
constant field
  Depends on details of the material.
  Define the magnetization M where
Bind=µ0M
Magnetism in Matter
 Induced field Bind in response to an external B0: Bind = χB0
èthe net field inside: B = B0+Bind =(1+χ) B0 = (µm/µ0)B0 = KmB0
µm: magnetic permeability , χ: magnetic susceptibility,
K: relative permeability
Classification of Magnetic Matter
Type
Direction of
Binduced
Strength of
Binduced
χ=µm/µ0-1
Contributing
Elements
Ferromagnetic
(e.g. Fe, Co, Ni…)
Same as B0
Strong
>>0
(~103)
Domain of
Magnetic Dipole
Paramagnetic
(e.g. Al, Ca,…)
Same as B0
Weak
>0
(~10-5)
µatoms
Diamagnetic
(e.g. Cu, Au,…)
Opposite
Weak
<0
(~-10-5)
Magnetic
Quadruple
Superconductor
Opposite
-1
Quantum eff.
=-B0
Permanent Magnetic Moments (domains)
Inside Ferromagnetic Material
polarized domains
No external B field,
permanent mag. moments
exist, but oriented randomly
à no induced B field
B0 applied , permanent
magnetic moments line up
in the direction of B0
à strong induced B field
Note: Inductance with a ferromagnetic core: B= µm/µ0B0 >>B0
Meissner Effect
 Certain superconductors (type I) exhibit perfect
diamagnetism in superconducting state:
no magnetic field allowed inside (Meissner Effect)