Download Divergence and circulation

Divergence and
circulation
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A vector field is generally characterized by
1) how field lines possibly diverge away
from or converge upon (point) sources
plus 2) how field lines circulate, closing
upon themselves.
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An electrostatic field diverges but has no
circulation implying the force field is
conservation. A test charge gains no net
energy in traversing a closed circuit.
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This follows from Coulomb’s Law for a
point charge, and, by superposition, for
any collection of static charges.
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We will see later nonconservative electric
fields may appear in nonstatic
circumstances.
Surface S
Curve C
True for static fields only.
Ampere’s Law
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Gauss’s Law for magnetic fields says there
are no magnetic monopoles.
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Ampere’s Law relates magnetic field
circulation around any loop to the total
current through the loop.
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In Ampere’s Law, S is an open surface with
closed curve C as as its boundary and j is
the electric current density (A/m2). I is the
total current through any such open
surface S, any because, by implicit
assumption, current is conserved and if it
flows through one such surface it flows
through any other.
Ampere’s law applied to an
infinitely long wire
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Integrate around a circle (a) enclosing the wire. The right-hand fingers follow the
curve. The thumb gives the direction of positive current in Ampere’s Law.
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In (b) the orientation of the integration is reversed and the current negative.
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In ( c), it is easy to show the integral vanishes as it must according to Ampere’s Law.
Magnetic field outside
and inside a wire
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We can use Ampere’s Law to
DERIVE the field outside a
straight wire carrying current.
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Assuming that the current
density is uniform within the
wire, we can also find the field
within the wire.
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The field vanishes at the center.
Coaxial cable
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Question: An infinitely long
coaxial cable carries equal and
opposite currents in an inner
and outer conductor as shown.
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Find the magnetic field
everywhere in space.
Coaxial cable
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Applying Ampere’s law to a
concentric circle,ignoring the
outer conductor and outer
current gives the field inside
the outer conductor as for an
infinite straight wire.
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Applying Ampere’s Law to a
circular path outside the outer
conductor, we find the
azimuthal field vanishes as our
path has vanishing net current
through it.
Toroidal field
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The toroidal shape coil
contains a magnetic field
within itself and is a
common “inductive”
element in electronic
circuits.
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The field strength is
proportional to the number
of turns N.
Magnetic field
inside a solenoid
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A long solenoid has a strong field inside. The
flux is returned over a large volume so the field
strength is small outside.
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Applying Ampere’s Law assuming uniform field
inside and vanishing field outside implies the
field strength is proportional to the number of
turns per unit length N/L.
Ferromagnetism
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The magnetic field of a
“permanent magnet” is
associated with the spin of
charged electrons in atoms.
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The electrons behave like a
collection of tiny permanent
magnets. When these tiny
magnets aligned, not randomly
oriented, a net macroscopic
magnetization and magnetic
field is observed.
Classical orbital
magnetic moment
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A classical electron in a circular
orbit has a time averaged current
proportional to speed and a
magnetic moment m=IA
proportional to its angular
momentum L=me rxv.
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m = (-e/2me)L
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In quantum mechanics, it is found
that orbital angular momentum
and magnetism is quantized, Lz=
n h/(2 pi ), where h is Planck’s
constant. (hbar is h over 2 pi.)
Electron spin
magnetic moment
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The electron is found to behave
as if it had an intrinsic
quantized spin angular
momentum and an associated
magnetic moment with
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mz = (+ or -)(-e hbar)/(2me)
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The magnitude is 9.27e-24 J/T.
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The magnetic energy U~mzB in
a 1 T field is 9.27e-24 J, much
smaller than 1 eV=1.602e-19 J.
Atomic magnetic
moments
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The proton magnetic moment is
about equal to the Bohr magneton
with the proton mass substituted
for the electron mass and ~600
times smaller. Nuclear magnetic
moments are often negligible
compared to electronic moments.
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The H atom magnetic moment is
approximately the electron spin
magnetic moment.
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In He, the two electron spins and
magnetic moments cancel.
Ferromagnets
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In some materials, below a critical
(Curie) temperature, unpaired
electrons spontaneously align
within small domains.
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Application of an external field
will grow domains aligned with
the field and shrink others
resulting in net bulk averaged
alignment and magnetism.
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When the external field is
removed, a remnant magnetization
remains. The material is
“permanently” magnetized.
Curie
temperature
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Spontaneously alignment is a
“solid phase” for spin.
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Above a material dependent
Curie temperature, the spins
are randomly aligned.
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Other “spin phases” including
antiferromagnetism are
possible.
Paramagnetism
and diamagnetism
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Paramagnetic substances have small but positive induced magnetism
that results from the presence of atoms that have permanent (electron
spin) magnetic moments which, when placed in an external magnetic
field, tend to line up with the field. The alignment process competes
with thermal motion which randomizes the moment orientations.
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A diamagnetic substance is one in which electron spins cancel in
pairs as explained by quantum mechanics dynamics. When an
external magnetic field is applied to a diamagnetic substance, a weak
orbital magnetic moment is induced in the direction opposite the
applied field.
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Diamagnetic substances are weakly repelled by a magnet.
Meissner effect
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A superconductor is a material which,
below a critical temperature, ceases to
resist current flow.
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When a superconductor is placed in a
magnetic field, currents appear so as to
EXCLUDE magnetic field from the
interior, rather like, when a
superconductor is placed in an electric
field, charges appear so as to exclude
electric field from the interior.
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The direction of the currents is such that
the superconductor is repelled in a
nonuniform field.
Force in a
uniform field
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Recall that the magnetic forces
on elements of a circuit in a
uniform field result in no net
force but, a net torque that
depends on the orientation.
Force in
nonuniform field
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In a nonuniform magnetic field, a net force as well as a net
torque acts on a circuit.
Loudspeaker
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An acoustic speaker operates by energizing an electromagnet with
an electrical signal. The magnetic force on a permanent magnet
attached to a diaphragm causes the diaphragm to move.
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The inverse (motion of diaphragm causing electric current) is
explained by Faraday’s Law of Induction.
Motor
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Can you explain this to a newbie? What about a generator? Can
you explain that to someone? Or on an examination?