Magnetostatics

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Magnetostatics
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Current in opposite directions repel Current in
the same direct attract
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Magnetic Forces do no work
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Magnetic Forces do no work

• Magnetic forces may alter the direction a
  particle moves but can not speed it up or slow it
  down

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Particle in Magnetic field
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• Assume first v perpendicular to B then

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• So force is always directed along the ve Y axis
• For a uniform field B force is constant
• Magnetic force does no work so we cant change
  v
• Particle moves in a circle of radius, R

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Suppose
B
V
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Earths Magnetic field
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The Zeeman Effect(see http//csep10.phys.utk.edu/a
str162/lect/sun/magnetic.html) The magnetic field
of the Sun can be probed in a rather precise and
direct manner because in the presence of a
magnetic field the energy levels of atoms (and
ions and molecules) are split into more than one
level. This causes spectral transition lines to
also be split into more than one line, with the
amount of splitting proportional to the strength
of the magnetic field. This is called the Zeeman
Effect, and the corresponding increase in the
number of spectral lines is called Zeeman
splitting. Thus, we can infer the presence of
magnetic fields if we observe Zeeman splitting in
the spectrum, and we can measure the strength of
the field by measuring quantitatively the amount
of Zeeman splitting.
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• Sunspots and Magnetic Fields
• Measurement of the light from sunspots (obtained
  by masking off the light from parts of the Sun
  not in the sunspot) indicate significant Zeeman
  splitting of the spectral lines. Thus, sunspots
  are associated with strong magnetic fields.
  Furthermore, it is observed that
• When sunspots come in pairs, one tends to have a
  magnetic field polarity that is opposite that of
  the other (that is, one behaves magnetically like
  the north pole of a bar magnet and the other
  behaves magnetically like the south pole of a bar
  magnet).
• During a given sunspot cycle, the leading
  sunspots in groups in the northern hemisphere of
  the Sun all tend to have the same polarity, while
  the same is true of sunspots in the southern
  hemisphere, except that the common polarity is
  reversed from that of sunspots in the northern
  hemisphere.

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3. During the next sunspot cycle, the
regularities noted in the previous point reverse
themselves the polarity of the leading spots in
each hemisphere is opposite from what it was in
the previous cycle.
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• The Solar Magnetic Field
• The a image shows the distribution of magnetic
  field on the solar surface from the Michelson
  Doppler Imager experiment on SOHO (January 27,
  1998). Black denotes a negative polarity
  (magnetic field pointing into the Sun) while
  white denotes a positive polarity (magnetic field
  pointing out of the Sun). Large concentrations of
  both polarities are found near active regions and
  sunspots.

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• The Solar magnetic field has a 22 year cycle,
  exactly twice that of the sunspot cycle, because
  the polarity of the field returns to its original
  value every two sunspot cycles. Thus, the
  fundamental period governing solar activity is
  actually the 22 year magnetic cycle, and the
  sunspot cycle (which is exactly half that) is
  just a special manifestation of the magnetic
  cycle. As we shall see, the magnetic field plays
  an important role in most aspects of the active
  Sun (sunspots, prominences, flares, the solar
  wind, and the nature of the corona), so the 22
  year magnetic cycle is central to the periodicity
  of the activite sun

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• Why are Sunspots Dark
• Well, because they are cooler than the rest of
  the surface. But that is only a partial
  explanation. Why are they cooler? The answer is
  the strong magnetic fields associated with the
  sunspots. In the region below the photosphere,
  convective cells are largely responsible for
  vertical motion of large packets of gas and that
  this bubbling activity carries heat from the
  interior to the solar surface.Magnetic fields
  exert forces on charged particles, and because
  this solar material is highly ionized, the
  magnetic fields influence the convective motion.
• Detailed considerations indicate that the
  magnetic forces hinder the convection of heat to
  the surface by making it harder for the hot gases
  to rise. Thus, the region in sunspots having
  strong magnetic fields tends to be cooler than
  the surrounding region and thus appears darker
  than the surrounding regions at higher
  temperature.

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.
March 22 2007 This image taken by Japan's Hinode
spacecraft studying the Sun reveals the
structure of the solar magnetic field rising from
a sunspot
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• Van Allen Radiation Belts
• A fundamental property of magnetic fields is
  that they exert forces on moving electrical
  charges. Thus, a magnetic field can trap charged
  particles such as electrons and protons as they
  are forced to execute a spiraling motion back and
  forth along the field lines.
• As illustrated in the adjacent figure, the
  charged particles are reflected at "mirror
  points" where the field lines come close together
  and the spirals tighten. One of the first fruits
  of early space exploration was the discovery in
  the late 1950s that the Earth is surrounded by
  two regions of particularly high concentration of
  charged particles called the Van Allen radiation
  belts.

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Currents

• The current in a wire is the charge per unit time
  passing through a given point
• Convention current flows from ve to -ve
• In reality it is usually electrons which do the
• moving in the opposite direction to the current
• (just a silly convention)

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units

• Current is measured in Amperes(A)
• 1A1C/s(Coulombs per second)

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• A neutral wire, of course, contains as many
  stationary positive charges as moving ve
  charges.
• The former do not contribute to the current
• In the situation where both types move(e.g.
  plasma)

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The magnetic force on a current carrying conductor
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Example

• A rectangular loop of wire supporting a mass m
  hangs vertically with one end in a uniform
  magnetic field ,B.For what current in the loop ,
  would the magnetic force upwards exactly cancel
  the gravitational force down

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• The current must circulate clockwise
• In order for to point upwards
• The force terms in the two vertical segments
  cancel
• The force on the upper horizontal segment is
  acting up and its magnitude is

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Who does the work?

• We are lifting a weight against gravity so work
  must be done but magnetic forces do no work so
  what is working?

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• When the loop starts to rise the charges in the
  wire are no longer moving horizontally
• Their velocity now acquires an upward
  component,u,the speed of the loop in addition to
  the horizontal component,w,associated with the
  current(I?w)

quB
The magntic force which is always perpendicular
to the velocity, no longer points straight up, It
does have a vertical component(qwB) The net
vertical force on all the charge (?a) is ?awB
qwB
Fm
v
u
q
w
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• But it now has a horizontal component which
  opposes the flow of current
• The battery which is responsible for maintaing
  the current must work against this force,i.e.
• It must counter the force
• Fhorizontal?auB

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• In a time dt the charges move a horizontal
  distance wdt
• hence work done by battery is

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dt
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Steady Currents
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• Note a moving point charge can not constitute a
  steady current
• When a steady current flows in a wire, its
  magnitude,I,must be the same all along the wire,
  otherwise charge would be pilling up somewhere
  and ? would not be a constant in time

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• Stationary charges electrostatics
• Steady current magnetostatics

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Biot-Savart Law
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Biot-Savart Law

• Law only applies to steady currents
• It does not apply to moving point charges

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• Find the magnetic field a distances from a long
  straight wire carrying a steady current

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(1)
(2)
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where J is constained to be within the volume,W
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Ampères Law
Ampères Law
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http//www.math.umn.edu/nykamp/m2374/readings/sto
kesidea/
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In applying Amperes law, we integrate around a
closed loop The surface bounded by the loop is
not unique
I2
I1
I3
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The surface bounded by the loop has been
stretched upwards, I2 Now passes through the new
surface
I3
I2
I1
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The magnetic field B depends on I2 But B.dl
changes sign as we go around loop and the ve and
ve contributions cancel
I3
I2
I1
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A right hand rule is used to assign signs to
currents with the fingers of your right hand in
the direction in whivch the lop is traveled then
your thumb defines the ve direction
I3
I2
I1
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I4 penitrates the new surface twice, once moving
down and once moving up So contributes nothing
I3
I2
I1
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Solenoid Field from Ampère's Law

• A solenoid is a long wire wound in a closed pack
  helix,carrying a current I.The solenoid is the
  vector sum of the fields set up by all the turns.

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For an ideal solenoid we assume B zero for all
points external to solenoid
B perpendicular to path
d
c
b
a
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Magnetic Field of Toroid

• Finding the magnetic field inside a toroid is a
  good example of the power of Ampere's law. The
  current enclosed by the dashed line is just the
  number of loops times the current in each loop.
  Amperes law then gives the magnetic field by

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Toroid Detail

• All of the loops of wire which make up a toroid
  contribute magnetic field in the same direction.
  The sense of the magnetic field is that given by
  the right hand rule

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The Tokamak

• This is magnetic confinement device is called
  the tokamak, a word formed from the Russian words
  "TOroidalnaya KAmera ee MAgnitnaya Katushka," or
  "Toroidal Chamber and Magnetic Coil". Tokamaks
  were originally designed and used in Russia. In
  this design, the chamber is toroidal, or
  doughnut-shaped, thus having no open ends. The
  magnetic field is generated through the current
  running in the coils that are wrapped around the
  reactor. The field is stronger towards the
  center, causing the plasma to tend towards the
  outer wall. However, another magnetic field
  generated by a current going through the plasma
  itself combines with the coils' magnetic field to
  create magnetic lines that spiral around the
  torus. This spiralling counteracts the drifting
  effect on the plasma because of the strong inner
  field, and effectively traps the plasma.

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JET Pulse 64159 - View of a plasma from the KL1
CCD video camera (from behind a quartz window).
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The Hall Effect
w
Consider a flat strip of material,
width,w Carrying a current I. By convention the
current flows from ve to ve. Suppose the
current is carried by carriers, charge,q.A
uniform magnetic field,B is established
perpendicular to the plane of the strip.
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The Hall Effect
w
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Magnetic Vector Potentials
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Question
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• Hence

of Poissons
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• We have shown that there exists a solution A s.t.

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In cartesian coordinates
3 sets of Poissons equations
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• We may assume J goes to zero at infinity
• Then we can solve

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Multipole expansion of the vector potential

• Idea we are looking for an approximate formula
  for a localized current distribution
• We will write the potential in powers of 1/r
• Keep the highest non vanishing contribution

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2
2
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quadrapole
monopole
dipole
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Vector Potential a large distance from a closed
current loop
Monopole dipole
quadrapole
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3
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Electric dipole
Note
-1/2
dk
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Boundary Conditions for Magnetostatics

• Just as the electric field suffers a
  discontinuity at a surface charge
• So the magnetic field suffers a discontinuity
• at a surface current

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Recall
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Now our surface is 2 dimensional so we need 2
directions to describe it, just as above we had a
sheet in the x,y plane The i and j are tangential
to the surface. We will consider separately
component of B parallel to surface but
perpendicular to current and component of B
parallel to surface and parallel to current
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r
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