Magnetostatics

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Magnetostatics
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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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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

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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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