Magnetism

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Magnetism

• Magnetic Force

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Magnetic Force Outline

• Lorentz Force
• Charged particles in a crossed field
• Hall Effect
• Circulating charged particles
• Motors
• Bio-Savart Law

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Class Objectives

• Define the Lorentz Force equation.
• Show it can be used to find the magnitude and
  direction of the force.
• Quickly review field lines.
• Define cross fields.
• Hall effect produced by a crossed field.
• Derive the equation for the Hall voltage.

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Magnetic Force

• The magnetic field is defined from the Lorentz
  Force Law,

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Magnetic Force

• The magnetic field is defined from the Lorentz
  Force Law,
• Specifically, for a particle with charge q moving
  through a field B with a velocity v,
• That is q times the cross product of v and B.

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Magnetic Force

• The cross product may be rewritten so that,
• The angle is measured from the direction of
  the velocity to the magnetic field .
• NB the smallest angle between the vectors!

v x B
B
v
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Magnetic Force
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Magnetic Force

• The diagrams show the direction of the force
  acting on a positive charge.
• The force acting on a negative charge is in the
  opposite direction.

B
F
-
v
B

F
v
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Magnetic Force

• The direction of the force F acting on a charged
  particle moving with velocity v through a
  magnetic field B is always perpendicular to v and
  B.

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Magnetic Force

• The SI unit for B is the tesla (T) newton per
  coulomb-meter per second and follows from the
  before mentioned equation .
• 1 tesla 1 N/(Cm/s)

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Magnetic Field Lines

• Review

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Magnetic Field Lines

• Magnetic field lines are used to represent the
  magnetic field, similar to electric field lines
  to represent the electric field.
• The magnetic field for various magnets are shown
  on the next slide.

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Magnetic Field Lines

• Crossed Fields

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Crossed Fields

• Both an electric field E and a magnetic field B
  can act on a charged particle. When they act
  perpendicular to each other they are said to be
  crossed fields.

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Crossed Fields

• Examples of crossed fields are cathode ray tube,
  velocity selector, mass spectrometer.

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Crossed Fields

• Hall Effect

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Hall Effect

• An interesting property of a conductor in a
  crossed field is the Hall effect.

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Hall Effect

• An interesting property of a conductor in a
  crossed field is the Hall effect.
• Consider a conductor of width d carrying a
  current i in a magnetic field B as shown.

x
x
x
x
Dimensions Cross sectional area A Length x
d
x
x
x
x
i
i
x
x
x
x
x
x
x
x
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Hall Effect

• Electrons drift with a drift velocity vd as
  shown.
• When the magnetic field is turned on the
  electrons are deflected upwards.

x
x
x
x
d
x
x
x
x
vd
-
i
i
x
x
x
x
x
x
x
x
FB
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Hall Effect

• As time goes on electrons build up making on side
  ve and the other ve.

x
x
x
x
Low
- - - - -
d
x
x
x
x
vd
-
i
i
x
x
x
x

High
x
x
x
x
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Hall Effect

• As time goes on electrons build up making on side
  ve and the other ve.
• This creates an electric field from ve to
  ve.

x
x
x
x
Low
- - - - -
x
x
x
x
vd
-
i
i
x
x
x
x

High
x
x
x
x
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Hall Effect

• The electric field pushed the electrons
  downwards.
• The continues until equilibrium where the
  electric force just cancels the magnetic force.

x
x
x
x
Low
- - - - -
x
x
x
x
vd
-
i
i
x
x
x
x

High
x
x
x
x
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Hall Effect

• At this point the electrons move along the
  conductor with no further collection at the top
  of the conductor and increase in E.

x
x
x
x
Low
- - - - -
x
x
x
x
vd
-
i
i
x
x
x
x

High
x
x
x
x
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Hall Effect

• The hall potential V is given by, VEd

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Hall Effect

• When in balance,

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Hall Effect

• When in balance,
• Recall,

dx
A
A wire
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Hall Effect

• Substituting for E, vd into we
  get,

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A circulating charged particle
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Magnetic Force

• A charged particle moving in a plane
  perpendicular to a magnetic field will move in a
  circular orbit.
• The magnetic force acts as a centripetal force.
• Its direction is given by the right hand rule.

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Magnetic Force
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Magnetic Force

• Recall for a charged particle moving in a circle
  of radius R,
• As so we can show that,

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• Magnetic Force on a current carrying wire

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Magnetic Force

• Consider a wire of length L, in a magnetic field,
  through which a current I passes.

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Magnetic Force

• Consider a wire of length L, in a magnetic field,
  through which a current I passes.
• The force acting on an element of the wire dl is
  given by,

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Magnetic Force

• Thus we can write the force acting on the wire,

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Magnetic Force

• Thus we can write the force acting on the wire,
• In general,

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Magnetic Force

• The force on a wire can be extended to that on a
  current loop.

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Magnetic Force

• The force on a wire can be extended to that on a
  current loop.
• An example of which is a motor.

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Magnetic Force

• The force on a wire can be extended to that on a
  current loop.
• An example of which is a motor.
• The diagram on the next slide shows a simple
  motor made up of a rectangular loop of sides a
  and b carrying a current I.

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Magnetic Force
side1
side2
b
side4
side3
a
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Magnetic Force

• The loop is oriented so that S1 and S3
  perpendicular to the magnetic field and S2 and S4
  are not.
• The vector n is defined so that its
  perpendicular to the loops plane.

?
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Magnetic Force

• The net force acting on the loop is the sum of
  the forces on each side.
• Clearly F2 and F4 cancel.
• However F1 and F3 act together to produce a
  torque.

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• The torque acts to rotate the loop so that n
  lines up with B.
• The torque to each is given by Fx d. ie.
• The net torque,
• If there are N loops,

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Interlude

• Next.
• The Biot-Savart Law

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Biot-Savart Law
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Objective

• Investigate the magnetic field due to a current
  carrying conductor.
• Define the Biot-Savart Law
• Use the law of Biot-Savart to find the magnetic
  field due to a wire.

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Biot-Savart Law

• So far we have only considered a wire in an
  external field B. Using Biot-Savart law we find
  the field at a point due to the wire.

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Biot-Savart Law

• We will illustrate the Biot-Savart Law.

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Biot-Savart Law

• Biot-Savart law

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Biot-Savart Law

• Where is the permeability of free space.
• And is the vector from dl to the point P.

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Biot-Savart Law

• Example Find B at a point P from a long straight
  wire.

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Biot-Savart Law

• Sol

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Biot-Savart Law

• We rewrite the equation in terms of the angle the
  line extrapolated from makes with x-axis at
  the point P.
• Why?
• Because its more useful.

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Biot-Savart Law

• Sol
• From the diagram,
• And hence

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Biot-Savart Law

• Sol
• From the diagram,
• And hence

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Biot-Savart Law

• Hence,
• As well,
• Therefore,

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Biot-Savart Law

• For the case where B is due to a length AB,

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Biot-Savart Law

• For the case where B is due to a length AB,
• If AB is taken to infinity,