Rotations in R2

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Rotations in R2
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Rotations in R2
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(1)
Definition If the components of the quantity A
transform under a rotation accoring to (1) then A
is said to be a vector
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Matrix
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Rotation
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Classical Physics

• Newtons Laws
• 1.Every body remains at rest or in constant
  rectilinear motion unless acted on by a force
• 2

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Newtons third law
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Comments

• Makes perfect sense for electrostatics and
  gravity
• e.g

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• There is an implicit assumption of an absolute
  time here, i.e absolute simulantiety
• There is a suggestion of static interaction at a
  distance

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Inertial frames
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Maxwells equations together with force
law FqEvxB Contains all of Classical
Electrodynamics
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m/s
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m/s
Exactly the speed of light
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• Static charges and charges in motion at a
  constant rate do not radiate, accelerated charges
  radiate

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Consider the circuit shown . There is an energy
sourcethat restores the energy that is radiated
or lost as heat in the resistor. If the
resistance loses are small the current in the
circuit varies sinusodially with resonance
angular frequency w( ?1/?LC ). The oscillator is
coupled through a transformer to a transmission
line which carries the current to an anteena, in
this case it is a simple dipole antenna
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• An electric charge at rest sets up a pattern of
  electric field lines. A charge in motion sets up
  a pattern of magnetic field lines in addition to
  the electric field. Once a steady condition has
  been reached (after the charge is in motion and
  the fields are established in space)there is an
  energy density associated with the electric and
  magnetic fields but the energy density remains
  constant in time. However if you wiggle the
  charge back and forth you can send a signal.

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• Static charges and charges in motion at a
  constant rate do not radiate, accelerated charges
  radiate

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• The more important fundamental laws and facts of
  physical science have all been discovered, and
  these are now so firmly established that the
  possibility of their ever being supplanted in
  consequence of new discoveries is exceedingly
  remote . . . Our future discoveries must be
  looked for in the sixth place of decimals.
• Albert A. Michelson, 1894

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Lorentz Transformation
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Lorentz Transformation
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Lorentz Transformation
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• There is no motion in the y or z directions hence

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• Now assume that observers in S and S
• Measure a flash of light moving along the
• x(x) directions
• The speed of light is constant so

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• For a general outgoing spherical pulse

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hence
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Linear Transformation equation
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Speed of light is constant
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Lorentz transformation
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Inverse Lorentz transformation
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Properties and Consequences of the Lorentz
transformation
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• Then

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If the differences are infinitesmalwe can go
over to differentials

• Then

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Length Contraction
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Length Contraction
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Simulaniety

• Suppose We have two diodes fixed at a distance L
  apart in and fixed to a single circuit so that
  both light at the same time

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Causality
Causality is perserved!!
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Time Dilation
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• Be careful to think operationally
• What we mean is time goes slower
• As measured by the observer at rest when looking
  at an event occurring in the frame
• S
• If you run very fast you will appear to live
  longer to an observer at rest but to you you
  will live exactly the same amount of time as if
  you had stayed at rest!

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Properties and Consequences of the Lorentz
transformation
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• Then

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If the differences are infinitesmalwe can go
over to differentials

• Then

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Length Contraction
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Length Contraction
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Simulaniety

• Suppose We have two diodes fixed at a distance L
  apart in and fixed to a single circuit so that
  both light at the same time

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Causality
Causality is perserved!!
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Time Dilation
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• Be careful to think operationally
• What we mean is time goes slower
• As measured by the observer at rest when looking
  at an event occurring in the frame
• S
• If you run very fast you will appear to live
  longer to an observer at rest but to you you
  will live exactly the same amount of time as if
  you had stayed at rest!