e.m. Field tensor & covariant equation of motion Define the - - PowerPoint PPT Presentation

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e.m. Field tensor & covariant equation of motion Define the - - PowerPoint PPT Presentation

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e.m. Field tensor & covariant equation of motion Define the tensor of dimension 2 4 potential 4 potential F , is the e.m. field tensor. It is easily found to be In SI units, F is obtained by E E/c The equation of

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SLIDE 1
  • P. Piot, PHYS 571 – Fall 2007

e.m. Field tensor & covariant equation of motion

4 potential 4 potential

  • Define the tensor of dimension 2
  • F, is the e.m. field tensor. It is easily found to be
  • In SI units, F is obtained by E → E/c
  • The equation of

motion is

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SLIDE 2
  • P. Piot, PHYS 571 – Fall 2007

Invariant of the e.m. field tensor

  • Consider the following invariant quantities
  • Usually one redefine these invariants as
  • Which can be rewritten as

where

  • Finally note the identities
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SLIDE 3
  • P. Piot, PHYS 571 – Fall 2007

Eigenvalues of the e.m. field tensor

  • The eigenvalues are given by
  • Characteristic polynomial
  • With solutions
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SLIDE 4
  • P. Piot, PHYS 571 – Fall 2007

Motion in an arbitrary e.m. field I

  • We now attempt to solve directly the equation

following the treatment by Munos. Let

  • The equation of motion reduces to
  • Where the matrix exponential is defined as

We consider a time independent e.m. field

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SLIDE 5
  • P. Piot, PHYS 571 – Fall 2007

Motion in an arbitrary e.m. field II

  • The main work is now to compute the matrix exponential.
  • To compute the power series of F one needs to recall the identities
  • Because of this one can show that any power of F can be written as

linear combination of F, F, F2 and I:

  • This means
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SLIDE 6
  • P. Piot, PHYS 571 – Fall 2007
  • Consider

.

  • We need to compute the coefficient of the expansion

Motion in an arbitrary e.m. field III

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SLIDE 7
  • P. Piot, PHYS 571 – Fall 2007
  • The solution for the coefficient is

.

  • Now let evaluate the ti’s
  • The Trace is invariant upon change of basis. So consider a basis

where F is diagonal, let F’ be the diagonal form then

  • Recall than

Motion in an arbitrary e.m. field IV

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SLIDE 8
  • P. Piot, PHYS 571 – Fall 2007
  • The traces are then

.

  • Now let evaluate the ti’s

Motion in an arbitrary e.m. field V

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SLIDE 9
  • P. Piot, PHYS 571 – Fall 2007
  • The traces are then

.

  • Substitute in the power expansion to yield

Motion in an arbitrary e.m. field VI

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SLIDE 10
  • P. Piot, PHYS 571 – Fall 2007
  • Remember that
  • Integrate for

Motion in an arbitrary e.m. field VII

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SLIDE 11
  • P. Piot, PHYS 571 – Fall 2007
  • Let’s consider the special case
  • Then
  • So we just take the limit in the equation motion derived in

the previous slide which means:

  • So we obtain

Motion in an arbitrary e.m. field VIII

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SLIDE 12
  • P. Piot, PHYS 571 – Fall 2007
  • With
  • compute

ExB drift

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SLIDE 13
  • P. Piot, PHYS 571 – Fall 2007
  • With
  • compute

ExB drift I

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SLIDE 14
  • P. Piot, PHYS 571 – Fall 2007
  • The “projected” equation of motions
  • …..
  • This is the so-called ExB drift and the drift velocity of the particle is

ExB drift II

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SLIDE 15
  • P. Piot, PHYS 571 – Fall 2007

ExB drift III