# NIU PHYS 671 - LECTURE (15 pages)

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## LECTURE

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- Pages:
- 15
- School:
- Northern Illinois University
- Course:
- Phys 671 - Electromagnetic Theory II

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e m Field tensor covariant equation of motion Define the tensor of dimension 2 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 motion is 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 P Piot PHYS 571 Fall 2007 Eigenvalues of the e m field tensor The eigenvalues are given by Characteristic polynomial With solutions P Piot PHYS 571 Fall 2007 Motion in an arbitrary e m field I We now attempt to solve directly the equation We consider a time independent e m field following the treatment by Munos Let The equation of motion reduces to Where the matrix exponential is defined as 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 P Piot PHYS 571 Fall 2007 Motion in an arbitrary e m field III Consider We need to compute the coefficient of the expansion P Piot PHYS 571 Fall 2007 Motion in an arbitrary e m field IV 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 P Piot PHYS 571 Fall 2007 Motion in an arbitrary e m field V The traces are then Now let evaluate the ti s P Piot PHYS 571 Fall 2007 Motion in an arbitrary e m field VI The traces are then Substitute in the power expansion to yield P Piot PHYS 571 Fall 2007 Motion in an arbitrary e m field VII Remember that Integrate for P Piot PHYS 571 Fall 2007 Motion in an arbitrary e m field VIII Let s consider the special case Then So we just take the limit the previous slide which means in the equation motion derived in So we obtain P Piot PHYS 571 Fall 2007 ExB drift With compute P Piot PHYS 571 Fall 2007 ExB drift I With compute P Piot PHYS 571 Fall 2007 ExB drift II The projected equation of motions This is the so called ExB drift and the drift velocity of the particle is P Piot PHYS 571 Fall 2007 ExB drift III P Piot PHYS 571 Fall 2007

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