P. Piot, PHYS 571 – Fall 2007e.m. Field tensor & covariant equation of motion4 potential4 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 isP. Piot, PHYS 571 – Fall 2007Invariant of the e.m. field tensor • Consider the following invariant quantities• Usually one redefine these invariants as• Which can be rewritten aswhere• Finally note the identitiesP. Piot, PHYS 571 – Fall 2007Eigenvalues of the e.m. field tensor • The eigenvalues are given by• Characteristic polynomial•With solutionsP. Piot, PHYS 571 – Fall 2007Motion in an arbitrary e.m. field I• We now attempt to solve directly the equationfollowing 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. fieldP. Piot, PHYS 571 – Fall 2007Motion 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, F2and I:• This meansP. Piot, PHYS 571 – Fall 2007• Consider. • We need to compute the coefficient of the expansionMotion in an arbitrary e.m. field IIIP. 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 basiswhere F is diagonal, let F’ be the diagonal form then• Recall thanMotion in an arbitrary e.m. field IVP. Piot, PHYS 571 – Fall 2007• The traces are then. • Now let evaluate the ti’sMotion in an arbitrary e.m. field VP. Piot, PHYS 571 – Fall 2007• The traces are then. • Substitute in the power expansion to yieldMotion in an arbitrary e.m. field VIP. Piot, PHYS 571 – Fall 2007• Remember that• Integrate forMotion in an arbitrary e.m. field VIIP. 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 obtainMotion in an arbitrary e.m. field VIIIP. Piot, PHYS 571 – Fall 2007•With• computeExB driftP. Piot, PHYS 571 – Fall 2007•With• computeExB drift IP. 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 IIP. Piot, PHYS 571 – Fall 2007ExB drift