4-current density . Lets suppose J is the 4-current density, let - - PowerPoint PPT Presentation
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4-current density . Lets suppose J is the 4-current density, let - - PowerPoint PPT Presentation
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4-current density . Lets suppose J is the 4-current density, let J =(c , J ) So Using the property Smooth function we can rewrite J as P. Piot, PHYS 571 Fall 2007 Charge continuity equation
SLIDE 1
P. Piot, PHYS 571 – Fall 2007
4-current density
….
Let’s suppose Jα is the 4-current density, let Jα=(cρ, J)
So
Using the property
we can rewrite Jα as
Smooth function
SLIDE 2
P. Piot, PHYS 571 – Fall 2007
Charge continuity equation
Consider the divergence of J
So charge continuity can be written as
SLIDE 3
P. Piot, PHYS 571 – Fall 2007
4-gradient
In previous slide we introduce the 4-gradient operator
This operator transforms as
Note that
Can define the covariant form
The self-contraction yields the d’Alembertian:
SLIDE 4
P. Piot, PHYS 571 – Fall 2007
4-potential
Define
This is precisely the equation we solved to get the field of a moving
charge three lessons ago…
In SI unit
In SI unit: φ → cφ
SLIDE 5
P. Piot, PHYS 571 – Fall 2007
Covariance of Maxwell equations
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 covariant form is
4 potential 4 potential
SLIDE 6
P. Piot, PHYS 571 – Fall 2007
Inhomogeneous Maxwell’s eqns
Consider
Similarly
The inhomogeneous Maxwell’s equations can be caster under
the equation
SLIDE 7
P. Piot, PHYS 571 – Fall 2007
Homogeneous Maxwell’s eqns I
Consider the Levi-Civita tensor (rank 4)
And consider
SLIDE 8
P. Piot, PHYS 571 – Fall 2007
Homogeneous Maxwell’s eqns II
Consider the Levi-Civita tensor (rank 4)
The homogeneous Maxwell’s equations can be caster under
the equation
With the Dual field tensor defined as
SLIDE 9
P. Piot, PHYS 571 – Fall 2007
Covariant form of Maxwell’s equation
To introduce H and D field introduce the rank 2 tensor:
Then Maxwell’s equation writes
F is a rank 2 tensor that conforms to Lorentz transformation so the
(E,B) field can be computed in an other frame by
r, in matrix notation
SLIDE 10
P. Piot, PHYS 571 – Fall 2007
Covariant form of Maxwell’s equation
Example consider the Lorentz boost along z- axis
Then from
We get the same matrix as [JDJ 11.148]
SLIDE 11
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
SLIDE 12
P. Piot, PHYS 571 – Fall 2007
Eigenvalues of the e.m. field tensor
The eigenvalues are given by
Characteristic polynomial
With solutions
SLIDE 13
P. Piot, PHYS 571 – Fall 2007
Equation of motion
The equation of motion (EOM) can be written
where
This is equivalent to defining a 4-force:
We need to solve EOM once we have specified the external e.m.
