[3] O.D. Jefimenko, Electricity and Magnetism. An introduction to the theory of electric and magnetic fields, Chap. 15, (Appleton-Century-Crofts, New York City, ...
Rapid computation of EM fields and conserved quantities using alternatives to Maxwell’s equations B. Thid´e1,2 , F. Tamburini2,3 1
2
Swedish Institute of Space Physics, Uppsala, Sweden Department of Physics and Astronomy, University of Padova, Padova, Italy 3 Twist-Off S.R.L., Padova, Italy
Maxwell’s axiomatic equations, in the microscopic version introduced by Lorentz, are the conventional starting point when solving for the EM fields and observables, including the conserved quantities energy, linear momentum and angular momentum of an electro-mechanical system such as a plasma. Inspired by the work by Stratton [1], Panofsky-Phillips [2], and Jefimenko [3], we have derived a new set of axiomatic equations which are equivalent to the Maxwell-Lorentz equations, are symmetric, and seem to be better adapted to numerical evaluation. These new equations can be written Z
E(t, x) =
V0
1 B(t, x) = c
d3x0 Θ (t 0 , x0 , x)
Z V0
d3x0
x − x0 × Θ (t 0 , x0 , x) |x − x0 |
(1a) (1b)
where t 0 = t − |x − x0 | /c is the retarded time relating the field point x and the source point x0 and Θ (t 0 , x0 , x) =
x − x0 4πε0 |x − x0 |2 |x − x0 | 1 x − x0 x − x0 0 0 + j(t , x ) · |x − x0 | |x − x0 | 4πε0 c |x − x0 |2 x − x0 1 x − x0 0 0 × + j(t , x ) × |x − x0 | |x − x0 | 4πε0 c |x − x0 |2 0 0 1 ˙j(t 0 , x0 ) × x − x × x − x + |x − x0 | |x − x0 | 4πε0 c2 |x − x0 | ρ(t 0 , x0 )
Tests have shown that a solver based on Eqs. 1 is at least five times faster than solvers used in commercial-grade computational electromagnetic packages based on Maxwell-Lorentz equations. Comparisons will be presented for the case of angular momentum (vorticity, helicity) and other physical observables.
References [1] J. A. Stratton, Electromagnetic Theory, Chap. VIII, (McGraw-Hill Book Co., New York City, NY, USA, 1941) [2] W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism, 2nd ed., Chap. 14, (Addison-Wesley Publishing Company, Reading, MA, USA, 1962) [3] O. D. Jefimenko, Electricity and Magnetism. An introduction to the theory of electric and magnetic fields, Chap. 15, (Appleton-Century-Crofts, New York City, NY, USA, 1966)
