better version of Maxwell to Helmholtz
- To: mathgroup at smc.vnet.net
- Subject: [mg42106] better version of Maxwell to Helmholtz
- From: CAP F <Ferdinand.Cap at eunet.at>
- Date: Thu, 19 Jun 2003 03:59:27 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Paul,
Here is code 55, Maxwell to Helmholtz in a
better format. Ferdinand
(* c55 From Maxwell's to vector Helmholtz equations,
see page 179 of the book. k=\[Omega]/c, c=1/Sqrt[eps],
eps=\[CurlyEpsilon] *)
<<Calculus`VectorAnalysis`
$Packages
Clear[H,El,t1,t2,t3,t4,t5,t6,t7,t8];
H[x_,y_,z_,t_]:={Hx[x,y,z]*Exp[I*k*t/Sqrt[eps]],
Hy[x,y,z]*Exp[I*k*t/Sqrt[eps]],Hz[x,y,z]*Exp[I*k*t/Sqrt[eps]]};
El[x_,y_,z_,t_]:={Ex[x,y,z]*Exp[I*k*t/Sqrt[eps]],
Ey[x,y,z]*Exp[I*k*t/Sqrt[eps]],Ez[x,y,z]*Exp[I*k*t/Sqrt[eps]]};
t1=Curl[El[x,y,z,t],Cartesian[x,y,z]];
t2=-D[H[x,y,z,t],t];
(* The induction equation is now: t1=t2 *)t1\[Equal]t2;
t3=Curl[H[x,y,z,t],Cartesian[x,y,z]];
t4=eps*D[El[x,y,z,t],t];
(* t3=t4 is now the other Maxwell equation *)t3\[Equal]t4;
t5=Curl[t1,Cartesian[x,y,z]]; (* Curl on curl El *)
t6=Curl[t2,Cartesian[x,y,z]];(* Curl on D H,t *)
t7=D[t2,t];(* Time derivative of Curl H *)
t8=D[t4,t];(* Second time derivative of El *)
Simplify[(t5-t8)]/Exp[I*k*t/Sqrt[eps]];(*
This gives the vector Helmholtz equation for the electric field *)
(* Now the vector equation for the magnetic field *)
t9=D[t1,t];
t10=D[t2,t];
t11=Curl[t3,Cartesian[x,y,z]];
t12= Curl[t4, Cartesian[x,y,z]];
Simplify[(t11- eps t10 )/Exp[I*k*t/Sqrt[eps]] ];(*
This gives the vector equation for the magnetic field. *)
(* Now the 6 coupled scalar equations.*)
Eq1=Simplify[(t5[[1]]-t8[[1]])/Exp[I*k*t/Sqrt[eps]]];
Eq2 =Simplify[( t5[[2]]-t8[[2]])/Exp[I*k*t/Sqrt[eps]]];
Eq3=Simplify[(t5[[3]]-t8[[3]])/Exp[I*k*t/Sqrt[eps]]];
Eq4=Simplify[(t11[[1]]-eps*t10[[1]])/Exp[I*k*t/Sqrt[eps]]];
Eq5=Simplify[(t11[[2]]-eps*t10[[2]])/Exp[I*k*t/Sqrt[eps]]];
Eq6=Simplify[(t11[[3]]-eps*t10[[3]])/Exp[I*k*t/Sqrt[eps]]];