-- Operator Declarations --
## Cartesian
h1 = 1
h2 = 1
h3 = 1
X = (x, y, z)
## Cylindrical
# h2 = 1
# h3 = r
#X = (r, theta, z)
## Spherical
#h2 = r
#h3 = r*sin(theta)
#X = (r, theta, phi)
## Toroidal
# htau = a/(cosh(tau) - cos(sigma))
# hsigma = htau
# hphi = htau*sinh(tau)
# X = (sigma, tau, phi)
W = h1*h2*h3
-- gradient
grad(v) = (d(v,X[1])/h1, d(v,X[2])/h2, d(v,X[3])/h3)
-- divergence
div(v) = ( d(v[1]*h2*h3,X[1]) + d(v[2]*h1*h3, X[2]) + d(v[3]*h1*h2, X[3]) )/W
-- curl
curl(f) = (d(f[3]*h3,X[2]) - d(f[2]*h2,X[3]),
d(f[1]*h1,X[3]) - d(f[3]*h3,X[1]),
d(f[2]*h2,X[1]) - d(f[1]*h1,X[2]))/(h1*h2)
-- laplacian
laplacian(f) = ( d(d(f*h2*h3/h1,X[1]),X[1]) + d(d(f*h1*h3/h2,X[2]),X[2]) + d(d(f*h1*h2/h3,X[3]),X[3]) )/W
cross(u,v) = (h1*(u[2] v[3] - u[3] v[2]),
h2*(u[3] v[1] - u[1] v[3]),
h3*(u[1] v[2] - u[2] v[1]))/W
"##############################################################"
"# The problem of three orthogonally crossed, center-fed #"
"# Electric Dipoles, centered inside three orthogonally #"
"# crossed Loop Antennas #"
"# T. E. Raptis (c) 2013 (v.3) #"
"# email: rtheo@dat.demokritos.gr #"
"##############################################################"
"Two decades ago, it was found that Maxwell equations allow self-dual"
"solutions with parallel E and B fields. The relevant papers were"
"[1] Chu, Ohkawa, PRL, V48(13), 837, 1982"
"[2] Zaghloul, Buckmaster,Am. J. Phys. V56(9), 801, 1988"
"[3] Shimoda et al, J. Phys. Soc. Jap. V 58(10),3570, 1989 "
"[4] Shimoda et al, Am. J. Phys. V58(4), 394, 1990"
"[5] N. A. Salingaros, Phys. Rev. A, V45(12), 8811, 1992"
"[6] J. E. Gray, J. Phys. A: Math. Gen. V25, 5373, 1992"
"[7] A. Chubykalo et al, Am. J. Phys. V78(8), 858, 2010"
"No actual experimental realization of most of them with artifical means"
"appears anywhere although some similar formation known as 'force-free'"
"flows appear in MHD studies and simulations. The below seem to be one"
"of the simplest condigurations using RF elements with similar properties."
"The overall result appears to be a set of overlapping spherical Bessel fields"
"creating a standing focus wave mode which accumulates energy. Such a field"
"could prove useful in fusion research and the study of excited vacuum states"
"in near field electrodynamics. At the end, Lorentz invariance is commented."
"##############################################################"
"We start with the problem of an electric dipole antenna placed at the center"
"of a magnetic loop antenna. Denote unit vector and polarization vector as"
r = (xr, yr, zr)
px = (p,0,0)
r
px
"p taken as"
I*L/omega/2
"Denote the full radiation fields for the electric dipole as"
"E = -a(k,R)r x(r x p)+ b(k,R)[3*r(r*p) - p]"
"B = g(k,R)(r x p)"
a(k,R) = k^2*exp(i*k*R)/R
"alpha:"
a(k,R)
b(k,R) = (1 - i*k*R)*exp(i*k*R)/R^3
"beta:"
b(k,R)
c(k,R) = (k^3*R + i)/(c0*k*R)*exp(i*k*R)
"gamma:"
c(k,R)
B = gamma*cross(r,px)
E = alpha*cross(r,B/gamma) - beta*(3*r*dot(r,px) - px)
E
B
"E is further simplified using xr^2 + yr^2 + zr^2 = 1"
E[1] = kappa*p*xr^2 - alpha*p
E
"with kappa = alpha - 3*beta."
KK = a(k,R) - 3*b(k,R)
simplify(KK)
"Add the dual loop fields as E -> d*H, H -> -d*E"
E1 = B
B1 = -E
Ex = E + delta*E1
Bx = B + delta*B1
Ex
Bx
"Find the complex delta factor so as to cancel radiation"
"in the final E x B product"
g = simplify(cross(Ex,Bx))
factor(g[1]/(gamma*xr*p^2), alpha)
g[2]/(gamma*yr*p^2)
g[3]/(gamma*zr*p^2)
"From the first element we see that delta should be"
delta = i
delta
"hence we get"
Ex
Bx
"Test all elements"
g
"Hence the Loop Antenna signal must be phase shifted. The total"
"fields will be alingned as E = iB. The phase shift corresponds to"
"Ei = Ei(r)cos(w*t), Bi = Bi(r)sin(w*t). The equations of motion for"
"a charged particle after taking real parts in such a configuration"
"will be as below (r.h.s. = 0)"
u = (ux(), uy(), uz())
F =(E*cos(omega*t) + cross(u,E)*sin(omega*t))/p- d(u,t)*kqm
F[1]
F[2]
F[3]
kqm = m/p/q
kqm
"We then proceed to construct two more pairs with polarizations"
"along the two remaining axes and extract the same field components."
"---------- Ey - By pair --------------------------------------"
py = (0, p, 0)
By = gamma*cross(r,py)
Ey = alpha*cross(r,By/gamma) - beta*(3*r*dot(r,py) - py)
Ey[2] = kappa*p*yr^2 - alpha*p
E1 = By
B1 = -Ey
Ey = Ey + i*E1
By = By - i*B1
Ey
By
" "
"---------- Ez - Bz pair --------------------------------------"
pz = (0, 0, p)
Bz = gamma*cross(r,pz)
Ez = alpha*cross(r,Bz/gamma) - beta*(3*r*dot(r,pz) - pz)
Ez[3] = kappa*p*zr^2 - alpha*p
E1 = Bz
B1 = -Ez
Ez = Ez + i*E1
Bz = Bz - i*B1
Ez
Bz
"Superposition results in the total fields (factoring p)"
E = Ex + Ey + Ez
B = Bx + By + Bz
E/p
B/p
"These can be further simplified in a form that satisfies E = iB"
(kappa*F1 +i*gamma*F2 - alpha*I, gamma*F2 - i*kappa*F1 + i*alpha*I)
"F1 is the rowwise contraction of the dyadic r o r"
dot(outer(r,r), (1, 1, 1))
"F2 is a double permutation of the unit vector U o r"
U = ((0, 1, -1),
(-1, 0, 1),
(1, -1, 0))
U
"It is easy to verify that {F1, F2} satisfy the relation"
"R*F2 = -curl (F1*R^2) and F1 o F2 = 0."
"Taking real parts with real(kappa) and real(gamma)"
"(notice that Re(i*z) = -Im(z) & Im(i*z) = Re(z))"
"so that we get the common field E(r) = B(r)."
kappare(R)*F1(theta,phi) - gammaim(R)*F2(theta,phi) - alphare(R)*I
fe = cos(k*R)+i*sin(k*R)
aa = subst(fe, exp(i*k*R), a(k,R))
kk = subst(fe, exp(i*k*R), KK)
ge = subst(fe, exp(i*k*R), c(k,R))
alphare = real(aa)
alphare
alphaim = imag(aa)
kappare = real(kk)
kappare
kappaim = imag(kk)
gammare = real(ge)
gammaim = imag(ge)
gammaim
"The equations of motion in such a field are found to be"
F1 = dot(outer(r,r), (1, 1, 1))
F2 = dot(U,r)
A = kappar*F1 - gammai*F2 - alphar*(1,1,1)
F = A*cos(omega*t) + cross(u,A)*sin(omega*t) - d(u,t)*kmq
F[1]
F[2]
F[3]
" "
"#########################################################"
"Invariance of Self-Dual/Anti-Self-Dual configurations in"
"different Lorentz frames"
"#########################################################"
" "
"Start from an abstract velocity vector"
v = (vx, vy, vz)
v
"Create an abstarct pair of self-dual E B fields"
B0 = (Bx0, By0, Bz0)
E0 = i*B0
E0
B0
"Obtain the generic transformed vectors using"
"the full Lorentz transform given by"
"gamma(E0 + VxB0) - f*V(VoE0)"
"gamma(B0 - VxE0) - f*V(VoB0)"
"f(V) being"
gamma^2/(1+gamma)
"From the above general form one can verify that"
"E1 = iB1 by just factoring the imaginary unit."
Eb = dot(v, E0)
Be = dot(v, B0)
E1 = gamma*(E0 + cross(v, B0)) - f*v*Eb
B1 = gamma*(B0 - cross(v, E0)) - f*v*Be
E1
B1
"ExB"
u=cross(E1,B1)
u[1]
u[2]
u[3]
"The same will hold true for E = -iB."