p69h0.5 Galilean invariance requires that in this case
(circuit at rest v=0, Lucas04_23) Ep = E and Fp = F,
1. Starting point, 11Sep2015
04_12 E transformation particle to observer frames
/$ ET(r´,t´) = ET(r - v*t,t)
/* 04_16Derivative form of Ampere's law
Take early form
/$ 1/c*∂[∂(t): Bi(r - v*t,t)]
= v/c*r*sin(θ)*φ´* [ 3*q*v/c*(r*cos(θ) - v*t)/|r - v*t|^5
+ 1/r/c*∂[∂(t): Ei(r - v*t,t)]
]
/* doesn't make sense
2. try (4-12)&(4-19) rather than (4-12)&(4-16)
04_19E,B for symmetry point charge @v_const . add Faradays law to [Ampere]
/$ ∮[•d(l´),.over.L: (Ei´(r´,t´) - v/cBi(r´,t´)))
= -1/c*∮[dAreap,.over.Ap: (∂[∂(t): Bi(r - v*t,t)]•n)
/* doesn't make sense,
3. try (4-11)&(4-13) rather than (4-12)&(4-16)
from 04_11 E&B_fields_static_plus_induced
/$ ET(r,v,t) = E0(r,t) + Ei(r,v,t) B(r,v,t) = B0(r,t) + Bi(r,v,t)
/* given Ep = E
/$ Ei´(r´,v´,t´) = Ei(r,v,t) - E0(r,t)
/* from 04_13 Total B magnetic flux density as induced from E0 + Ei
/$ B(r´,t´) = (v/c)[ E0(r´,t´) + Ei(r´,t´) ]
/* re-express in observer frame
/$ B(r - v*t,t) = (v/c)[ E0(r - v*t,t) + Ei(r - v*t,t) ]
/* close, but (v/c) in wrong place...
4.**what is the "inverse cross-product (_inv)" function such that
/$ a=βc -> _inv(β,c) = c [like a=β*c -> /(a,β)=c ]
/* 5. **first look at dot product -> inverse dot product
/$ a=β•c -> •_inv(a,β) = c
a=β•c = b1*c1 + b2*c2 + b3*c3
c = (β•c) * [n1*c1/(β•c) + n2*c2/(β•c) + n3*c3/(β•c)]
= a * [n1*c1/ a + n2*c2/ a + n3*c3/ a ]
/* unfortunately, this requires explicit knowledge of c
Hold this for now (interesting challenge - but MAYBE under-constrained?)
6. Try dot-multiplying by vector to simplify
/$ B(r - v*t,t) = (v/c)[ E0(r - v*t,t) + Ei(r - v*t,t) ]
(v/c)B(r - v*t,t) = (v/c)( (v/c)[ E0(r - v*t,t) + Ei(r - v*t,t) ] )
using pattern a(βc) = (aβ)c
(v/c)B(r - v*t,t) = (v/c) (v/c) ( E0(r - v*t,t) + Ei(r - v*t,t) )
/* but (v/c)(v/c) = 0 as a vector is collinear with itself!?!
strange "wrong answer" of very basic nature - what am I doing wrong
with vector operations?
Interesting challenge, but taking too much time and Im stuck.
12Sep2015 08h37 - Ill come back to this later
23May2016 try multiplying by vX rather than v
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Possible_Lucas_error_or_omission
04_24
E&B for [Faradays + part/obs frameTrans] - towards Fu_Faradays_Law
/$ Ei(r´,v´,t´) = Ei(r - v*t,t) + 1/c*[vBi(r - v*t,t)]
/%^% EIpdv(POIo(t),t) = EIodv(POIo,t) + 1/c*[Vonv(PART)BIodv(POIo,t)]
/* !!!WRONG - On hold as I havent been able to "move" (v/c) to the right place..
AND I still must show that Ep = E and Fp = F !!