Faradays law (4-2) can be rewritten using Stokes theorem
Lucas (4-02) := Faradays_Law
/$ ∫E(Rpcv,t´)•∂(l) = -1/c*∂[∂(t): ∫B(r,t)•nda]
/*very similar to (4-18)?
Stokes theorem - Kreyszig1972 p364h0.6 Eqn (8.10-1)
/$ ∬[∂(Area): (curl(v))_n) = ∮[ds,.over.C: v*t)
/* where (curl v)_n = (curl v)•n is the component of curl v in the direction of a unit normal vector n of S; the integration around C is taken in the sense shown in Figure 166, and vt is the component of v in the direction of the tanjent vector of C in Fig 166 (Howell : right-hand curl!)
Faradays law as expressed in (4-2)≈ RHS of (4-18), so this is OK with a warning about the switched ordering of differentiation and integration
BUT its not clear what has been done on the LHS :
stated as :
/$ (Ei´(r´,t´) - 1/c*vBi(r´,t´))
/*BUT - this MUST have ∂[∂t: of B (differential or integral)
My GUESS is that E0 is needed as per (4-13)
/$ B(r´,t´) = (v/c)[ E0(r´,t´) + Ei(r´,t´) ]
/* NOTE : units dont work for (2-2) versus (2-3), and (4-19) appears to have the same problem.
Replacing Ei(r´,t´) with Ei´(Rpcv,t´) - E0(r´,t´)
/$1) ∫[Ei´(Rpcv,t´) - E0(r´,t´))•∂(l) = -1/c*∂[∂(t): ∫B(r,t)•nda]
/* failed-try :
For E0(r´,t´), for a point charge : see Jackson1999 p28h0.1
/$(1.7) ET•n*∂(Area) = q/4/π/ε0*cos(θ)/r^2*∂(Area)
ET•n = q/4/π/ε0*cos(θ)/r^2
/* or spherical surfaces centered on a particle
/$ ET = q/4/π/ε0/r^2
(1.8) ET•n*∂(Area) = q/4/π/ε0*dΩ
/* where dΩ = solid angle subtended by da at the position of the charge
/$ r^2*dΩ = cos(θ)*∂(Area)
/* Taking E0(r´,t´) = E in 1.7 = q/4/π/ε0/r^2, and substituting
Nyet
Oops
take Amperes Law (4-1)!!!
Lucas04_01 := Generalized_Amperes_Law, note change of refFrame
/$ B(r´,t´) = v/c × E0(Rpcv,t´)
/* so
/$2) E0(Rpcv,t´) = c/v*B(r´,t´)
/*OOPS!!! CROSS PRODUCT!
putting this into 1 above
/$3) ∫[E´(Rpcv,t´) - c/v*B(r´,t´))•∂(l) = -1/c*∂[∂(t): ∫B(r,t)•nda]
/*write out more specifically (Howell notation) :
/$4) ∮[•d(l´),.over.L: (E´(Rpcv,t´) - c/v*B(r´,t´)))
= -1/c*∂[∂(t): ∫[dArea,.over.A: B(r,t)•n)]
/*Notice that I didn't have to apply Stokes theorem!
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Question
04_19
( E,B for symmetry point charge @v_const
add Faradays law to [Ampere]
/$-Lucas ∮[•d(l´),.over.L: (Ei´(r´,t´) - v/cBi(r´,t´))) = -1/c* ∮[dAreap,.over.Ap: (∂[∂(t): Bi(r - v*t,t)]•n)
-Howell ∮[•d(l´),.over.L: (Ep(r´,t´) - c/v*Bi(r,t))) = -1/c*∂[∂(t): ∮[dArea,.over.A: B(r,t]•n)
/%^%-Lucas ∮[•d(l´),.over.closedcurveL: EIpdv(POIp) - Vonv(PART)/c*BIodv(POIp(t),t)) = -1/c*∮[dAreap,.over.Ap: (∂[∂(t): BIodv(POIo,t)]•n}
-Howell ∮[•d(l´),.over.closedcurveL: EIpdv(POIp) - c/Vonv(PART)*BIodv(POIp(t),t)) = -1/c *∂[∂(t): ∮[*∂(Area),.over.A: BIodv(POIo,t)]•n}
/* better to use ∬ rather than ∮ for Bi.over.A ??
PROBLEMS - cross-product correction to my deriv, v/cBi(r´,t´ versus c/v*Bi(r,t),
changed order of diff/integ- B(r,t) vs Bi(ro - vo*t,t),