Lucas "... using a spherical surface centered about the charge
distribution q with spherical coordinates and noting that E and n
will then be in the same direction ..."
/$ 4*π*q = ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0(r)*r*sin(θ)*(1 - λ(v))/(1 - β^2*sin(θ)^2)^(3/2)))
= 4*π*q*(1 - λ(v))/(1 - β^2)
/$ 4*π*q = ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0(r)*r*sin(θ)*(1 - λ(v))/(1 - β^2*sin(θ)^2)^(3/2)))
= 4*π*q*(1 - λ(v))/(1 - β^2)
/* Howell - Collecting Lucas's result for (4-39) and (4-40)
04_39 E(r,v) for constant velocity, non-point charge, observer reference frame
/$ ET(r,v) = (1 - λ(v))*E0(r)/(1 - β^2*sin(θ)^2)^(3/2)
/* 04_40 L(v) expression for Gauss law for electric charge
/$ 4*π*q = ∬[∂(Area): ET(r)nh)
/* Expressing (4-40) in terms of constant r, variable [O,P] to integrate
.over.the spherical surface :
/$1) 4*π*q
= ∬[∂(Area): ET(r)nh)
= ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: ET(r)nh)
/* Subbing (4-39) into (1) - BUT Lucas has extra "r*sinO" term
***Where did that extra term come from? - may be a hint for (4-37) etc!!!!
/$ 4*π*q
= ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: (1 - λ(v))*E0(r)/(1 - β^2*sin(θ)^2)^(3/2)))
= (1 - λ(v))*E0(r) *∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: (1 - β^2*sin(θ)^2)^( - 3/2)))
/* As there is no change with P :
/$ = (1 - λ(v))*E0(r) *∫[∂(φ),0 to 2*π: φ)*∫[∂(θ),0 to π: (1 - β^2*sin(θ)^2)^( - 3/2)))
1a) = (1 - λ(v))*E0(r)*2*π *∫[∂(θ),0 to π: (1 - β^2*sin(θ)^2)^( - 3/2)))
/* To assess the integral in (1a), expand the series as per (4-38)
/$ (1 - β^2*sin(θ)^2)^(-3/2) = (1 + 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 + 35/16*β^6*sin(θ)^6 + ...)
/* So
/$ ∫[∂(θ),0 to π: (1 - β^2*sin(θ)^2)^( - 3/2)))
= ∫[∂(θ),0 to π: (1 + 3/2*β^2*sin(θ)^2 + 15/8*β^4*sin(θ)^4 + 35/16*β^6*sin(θ)^6 + ...)
= ∫[∂(θ),0 to π: 1)
∫[∂(θ),0 to π: 3/2*β^2*sin(θ)^2)
∫[∂(θ),0 to π: 15/8*β^4*sin(θ)^4)
∫[∂(θ),0 to π: 35/16*β^6*sin(θ)^6)
∫[∂(θ),0 to π: + ...)
/* Ignore the last term, integrate the others
= π
/$1a1) 3/2 *β^2*∫[∂(θ),0 to π: sin(θ)^2)
1a2) 15/8 *β^4*∫[∂(θ),0 to π: sin(θ)^4)
1a3) 35/16*β^6*∫[∂(θ),0 to π: sin(θ)^6)
/* Using Howell - Variables, notations, styles for Bill Lucas, Universal Force.odt
Recurring integrals from http://integral-table.com
/$1a1a) ∫[∂(θ),0 to Of: sin(θ)^2) = O/2 - sin(2*O)/4
1a2a) ∫[∂(θ),0 to Of: sin^3(O)) = -3*cos(θ)/4 + cos(3*O)/12
/* This is NOT going to work!! From where did Lucas get the extra "r*sinO" term?
/* Now looking at
/$ ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: ET*(1 - λ(v))*r*sin(O)/(1 - β^2*sin(θ)^2)^(3/2)]}
/* [ET, L(v), b, r] are pseudo-constants wrt dO, so they come out, note above Lucas error with integral limits, but now ET instead of E0!
Also Lucas missing a second r for dP !!???!!
Correcting original :
/$ ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0*(1 - λ(v))*r*sin(O)/(1 - β^2*sin(θ)^2)^(3/2)]*r}
∫[∂(φ),0 to 2*π: E0*(1 - λ(v))*r^2*∫[∂(θ),0 to π: sin(O)/(1 - β^2*sin(θ)^2)^(3/2)]}
/* Now looking at
/$ ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: ET*(1 - λ(v))*r*sin(O)/(1 - β^2*sin(θ)^2)^(3/2)]}
/* [ET, L(v), b, r] are pseudo-constants wrt dO, so they come out, note above Lucas error with integral limits, but now ET instead of E0!
Also Lucas missing a second r for dP !!???!!
Correcting original :
/$ ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0*(1 - λ(v))*r*sin(O)/(1 - β^2*sin(θ)^2)^(3/2)]*r}
∫[∂(φ),0 to 2*π: E0*(1 - λ(v))*r^2*∫[∂(θ),0 to π: sin(O)/(1 - β^2*sin(θ)^2)^(3/2)]}
/* using result above for ∫[dO, 0 to π : sinO/(1 - b^2*sin^2(O))^(3/2) }
/$ = ∫[∂(φ),0 to 2*π: E0*(1 - λ(v))*r^2*2/(1 - β^2)}
/* E0s has constant scalar magnitude at constant r for integral
/$ = E0*(1 - λ(v))*r^2 *2/(1 - β^2) *∫[∂(φ),0 to 2*π: 1}
= E0*(1 - λ(v))*r^2 *2/(1 - β^2) *2*π
/* sub E0 = q/r^2
/$ = 4*π*Q/r^2*(1 - λ(v))*r^2/(1 - β^2)
= 4*π*Q*(1 - λ(v))/(1 - β^2)
/*++++++++++++++++++++++++++++++++++++++
/*add_eqn "Howell_incomplete
04_41
L(v) expression for Gauss law for electric charge
/$ 4*π*q
= ∫[∂(φ),0 to 2*π: ∫[∂(θ),0 to π: E0(r)*r*sin(θ)*(1 - λ(v))/(1 - β^2*sin(θ)^2)^(3/2)))
= 4*π*q*(1 - λ(v))/(1 - β^2)
/* havent done yet
PRIORITY PROBLEM - sin,cos terms