Howell - comparisons for dp[dt_BTodv(POIo,t)].odt
www.BillHowell.ca 30Mar2018 initial draft
Summary :
Table of Contents
From Lucas (4-16), sub-sub-section above :
3) dp[dt : BTpdv(POIo(t),t)] = dp[dt : BTodv(POIo,t)]
= Vons(particle)^2/c*sin(AOpc(POIo(t)))*cos(AOpc(POIo(t)))*Rodh(Vonv_X_Rpcv(POIo))
*{ 3*Q(particle) /Rpcs(POIo(t))^3 *RO0ocs(POIo)/RO0pcs(POI,t)
- dp[dt : EIpds(POIo(t))]/Vons(particle)/cos(AOpc(POIo(t)))
}
Howell's expression Section "dp[dt : BTodv(POIo,t)] = dp[dt : BTpdv(POIo(t),t)] subbing for Lenz's Law :
2) dp[dt : BTpdv(POIo(t),t)] = dp[dt : BTodv(POIo,t)]
= Vons(particle)^2/c*sin(AOpc(POIo(t)))^2*cos(AOpc(POIo(t)))*Rodh(Vonv_X_Rpcv(POIo))
*{ 3*Q(particle) /Rpcs(POIo(t))^3 *[1 + lambda(v)/3]
- dp[dt : EIpds(POIo(t))]/Vons(particle)/cos(AOpc(POIo(t)))
}
These are "somewhat similar". However, there are very important differences, as highlighted above.
As for the causes of the discrepancies, that is not fully clear :
• As per above "... While the factor of 3 IS EXPLAINED, this is NOT the same as (3) in the sub-sub-section immediately above, as I have in (6)* an extra term for dp[dt : E0] at right angles to Rodh(Vonv_X_Rpcv(POIo)), due to angular changes. ..."
• perhaps there is also a difference between point-particle and finite-sized derivations, as I have not yet done the latter? But this is not obvious from Lucas' book.
+--+
This is NOT the same result for dp[dt : BTpdv(POIo(t),t)] as Lucas p68h0.0 Equation (4-16) :
(4-16) dp(dt : Bi(ro - vo*t,t))
= vos/c*ros*sinOo*Pph
*[ 3*q*vos/c*(ros*cosOo - vos*t)/|ro - vo*t|^5
+ 1/ros/c*dp(dt : |Ei(ro - vo*t,t)|) ]
= vos/c*sinOo*Pph*
[ 3*q*vos/c*(ros*cosOo - vos*t)/|ro - vo*t|^4
+ 1/c*dp(dt : |Ei(ro - vo*t,t)|) ]
It seems clear to me that Lucas has :
- dropped the term "Vons(particle)*cos(AOpc(POIo(t)))*EIpds(POIo(t))/Rpcs(POIo(t))". But there should be an EIpds(POIo(t)) term!
- has too many "c"'s
- has the term "(ros*cosOo - vos*t)/|ro - vo*t|" in the first term in parenthesis.
- I have NO real idea of why the (ros*cosOo - vos*t) pops up anyways, and why the cos term all of a suddden (spherical coords does not explain this!)
If I replace "(ros*cosOo - vos*t)/|ro - vo*t|" with "1", and remove the extra "c"s, and used my nomenclature, then Lucas's expression becomes :
(4-16)Mod dp(dt : Bi(ro - vo*t,t))
= Vons(particle)/c*sin(AOpc(POIo(t)))*Rodh(Vonv_X_Rpcv(POIo))
*[ 3*Q(particle)*Vons(particle)/Rpcs(POIo(t))^3 + dp[dt : EIpds(POIo(t))] ]
This is still wrong, as it is missing my term "Vons(particle) *cos(AOpc(POIo(t)))*EIpds(POIo(t))/Rpcs(POIo(t))".
enddoc