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))".




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