From the post "Not To Be taken Too Seriously, Please" and "Time For The Missing Term..." both dated 15 May 2016,
\(Z= \left\{ \int _{ 1 }^{sec(\theta_0) }{ e^{ -\cfrac { x.y }{ a_{ e } } }\left\{ \sqrt { y^{ 2 }-1 } \right\} } { d\, y }+ \int _{ 1 }^{ cos(\theta_0) }{ e^{ -\cfrac { x }{ a_{ e }y } }\left\{ \cfrac { 1 }{ y^{ 4 } } \right\} } { d\,y }\right\}\)
but \(\theta_0\) is not small, in fact \(\cfrac{\pi}{2}\ge\theta_0\ge 0\), so the simplification \(y=sec(\theta_0)\approx1\) is not valid.
Consider,
\(II_2= \int _{ 1 }^{sec(\theta_0) }{ e^{ -\cfrac { x.y }{ a_{ e } } }\left\{ \sqrt { y^{ 2 }-1 } \right\} } { d\, y }= -\cfrac{a_e} {x}\int _{ 1 }^{sec(\theta_0) }{-\cfrac{x}{a_e}e^{ -\cfrac { x.y }{ a_{ e } } }\left\{ \sqrt { y^{ 2 }-1 } \right\} } { d\, y }\)
\(II_2=- \cfrac{a_e}{x}\int _{ 1 }^{sec(\theta_0) }{\left(e^{ -\cfrac { x.y }{ a_{ e } } }\right)^{'}\left\{ \sqrt { y^{ 2 }-1 } \right\} } { d\, y }\)
Since,
\(y=sec(\theta )\) and
\( \cfrac { r_{ e } }{ x } =tan(\theta )\)
\( \sqrt { y^{ 2 }-1 } =\sqrt { sec^{ 2 }(\theta )-1 } =tan(\theta )=\cfrac { r_{ e } }{ x } \)
\( II_{ 2 }=- \cfrac{a_e}{x}\int _{ 1 }^{sec(\theta_0) }{\left(e^{ -\cfrac { x.y }{ a_{ e } } }\right)^{'}\cfrac { r_{ e } }{ x } } { d\, y }\)
\(II_{2}=-\cfrac { a_{ e }r_{ e } }{ x^{ 2 } } \left[ e^{ -\cfrac { x.y }{ a_{ e } } } \right] _{ 1 }^{ sec(\theta _{ 0 }) }\)
Since,
\(B_{ o }=\cfrac {\mu_o q v }{4\pi r^2 }.\cfrac{ r^{ 3 }_{ e }}{a^3_e}.sin^{ 2 }(\phi ).Z\)
\(x^2=r^2_e+r^2_e-2r^2_ecos(\phi)=2r^2_e(1-cos(\phi))\)
\(x=r_e\sqrt{2(1-cos(\phi))}=2r_{ e }sin(\phi /2)\)
\(II_{2} =-\cfrac { a_{ e } }{ 4r_{ e }sin^2(\phi /2) } \left[ e^{ -\cfrac { 2r_{ e }sin(\phi /2)y }{ a_{ e } } }\right]_{ 1 }^{ sec(\theta _{ 0 }) } \)
So,
\(B_o=\cfrac { \mu _{ o }qv }{ 4\pi r^{ 2 } } .\cfrac { r^{ 3 }_{ e } }{ a^{ 3 }_{ e } } .sin^{ 2 }(\phi )\left\{ -\cfrac { a_{ e } }{ 4r_{ e }sin^{ 2 }(\phi /2) }\right\} \left[ e^{ -\cfrac { 2r_{ e }sin(\phi /2)y }{ a_{ e } } }\right]_{ 1 }^{ sec(\theta _{ 0 }) } \)
\(B_o=-\cfrac { \mu _{ o }qv }{ 4\pi r^{ 2 } } .\cfrac { r^{ 2 }_{ e } }{ a^{ 2 }_{ e } } .cos^{ 2 }(\phi /2) \left[ e^{ -\cfrac { 2r_{ e }sin(\phi /2)y }{ a_{ e } } }\right]_{ 1 }^{ sec(\theta _{ 0 }) } \)
At \(\phi=\cfrac{\pi}{2}\),
\(B_{ o }=-\cfrac { \mu _{ o }qv }{ 8\pi r^{ 2 } } .\cfrac { r^{ 2 }_{ e } }{ a^{ 2 }_{ e } } \left\{ e^{ -\cfrac { \sqrt { 2 }r_{ e } sec(\theta _{ 0 }) }{ a_{ e } } }-e^{ -\cfrac { \sqrt { 2 }r_{ e } }{ a_{ e } } } \right\} \)
when \(\theta\) is small, at \(\phi=\cfrac{\pi}{2}\),
\(B_o\approx0\)
Which is all well...And time is fixed, no more time correction!
Note: For every given position \(x\), \(\theta\) is evaluated up to \(\theta_0\). \(x\) is not dependent on \(\theta\) nor on \(y=sec(\theta)\).
