The Other Term...

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 $y=sec(\theta)$.