\(U_B=\cfrac{1}{2}\cfrac{B^2_o}{\mu_o}\)
\(U_B=\cfrac{1}{2\mu_o}\left\{\cfrac {\mu_o q v }{4\pi r^2 }.\cfrac{ r^{ 3 }_{ e }}{a^3_e}.sin^{ 2 }(\phi ).Z\right\}^2\)
\(U_B=\cfrac{\mu_o}{2}\left\{\cfrac { q v }{4\pi r^2 }.\cfrac{ r^{ 3 }_{ e }}{a^3_e}\right\}^2sin^{ 4 }(\phi ).Z^2\)
where \(Z\) has two terms,
\(Z= \left\{ \int _{ 1 }^{sec(\theta_0) }{ e^{ -\cfrac { x.y }{ a_{ e } } }\left\{ \sqrt { y^{ 2 }-1 } \right\} } { d\, y }+ \int _{ 1 }^{ cos(\theta_0) }{ e^{ -\cfrac { x }{ a_{ e }y } }\left\{ \cfrac { 1 }{ y^{ 4 } } \right\} } { d\,y }\right\}\)
As \(x\le 2r_e\) and \(\theta_0\) is small, \(y=sec(\theta_0)\approx 1\), \(\sqrt { y^{ 2 }-1 }\approx 0\) because \(0\le y\le sec(\theta_0)\).
We simplify by letting the first term of \(Z\) be zero.
\( \int _{ 1 }^{sec(\theta_0) }{ e^{ -\cfrac { x.y }{ a_{ e } } }\left\{ \sqrt { y^{ 2 }-1 } \right\} } { d\, y }=0\)
\(Z= \int _{ 1 }^{ cos(\theta_0) }{ e^{ -\cfrac { x }{ a_{ e }y } }\left\{ \cfrac { 1 }{ y^{ 4 } } \right\} } { d\,y }\)
Then we consider, the average of \(U_B\) over one period, ie its power,
\(\overline{U_B}=\cfrac{1}{T}\int^{T}_{0}{U_B}dt=\cfrac{1}{T}\int^{T}_{0}\cfrac{\mu_o}{2}\left\{\cfrac { q v }{4\pi r^2 }.\cfrac{ r^{ 3 }_{ e }}{a^3_e}\right\}^2sin^{ 4 }(\phi )\left(\int _{ 0 }^{ cos(\theta_0) }{ e^{ -\cfrac { x }{ a_{ e }y } }\left\{ \cfrac { 1 }{ y^{ 4 } } \right\} } { d\,y }\right)^2 dt\)
\(\overline{U_B}=\cfrac{\mu_o}{2T}\left\{\cfrac { q v }{4\pi r^2 }.\cfrac{ r^{ 3 }_{ e }}{a^3_e}\right\}^2\int^{T}_{0}sin^{ 4 }(\phi )\left(\int _{ 0 }^{ cos(\theta_0) }{ e^{ -\cfrac { x }{ a_{ e }y } }\left\{ \cfrac { 1 }{ y^{ 4 } } \right\} } { d\,y }\right)^2 dt\)
but
\(x^2=r^2_e+r^2_e-2r^2_ecos(\phi)=2r^2_e(1-cos(\phi))\)
\(x=r_e\sqrt{2(1-cos(\phi))}=2r_{ e }sin(\phi /2)\)
\(\overline{U_B}=\cfrac{\mu_o}{2T}\left\{\cfrac { q v }{4\pi r^2 }.\cfrac{ r^{ 3 }_{ e }}{a^3_e}\right\}^2\int _{ 0 }^{ T }sin^{ 4 }(\phi )\left(\int^{cos(\theta_0)}_{0}{ e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }y } }\left\{ \cfrac { 1 }{ y^{ 4 } } \right\} } { d\,y }\right)^2 dt\)
\(\cfrac{d\phi}{dt}=\omega\) is a constant.
\(\overline{U_B}=\cfrac{\mu_o}{2T\omega}\left\{\cfrac { q v }{4\pi r^2 }.\cfrac{ r^{ 3 }_{ e }}{a^3_e}\right\}^2\int _{ 0 }^{2\pi }sin^{ 4 }(\phi )\left(\int^{ cos(\theta_0)}_{0}{ e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }y } }\left\{ \cfrac { 1 }{ y^{ 4 } } \right\} } { dy }\right)^2 d\phi\)
where \(y=cos(\theta)\),
\(\overline { U_{ B } } =\cfrac { \mu _{ o } }{ 2T\omega } \left\{ \cfrac { qv }{ 4\pi r^{ 2 } } .\cfrac { r^{ 3 }_{ e } }{ a^{ 3 }_{ e } } \right\} ^{ 2 }\int _{ 0 }^{ 2\pi } sin^{ 4 }(\phi )\left( \int _{ 0 }^{ \theta_0 }{ -e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta) } }\left\{ \cfrac { sin(\theta ) }{ cos^{ 4 }(\theta ) } \right\} } { d\theta } \right) ^{ 2 }d\phi \)
Consider,
\(h=e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }\)
\( h^{ ' }_{ \theta }=e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }\left( +\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos^{ 2 }(\theta ) } \right) .(-sin(\theta ))\)
\( h^{ ' }_{ \theta }=-h.\left( \cfrac { 2r_{ e }sin(\phi /2)sin(\theta ) }{ a_{ e }cos^{ 2 }(\theta ) } \right) \)
\( h^{ ' }_{ \theta }=e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }\left( +\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos^{ 2 }(\theta ) } \right) .(-sin(\theta ))\)
\( h^{ ' }_{ \theta }=-h.\left( \cfrac { 2r_{ e }sin(\phi /2)sin(\theta ) }{ a_{ e }cos^{ 2 }(\theta ) } \right) \)
\(\overline { U_{ B } } =\cfrac { \mu _{ o } }{ 2T\omega } \left\{ \cfrac { qv }{ 4\pi r^{ 2 } } .\cfrac { r^{ 3 }_{ e } }{ a^{ 3 }_{ e } } \right\} ^{ 2 }\cfrac { a^2_{ e } }{ r^2_{ e } } \int _{ 0 }^{ 2\pi } sin^{ 2 }(\phi )\left( \int _{ 0 }^{ \theta _{ 0 } }{ -e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }\cfrac { r_{ e } }{ a_{ e } } \cfrac { 2sin(\phi /2)sin(\theta ) }{ cos^{ 2 }(\theta ) } \left\{ \cfrac { cos(\phi /2) }{ cos^{ 2 }(\theta ) } \right\} } { d\theta } \right) ^{ 2 }d\phi \)
\(\overline { U_{ B } } =\cfrac { \mu _{ o } }{ 2T\omega } \left\{ \cfrac { qv }{ 4\pi r^{ 2 } } .\cfrac { r^{ 3 }_{ e } }{ a^{ 3 }_{ e } } \right\} ^{ 2 }\cfrac { a^2_{ e } }{ r^2_{ e } }\cfrac { r }{ r_{ e } } \int _{ 0 }^{ 2\pi } sin^{ 2 }(\phi )\left( \int _{ 0 }^{ \theta _{ 0 } }{h^{' }\left\{ \cfrac {cos(\phi /2) }{ cos^{ 2 }(\theta ) } \right\} } { d\theta } \right) ^{ 2 }d\phi \)
\(\overline { U_{ B } } =\cfrac { \mu _{ o } }{ 2T\omega } \left\{ \cfrac { qv }{ 4\pi r^{ 2 } } .\cfrac { r^{ 3 }_{ e } }{ a^{ 3 }_{ e } } \right\} ^{ 2 }\cfrac { a^2_{ e } }{ r^2_{ e } } \int _{ 0 }^{ 2\pi } \left( sin(\phi )cos(\phi /2)\int _{ 0 }^{ \theta _{ 0 } }{ h^{ ' }_{ \theta }\left\{ \cfrac { 1 }{ cos^{ 2 }(\theta ) } \right\} } { d\theta } \right) ^{ 2 }d\phi \)--- (1)
Integrating by parts,
then, let,
\(\Pi=\left( sin(\phi )cos(\phi /2)\int _{ 0 }^{ \theta _{ 0 } }{ h^{ ' }\left\{ \cfrac { 1 }{ cos^{ 2 }(\theta ) } \right\} } { d\theta } \right) ^{ 2 }\)
\(\Pi=\left( { h\cfrac { sin(\phi )cos(\phi /2) }{ cos^{ 2 }(\theta ) } +2\cfrac { a_{ e } }{ r_{ e } } \int _{ 0 }^{ \theta _{ 0 } }{ h\cfrac { r_{ e } }{ a_{ e } } \cfrac { 2sin(\phi /2)sin(\theta ) }{ cos^{ 2 }(\theta ) } \cfrac { cos(\phi /2) }{ cos(\theta ) } } }d\theta \right) ^{ 2 }\)
\(\Pi=\left( { h\cfrac { sin(\phi )cos(\phi /2) }{ cos^{ 2 }(\theta ) } +2\cfrac { a_{ e }cos(\phi /2) }{ r_{ e } } \int _{ 0 }^{ \theta _{ 0 } }{ h^{ ' }_{ \theta }\cfrac { 1 }{ cos(\theta ) } } d\theta } \right) ^{ 2 } \)---(2)
\(\int _{ 0 }^{ \theta _{ 0 } }{ h^{ ' }_{ \theta }\cfrac { 1 }{ cos(\theta ) } } d\theta =h\cfrac { 1 }{ cos(\theta ) } +\int _{ 0 }^{ \theta _{ 0 } }{ h\cfrac { sin(\theta ) }{ cos^{ 2 }(\theta ) } } d\theta \)---(3)
and
\(\int _{ 0 }^{ \theta _{ 0 } }{ h\cfrac { sin(\theta ) }{ cos^{ 2 }(\theta ) } } d\theta =\cfrac { a_{ e } }{ 2r_{ e }sin(\phi /2) } \int _{ 0 }^{ \theta _{ 0 } }{ h\cfrac { r_{ e } }{ a_{ e } } \cfrac { 2sin(\phi /2)sin(\theta ) }{ cos^{ 2 }(\theta ) } } d\theta\)
\(=-\cfrac { a_{ e } }{ 2r_{ e }sin(\phi /2) } \int _{ 0 }^{ \theta _{ 0 } }{ h^{ ' }_{ \theta } } d\theta \)
\(=-\cfrac { a_{ e } }{ 2r_{ e }sin(\phi /2) } \int _{ 0 }^{ \theta _{ 0 } }{ h^{ ' }_{ \theta } } d\theta =-\left| \cfrac { a_{ e } }{ 2r_{ e }sin(\phi /2) } h \right| ^{ \theta }_{ 0 }\)---(4)
Substituting (4) into (3), (2) and (1), we have,
\(\overline { U_{ B } } =\cfrac { \mu _{ o } }{ 2T\omega } \left\{ \cfrac { qv }{ 4\pi r^{ 2 } } .\cfrac { r^{ 3 }_{ e } }{ a^{ 3 }_{ e } } \right\} ^{ 2 }\cfrac { a^2_{ e } }{ r^2_{ e } } \int _{ 0 }^{ 2\pi } \left( { \left| \cfrac { 1 }{ cos^{ 2 }(\theta ) } e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }2cos^{ 2 }(\phi /2)sin(\phi /2) \right| ^{ \theta _{ 0 } }_{ 0 }\\+\left| \cfrac { 2a_{ e } }{ r_{ e }cos(\theta ) } e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }cos(\phi /2) \right| ^{ \theta _{ 0 } }_{ 0 }\\-\left| \cfrac { a^{ 2 }_{ e } }{ r^{ 2 }_{ e } } cot(\phi /2)e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } } \right| ^{ \theta _{ 0 } }_{ 0 } } \right) ^{ 2 }d\phi \)
with the substitution, \(h=e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }\)
Let \(III|_{ 0 }^{ 2\pi }\) be the integral,
\(III|_{ 0 }^{ 2\pi }=\int _{ 0 }^{ 2\pi } \left( { \left| \cfrac { 1 }{ cos^{ 2 }(\theta ) } e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }2cos^{ 2 }(\phi /2)sin(\phi /2) \right| ^{ \theta _{ 0 } }_{ 0 }+\left| \cfrac { 2a_{ e } }{ r_{ e }cos(\theta ) } e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }cos(\phi /2) \right| ^{ \theta _{ 0 } }_{ 0 }\\-\left| \cfrac { a^{ 2 }_{ e } }{ r^{ 2 }_{ e } } cot(\phi /2)e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } } \right| ^{ \theta _{ 0 } }_{ 0 } } \right) ^{ 2 }d\phi \)
and consider,
\((a+b+c)^{ 2 }=a^{ 2 }+b^{ 2 }+c^{ 2 }+2ab+2ac+2bc\)
let,
\(a= \left| \cfrac { 1 }{ cos^{ 2 }(\theta ) } e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }2cos^{ 2 }(\phi /2)sin(\phi /2) \right| ^{ \theta _{ 0 } }_{ 0 } \)
\(b=\left| \cfrac { 2a_{ e } }{ r_{ e }cos(\theta ) } e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }cos(\phi /2) \right| ^{ \theta _{ 0 } }_{ 0 }\)
\(c=-\left| \cfrac { a^{ 2 }_{ e } }{ r^{ 2 }_{ e } } cot(\phi /2)e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } } \right| ^{ \theta _{ 0 } }_{ 0 } \)
then consider,
\(h=e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }\)
\( h^{ ' }_{ \phi }=e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }\left( -\cfrac { 2r_{ e }cos(\phi /2) }{ a_{ e }cos(\theta ) } \right) \cfrac { 1 }{ 2 } \)
\( h^{ ' }_{ \phi }=-h.\cfrac { r_{ e }cos(\phi /2) }{ a_{ e }cos(\theta ) } \)
This is a long tedious way home...
Note:
\(\Pi=\left( { h\cfrac { sin(\phi )cos(\phi /2) }{ cos^{ 2 }(\theta ) } +2\cfrac { a_{ e } }{ r_{ e } } \int _{ 0 }^{ \theta _{ 0 } }{ h\cfrac { r_{ e } }{ a_{ e } } \cfrac { 2sin(\phi /2)sin(\theta ) }{ cos^{ 2 }(\theta ) } \cfrac { cos(\phi /2) }{ cos(\theta ) } } }d\theta \right) ^{ 2 }\)
\(\Pi=\left( { h\cfrac { sin(\phi )cos(\phi /2) }{ cos^{ 2 }(\theta ) } +2\cfrac { a_{ e }cos(\phi /2) }{ r_{ e } } \int _{ 0 }^{ \theta _{ 0 } }{ h^{ ' }_{ \theta }\cfrac { 1 }{ cos(\theta ) } } d\theta } \right) ^{ 2 } \)---(2)
\(\int _{ 0 }^{ \theta _{ 0 } }{ h^{ ' }_{ \theta }\cfrac { 1 }{ cos(\theta ) } } d\theta =h\cfrac { 1 }{ cos(\theta ) } +\int _{ 0 }^{ \theta _{ 0 } }{ h\cfrac { sin(\theta ) }{ cos^{ 2 }(\theta ) } } d\theta \)---(3)
and
\(\int _{ 0 }^{ \theta _{ 0 } }{ h\cfrac { sin(\theta ) }{ cos^{ 2 }(\theta ) } } d\theta =\cfrac { a_{ e } }{ 2r_{ e }sin(\phi /2) } \int _{ 0 }^{ \theta _{ 0 } }{ h\cfrac { r_{ e } }{ a_{ e } } \cfrac { 2sin(\phi /2)sin(\theta ) }{ cos^{ 2 }(\theta ) } } d\theta\)
\(=-\cfrac { a_{ e } }{ 2r_{ e }sin(\phi /2) } \int _{ 0 }^{ \theta _{ 0 } }{ h^{ ' }_{ \theta } } d\theta \)
\(=-\cfrac { a_{ e } }{ 2r_{ e }sin(\phi /2) } \int _{ 0 }^{ \theta _{ 0 } }{ h^{ ' }_{ \theta } } d\theta =-\left| \cfrac { a_{ e } }{ 2r_{ e }sin(\phi /2) } h \right| ^{ \theta }_{ 0 }\)---(4)
Substituting (4) into (3), (2) and (1), we have,
\(\overline { U_{ B } } =\cfrac { \mu _{ o } }{ 2T\omega } \left\{ \cfrac { qv }{ 4\pi r^{ 2 } } .\cfrac { r^{ 3 }_{ e } }{ a^{ 3 }_{ e } } \right\} ^{ 2 }\cfrac { a^2_{ e } }{ r^2_{ e } } \int _{ 0 }^{ 2\pi } \left( { \left| \cfrac { 1 }{ cos^{ 2 }(\theta ) } e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }2cos^{ 2 }(\phi /2)sin(\phi /2) \right| ^{ \theta _{ 0 } }_{ 0 }\\+\left| \cfrac { 2a_{ e } }{ r_{ e }cos(\theta ) } e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }cos(\phi /2) \right| ^{ \theta _{ 0 } }_{ 0 }\\-\left| \cfrac { a^{ 2 }_{ e } }{ r^{ 2 }_{ e } } cot(\phi /2)e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } } \right| ^{ \theta _{ 0 } }_{ 0 } } \right) ^{ 2 }d\phi \)
with the substitution, \(h=e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }\)
Let \(III|_{ 0 }^{ 2\pi }\) be the integral,
\(III|_{ 0 }^{ 2\pi }=\int _{ 0 }^{ 2\pi } \left( { \left| \cfrac { 1 }{ cos^{ 2 }(\theta ) } e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }2cos^{ 2 }(\phi /2)sin(\phi /2) \right| ^{ \theta _{ 0 } }_{ 0 }+\left| \cfrac { 2a_{ e } }{ r_{ e }cos(\theta ) } e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }cos(\phi /2) \right| ^{ \theta _{ 0 } }_{ 0 }\\-\left| \cfrac { a^{ 2 }_{ e } }{ r^{ 2 }_{ e } } cot(\phi /2)e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } } \right| ^{ \theta _{ 0 } }_{ 0 } } \right) ^{ 2 }d\phi \)
and consider,
\((a+b+c)^{ 2 }=a^{ 2 }+b^{ 2 }+c^{ 2 }+2ab+2ac+2bc\)
let,
\(a= \left| \cfrac { 1 }{ cos^{ 2 }(\theta ) } e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }2cos^{ 2 }(\phi /2)sin(\phi /2) \right| ^{ \theta _{ 0 } }_{ 0 } \)
\(b=\left| \cfrac { 2a_{ e } }{ r_{ e }cos(\theta ) } e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }cos(\phi /2) \right| ^{ \theta _{ 0 } }_{ 0 }\)
\(c=-\left| \cfrac { a^{ 2 }_{ e } }{ r^{ 2 }_{ e } } cot(\phi /2)e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } } \right| ^{ \theta _{ 0 } }_{ 0 } \)
then consider,
\(h=e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }\)
\( h^{ ' }_{ \phi }=e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }\left( -\cfrac { 2r_{ e }cos(\phi /2) }{ a_{ e }cos(\theta ) } \right) \cfrac { 1 }{ 2 } \)
\( h^{ ' }_{ \phi }=-h.\cfrac { r_{ e }cos(\phi /2) }{ a_{ e }cos(\theta ) } \)
This is a long tedious way home...
Note:
\(\sigma =\cfrac { \pi -\phi }{ 2 } \)
\( v=\cfrac { d }{ d\, t } xcos(\pi /2-\sigma )=\cfrac { d }{ d\, t } xsin(\sigma )=\cfrac { d }{ d\, t } xcos(\phi /2)\)
\(v=cos(\phi /2)\cfrac { dx }{ d\, t } -xsin(\phi /2)\cfrac { 1 }{ 2 } \cfrac { d\phi }{ d\, t } \)
\(\left\{ r_{ e }+\cfrac { x }{ 2 } sin(\phi /2) \right\} \cfrac { d\phi }{ d\, t } =cos(\phi /2)\cfrac { dx }{ d\, t } \)
