\(U_B=\cfrac{\mu_o}{32}\left( \cfrac { e }{ \pi a_{ e } } \right) ^{ 2 }\left( sin(2\phi )sin(\phi /2)\cfrac { d\phi }{ dt } \right) ^{ 2 }\left( \int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 4 }(\theta ) } h(\phi ,\theta ) } { d\, (cos(\theta )) } \right) ^{ 2 }\)
where,
\(h(\phi ,\theta )=e^{ -\cfrac { 2r_{ e }sin(\phi /2) }{ a_{ e }cos(\theta ) } }\)
What interest us here is the term,
\( \left(sin(2\phi )sin(\phi /2)\cfrac { d\phi }{ dt }\right)^2 \)
where \(\phi\) is revolves along the orbit of the charge. Imagine a cylinder defined by this motion at a height of \(y\) perpendicular to the plane of the orbit. The wall of such a cylinder cuts the \(B\) field established by the moving charge perpendicularly, at the highest point of the loop center at the circumference perpendicular to the tangent of the orbit.
Considering the \(B\) field alone, the force line does no work as the motion is in a perpendicular direction, and the charge in constant \(\cfrac{d\,\phi}{d\,t}\) around the orbit experience no change in kinetic energy.
But,
\(\cfrac{d\,U_B}{d\,\phi}=A\cfrac{d}{d\,\phi}\left\{\left( sin(2\phi )sin(\phi /2)\cfrac { d\phi }{ dt } \right) ^{ 2 }\left( \int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 4 }(\theta ) } h(\phi ,\theta ) } { d\, (cos(\theta )) } \right) ^{ 2 }\right\}\)
where \(A\) is the preceding constant and \(\theta_o\) is up to where the cylinder cuts the \(B\) field at \(y\).
Consider
\(\Omega=\cfrac{d}{d\,\phi}\left\{\left( sin(2\phi )sin(\phi /2)\cfrac { d\phi }{ dt } \right) ^{ 2 }\left( \int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 4 }(\theta ) } h(\phi ,\theta ) } { d\, (cos(\theta )) } \right) ^{ 2 }\right\}\)
Consider the first term,
\(2sin(2\phi )sin(\phi /2)\cfrac { d\phi }{ dt }\left \{2cos(2\phi)sin(\phi /2)\cfrac { d\phi }{ dt }+\cfrac{1}{2}sin(2\phi )cos(\phi /2)\cfrac { d\phi }{ dt }\right\}\)
\(=\left\{ sin(4\phi )sin^{ 2 }(\phi /2)+\cfrac { 1 }{ 2 } sin^{ 2 }(2\phi )sin(\phi ) \right\} \left( \cfrac { d\phi }{ dt } \right) ^{ 2 }\)
Consider the second term,
\(2\int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 4 }(\theta ) } h(\phi ,\theta ) } { d\, (cos(\theta )) } \cfrac{d}{d\,\phi}\int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 4 }(\theta ) } h(\phi ,\theta ) } { d\, (cos(\theta )) }\)
\(=2\int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 4 }(\theta ) } h(\phi ,\theta ) } { d\, (cos(\theta )) } \cfrac{d}{d\,\phi}\int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 4 }(\theta ) } h(\phi ,\theta ) } { (cos(\phi )sin^{ 3 }(\theta )d\phi ) }\)
From the post "Catastrophe! No Small Change",
\(cos(\phi )sin^{ 3 }(\theta )d\phi =dcos(\theta )\), so,
\(=2\int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 4 }(\theta ) } h(\phi ,\theta ) } { d\, (cos(\theta )) }.{ \cfrac { 1 }{ cos^{ 4 }(\theta_o ) } h(\phi ,\theta_o ) } {cos(\phi )sin^{ 3 }(\theta_o ) }\)
Therefore,
\(\Omega=\left( sin(2\phi )sin(\phi /2)\cfrac { d\phi }{ dt } \right)\left( \int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 4 }(\theta ) } h(\phi ,\theta ) } { d\, (cos(\theta )) } \right). \\\left[\cfrac { d\phi }{ dt } \left \{2cos(2\phi)sin(\phi /2)+\cfrac{1}{2}sin(2\phi )cos(\phi /2)\right\}\left( \int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 4 }(\theta ) } h(\phi ,\theta ) } { d\, (cos(\theta )) } \right)+{ \cfrac { 2 }{ cos^{ 4 }(\theta_o ) } h(\phi ,\theta_o ) } {cos(\phi )sin^{ 3 }(\theta_o ) }\left( sin(2\phi )sin(\phi /2)\cfrac { d\phi }{ dt } \right)\right] \)
And we have the change in energy with time as,
\(\cfrac{d\,U_B}{d\,t}=\cfrac{d\,U_B}{d\,\phi}\cfrac { d\phi }{ dt }=A\Omega\cfrac { d\phi }{ dt } \)
Clearly \(\Omega\ne0\) for all \(\phi\). This implies that the charge while in orbit constantly absorb and radiate energy. A plot of,
\(O=\left(sin(2\phi )sin(\phi /2)\right)^2\) and \(sin(\phi/2)\) below clearly shows the periodicity in \(U_B\) of \(2\pi\).
And that the net change in \(U_B\) over one period is zero as the graph returns to the same point (zero) at the end of the period.
Together with the previous post "Diffusion, Only If...", we have a non zero expression for \(A(x,\dot{x})\),
\(\cfrac { \partial\,P }{ \partial\,t }=-\cfrac{1}{2}A(x,\dot{x})=\cfrac{d\,U_B}{d\,t}\)
This is the mechanism by which energy is propagated throughout a material. The orbiting charges are constantly radiating and absorbing energy, with no net change over a period of the orbiting motion.
