Passing It On...

From the post "Catastrophe! No Small Change",

$U_B=\cfrac{\mu_o}{32}\left( \cfrac { e }{ \pi a_{ e } }  \right) ^{ 2 }\left( sin(2\phi )sin(\phi /2)\cfrac { d\phi  }{ dt }  \right) ^{ 2 }\left( \int _{ 0 }^{ \theta _{ o } }{ \cfrac { 1 }{ cos^{ 4 }(\theta ) } h(\phi ,\theta ) } { d\, (cos(\theta )) } \right) ^{ 2 }$

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.