Temperature Redefined Old

From previously, the perpendicular temperature field due to temperature particles distributed over a surface is,

$p_{ d }=T_{ E }=\cfrac { \rho _{ { T } } }{ \tau _{ o } }$

where $\rho _{ { T } }$ is the temperature charge density per unit area,  $\tau _{ o }$ is the analogue to $\varepsilon_o$.

The force on a temperature charge, $T^{-}$ due to two temperature charge distributions, between a hot and cold surface is,

$T^{ - }.\left( p_{ dh }-p_{ dc } \right) =m_{ T }\cfrac { dv }{ dt }$

where $m_T$ is the mass of the temperature charge.  Over a contact surface, $A_c$,

$T^{ - }\int _{ A_{ c } }{ p_{ dh }-p_{ dc } } dA=T^{ - }\int _{ A_{ c } }{ \cfrac { \rho _{ h } }{ \tau _{ h } } -\cfrac { \rho _{ c } }{ \tau _{ c } }  } dA=\int _{ A_{ c } }{ m_{ T }(A)\cfrac { dv }{ dt }  } dA$

where $m_T(A)$ is the area mass distribution of the temperature charge in area $A_c$.  When we consider an area the size of one temperature charge, $A_T$,

$\cfrac { T^{ - } }{ m_{ T } } \int _{ A_{ T } }{ \cfrac{ \rho _{ h } }{\tau_h}-\cfrac{ \rho _{ c } } {\tau_c}} dA=\int _{ A_{ T } }{ \cfrac { dv }{ dt }  } dA$

$\cfrac { A_{ T }T^{ - } }{ m_{ T } } \left(  \cfrac{ \rho _{ h } }{\tau_h}-\cfrac{ \rho _{ c } } {\tau_c} \right) =\cfrac { dv }{ dt } A_{ T }$

When we replace, $\rho _{ h }$ with $T^{-}\rho _{ n }$,

$\rho =T^{-}.\rho _{ n }$

where $\rho _{ n }$ is the number density of temperature particle per unit area, the velocity of one temperature charge crossing from the hot body to the cold body is,

$\cfrac { dv }{ dt } =\cfrac { (T^{ - })^{ 2 } }{ m_{ T } } \left( \cfrac{ \rho _{ nh } }{\tau_h}-\cfrac{ \rho _{ nc } } {\tau_c} \right)$

At thermal equilibrium,

$\cfrac { dv }{ dt }=0\,\,\,\,\implies\,\,\,\,  \cfrac{ \rho _{n h } }{\tau_h}=\cfrac{ \rho _{n c } } {\tau_c}$  --- (*)

Only if both the hot and cold bodies are of the same material, in particular,

$\tau_h=\tau_c$    then,

${ \rho _{ nh } }={ \rho _{ nc } }$

If everything attain the same temperature on thermal contact at equilibrium, then there cannot be any sort of thermal insulation.  Expression (*) allows for materials of different $\tau$ to attain different $\rho_n$.

This points to $\rho_n$, the surface area number density of temperature particles as the temperature we measure with a contact thermometer.