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.
