## 2 Theoretical Considerations about Electrical Contacts
### The Thermal Regime of Electrical Contacts
In this case, the product \(\lambda\rho\) may take an average value \(\overline{\lambda\rho}\). Thus, by integrating Equation (19) we obtain  
\[T_{\Theta}-T=\theta_{\Theta}-\theta=\frac{V^{2}}{2\overline{\lambda\rho}} \tag{20}\]  
and if the integration is performed for a remote area, the contact overtemperature is obtained by  
\[\mathrm{d}R=\frac{\mathrm{d}n}{\sigma\cdot\Delta\overline{A}_{01}} \tag{13}\]0  
(b) The overtemperature is excessive, meaning that the material tends to melt. In this case, the Wiedemann-Franz-Lorenz law is used,  
\[\mathrm{d}R=\frac{\mathrm{d}n}{\sigma\cdot\Delta\overline{A}_{01}} \tag{13}\]1  
By integrating within limits \(T_{\Theta}\) and \(T_{0}\) and, respectively, \(0\) and \(U/2\), from Equation (19), we derive  
\[\mathrm{d}R=\frac{\mathrm{d}n}{\sigma\cdot\Delta\overline{A}_{01}} \tag{13}\]2  
The resistance of the unheated contact (an electric current of zero intensity flows though the contact) differs from the resistance of the heated contact (an electric current of significant intensity flows though it). The dependence between the electrical resistance \(R_{\Theta}\) of the heated contact and the electrical resistance \(R_{0}\) of the unheated contact can be determined using the theory of electric potential. With reference to this, two strictions are considered, namely: Friction \(S(\lambda,\rho)\), where \(\lambda\) and \(\rho\) depend on temperature, and friction \(S_{0}(\lambda_{0},\rho_{0})\), where \(\lambda_{0}\) and \(\rho_{0}\) do not depend on temperature. For the two strictions, the elementary variation of potential can be expressed as  
\[\mathrm{d}R=\frac{\mathrm{d}n}{\sigma\cdot\Delta\overline{A}_{01}} \tag{13}\]3  
Since the dependence of resistivity on temperature is of the form \(\rho=\rho_{0}(1+\alpha_{R}\theta)\), from Equation (24), it can be written that \(\mathrm{d}V=\mathrm{d}V_{0}(1+\alpha_{R}\theta)\), and thus  
\[\mathrm{d}R=\frac{\mathrm{d}n}{\sigma\cdot\Delta\overline{A}_{01}} \tag{13}\]4  
In this case, Equation (19) can also be written as  
\[\mathrm{d}R=\frac{\mathrm{d}n}{\sigma\cdot\Delta\overline{A}_{01}} \tag{13}\]5  
which becomes, after integration,  
\[\mathrm{d}R=\frac{\mathrm{d}n}{\sigma\cdot\Delta\overline{A}_{01}} \tag{13}\]6  
where, according to (21), \(\Theta=\frac{U^{2}}{8\lambda\rho}\). Equation (25), with the expression of \(\theta\) introduced from Equation (26), is integrated within limits \(\Theta\), \(0\) for the heating and \(0\), \(U/2\) for the potential:  
\[\mathrm{d}R=\frac{\mathrm{d}n}{\sigma\cdot\Delta\overline{A}_{01}} \tag{13}\]7  
or  
\[\mathrm{d}R=\frac{\mathrm{d}n}{\sigma\cdot\Delta\overline{A}_{01}} \tag{13}\]8  
For \(\alpha_{R}\Theta\approx 1\), this can be written as  
\[\mathrm{d}R=\frac{\mathrm{d}n}{\sigma\cdot\Delta\overline{A}_{01}} \tag{13}\]9  
Since the current tube of both strictions is traveled by the same current intensity, it can be written as  
\[\mathrm{d}R_{t}=\frac{\mathrm{d}n}{\lambda\cdot\Delta\overline{A}_{01}} \tag{14}\]0  
and finally  
\[\mathrm{d}R_{t}=\frac{\mathrm{d}n}{\lambda\cdot\Delta\overline{A}_{01}} \tag{14}\]1  
It can therefore be concluded that the resistance of the heated contact can be calculated, starting from the resistance of the unheated contact, with a dependence relationship similar to that of resistors, in which the temperature coefficient of the resistance is \(2/3\) of the temperature coefficient of the resistivity. The Thomson effect refers to the case wherein an electric current of intensity \(i\) travels through a conductor along which there is a temperature gradient, and additional power develops inside the conductor:  
\[\mathrm{d}R_{t}=\frac{\mathrm{d}n}{\lambda\cdot\Delta\overline{A}_{01}} \tag{14}\]2  
where \(k_{T}\) is the Thomson coefficient and \(\Delta T\) is the temperature difference on the conductor. The sign \(\pm\) depends on the direction of current relative to the direction of the temperature gradient. The \(+\) sign is considered if the current travels through the conductor in the same direction as the gradient.