**Thomson effect**, discovered by William Thomson (Lord Kelvin) in 1854, is a thermoelectric effect that is manifested with the onset of a gradient of electric potential (and therefore an electromotive force, called in this case Thomson electromotive force) in a conducting material, when it is subjected to a temperature gradient.

In qualitative terms, the explanation of this effect consists in the fact that charge carriers, diffusing inside the conducting material, reach a lower density in the higher temperature areas; therefore, if the charges in question are electrons, it will be the hotter end to be at higher potential (positive Thomson coefficient), while if they are gaps it will be the colder end to be at higher potential (negative Thomson coefficient).

Instead in quantitative terms, we can say that in a conducting material subjected to a temperature gradient (therefore not isothermal) the Fermi level is not homogeneous but it follows the temperature trend, because of the fact that Fermi energy for charge carriers is a slightly decreasing function of temperature, this involves, therefore, the emergence of electromotive force (Thomson).

Thomson electromotive force is directly proportional to the temperature difference \(T_2-T_1\) between the ends of the conductor, and it does not depend neither from geometrical shape, nor from extension of conductor, nor from the way temperature varies along the conductor (law of independence from thermal gradient).

Therefore thinking to analyze the phenomenon from a mathematical point of view, considering a portion of a threadlike conductor along \(dl\) subjected to a temperature gradient, the temperature variation will be linear \(dT\) and the electromotive force \(E\) at its extremes will be equal to:

\[dE=\tau_{M,T}dT\]

Overall instead:

\[E_{M,T_2,T_1}=\int_{T_1}^{T_2} \tau_{M,T} dT\]

where \(\tau_{M,T}=[\mu V/K]\) is the Thomson coefficient, dependent on the nature of the material (M) and the temperature (T) (e.g. for copper \(\tau_{Cu}=+2\), for iron \(\tau_{Fe}=-8\), for platinum \(\tau_{Pt}=+13\), for constantan \(\tau=-25)\); to positive values of \(\tau\) corresponds an electromotive field directed in the same direction of the temperature gradient, i.e. in the conductor there is a migration of negative charges towards the end at lower temperature.

Thomson electromotive force has a very low value, comparable with that of electromotive forces due to thermal noise, so it is convenient to use an indirect measurement of their value, based on the fact that if the conductor is crossed by a flow of electric current, this will give life to Joule effect, that is will develop heat that will be exchanged with the surrounding environment, and whose direction is concordant with the direction of the current. The heat developed in this way is called Thomson heat, and corresponds to the energy of Thomson electromotive field; it is worth:

\[dQ=I\tau \Delta T \Delta t\]

where \(I\) is the current intensity assumed positive if of concordant direction in the direction of increasing temperatures, \(\Delta T\) the temperature difference and \(\Delta t\) the time interval considered.