# Thomson effect: what is it?

This post is the second part about thermoelectricity. Thomson effect is a reversible thermoelectric phenomenon and related to Seebeck and Peltier effects.

I recommend that you read the post about Seebeck and Peltier effects before continue.

### Seebeck coeficient

Adding important information about the other two reversible thermoelectric phenomena. The equation of current density $J$ in a conductor during Seebeck effect is:

$J=\sigma (-\Delta V+E_{emp})$

• $\sigma$ is material’s conductivity.
• $\Delta V$ is potential difference.
• $E_{emp}$ is electromotive force.

Electromotive force calculation.

$E_{emp}=-S\cdot \Delta T$

• $S$ is Seebeck coefficient in V/K (volt/kelvin).
• $\Delta T$ is temperature difference between junctions.

Seebeck coefficient can also be calculated in this form.

$S=-\frac{\delta V}{\delta T}$

• $\delta V$ and $\delta T$ are voltage and temperature variations respectively.

### Peltier coeficient

Below is the equation absorbed ou transmitted $Q$ by time unit in the junction by Peltier effect.

$Q=(\Pi _{A}-\Pi _{B})\cdot I$

$\Pi _{A}$ and $\Pi _{B}$ are Peltier coefficients of materials A and B, represent the quantity of heat by electric charge in Joules/Coulomb (J/C). $I$ is electric current. Peltier coefficient $\Pi$ can be calculated from Seebeck coefficient.

$\Pi _{AB}=\Pi _{A}-\Pi _{B}=S_{AB}\cdot \Delta T$

• $\Delta T$ is temperature difference between the junctions.
• $S_{AB}=S_{A}-S_{B}$.

### The Thomson effect

Was discovered in the 1850s, by the British physicist, mathematician, and engineer Willian Thomson Kelvin. The same one who introduced the Kelvin scale temperature. According to the Thomson effect, when an electric current passes through a homogeneous conductor, the latter have temperature gradient, or difference in extremities, the conductor absorbs or emits heat. Depending on directions of electric current and temperature gradient. J_{C} is the current density. Heat flows from hot to cold. Source: (Uchida, 2020).

What if you heat up the middle of a conductor? On one side the heat will be absorbed and on another, emitted. This is part of the experiment diagram that confirmed the Thomson effect. An iron rod in “U” shape is heated on C point. R_{1} and R_{2} resistances are linked to Wheatstone bridge. With heat absorption and emission, resistances change and the bridge becomes unbalanced. Source: David Darling.

#### Positive and negative Thomson effect

Considers electric current flowing from A to B in a conductor, with a heat source on C. In materials with a positive Thomson effect, heat from A to C is absorbed but liberated on another conductor’s side. Metals that show the positive effect are copper (Cu), cadmium (Cd), zinc (Zn), antimony (Sb), silver (Ag), and stanium (Sn).

While on negative Thomson effect, heat is liberated from A to C, but absorbed from C to B. Metals that show the negative effect are: iron (Fe), cobalt (Co), nickel (Ni), bismuth (Bi), platinum (Pt) and mercury (Hg).

Without an electrical current, M and N points have the same temperature. However, when there is a current, N will have a higher temperature than M in the positive effect. While on negative effect, the temperature on point M will be bigger than N.

#### Heat equation and Thomson coefficient

Equation of produced heat $q$ by volume unit.

$q=\rho J^2-\mu J\frac{dT}{dx}$

• $\rho$, resistivity.
• $J$, current density.
• $\mu$ is Thomson coefficient in volts per degree, can be positive or negative, depending on the material.
• $\frac{dT}{dx}$, temperature gradient.

The first term of equation is simply heating by Joule law, which isn’t reversible. The second term is Thomson heat, which changes signal when J changes direction. 