= Seebeck effect
{c}
The <Seebeck effect> is the generation of a thermoelectric <electromotive force> by a <temperature> difference. In a <thermocouple>, two different homogeneous <electrical conductors> A and B connect a measuring junction at $T_h$ to two reference connections at a common $T_c$. With negligible <electric current>, the measured terminal voltage is determined by the difference of their <Seebeck coefficients>. A hot junction alone does not specify the reading: the reference <temperature>, the second material, and the terminal order matter.
\Image[/seebeck-thermocouple-experiment.png]
{title=Two-material <thermocouple>: junction <temperatures> and the voltage sign}
{description=An idealized open-circuit <Seebeck effect> experiment. The A and B wires meet at the measuring junction; each reference connection meets an identical meter lead at the same $T_c$. The positive meter input connects to A. The shown $20\,{}^\circ\mathrm C$, $70\,{}^\circ\mathrm C$, and $+2\,\mathrm{mV}$ belong to the explicitly toy constant-<Seebeck coefficient> example below, not measured data.}
For a motivating experiment, join one end of A to one end of B, insulating the wires elsewhere. Connect their free ends to identical meter leads at reference <temperature> $T_c$, for example in an ice-water <heat bath>. A meter with input <electrical resistance> much larger than the <thermocouple>'s <electrical resistance> approximates an open circuit, meaning $I\approx0$. Keep the measuring junction in a separately controlled <heat bath> at $T_h$ and wait for stable junction <temperatures>. Both reference connections must share $T_c$; identical leads with identical endpoint <temperatures> have cancelling thermoelectric contributions, allowing the meter to be elsewhere. https://www.nist.gov/system/files/documents/director/planning/report97-1.pdf[Marx, Link and Scott, Economic Assessment of the NIST Thermocouple Calibration Program (1997), §2.1]{ref}.
Define the reading as $V_{AB}=V_A-V_B$, with the positive input on the A reference terminal and the negative input on B. Use the <Seebeck coefficient> convention in which the voltage change along a homogeneous wire at zero <electric current> is $dV=-S(T)\,dT$. Following the path from A at $T_c$ to the common junction at $T_h$, then back along B to $T_c$, gives
$$
V_{AB}(T_h,T_c)=\int_{T_c}^{T_h}\bigl[S_A(T)-S_B(T)\bigr]\,dT.
$$
This formula assumes homogeneous materials whose <Seebeck coefficients> depend only on the local <temperature>, and no additional uncompensated lead contributions. Here voltage is the electrical transport voltage associated with <electrochemical potential>, rather than a direct measurement of an isolated equilibrium contact <electric potential>.
Start with $T_h=T_c$, then increase $T_h$ while holding $T_c$ fixed and record $V_{AB}$ at each stable setting. In the ideal model the equal-<temperature> reading is zero. If $S_A-S_B>0$ throughout the interval, A reads positive when $T_h>T_c$. Exchanging meter inputs reverses the reading; exchanging the two junction <temperatures> reverses the oriented <integral>. Repeating the run with identical homogeneous materials gives zero because $S_A=S_B$. These controls test the stated model; a nonzero experimental baseline should be measured rather than silently removed by assuming it is zero.
For an explicitly toy linear example, stipulate constant <Seebeck coefficients> $S_A=25\,\mu\mathrm{V/K}$ and $S_B=-15\,\mu\mathrm{V/K}$. Choose reference <temperature> $T_c=20\,{}^\circ\mathrm C$ and measuring-junction <temperature> $T_h=70\,{}^\circ\mathrm C$. A <temperature> difference has the same numerical size in <kelvins> and on the <degree Celsius> scale, so
$$
S_A-S_B=40\,\mu\mathrm{V/K},\qquad \Delta T=50\,\mathrm K,
\qquad V_{AB}=(40\,\mu\mathrm{V/K})(50\,\mathrm K)=+2.00\,\mathrm{mV}.
$$
Swapping the inputs gives $-2.00\,\mathrm{mV}$. Bringing both junction regions to the same <temperature> gives zero. With <temperature>-dependent <Seebeck coefficients>, shifting both junction <temperatures> equally while keeping their difference fixed can change the reading. No actual material pair is claimed to have the stipulated coefficients over this interval.
\Image[/seebeck-toy-voltage-slope.png]
{title=Toy <Seebeck effect> calibration: voltage slope and reversed inputs}
{description=Calculated lines, not measured data. The solid line is $V_{AB}=(40\,\mu\mathrm{V/K})\Delta T$; the dashed line shows the same device with reversed meter inputs. The <slope> is the difference of the two <Seebeck coefficients>, not either coefficient separately. The reference <temperature> is fixed at $20\,{}^\circ\mathrm C$; negative $\Delta T$ means the measuring junction is colder than the reference connections.}
Real <thermocouples> need a calibrated voltage-versus-<temperature> relation: the constant-<Seebeck coefficient> approximation is useful only where the coefficient difference changes little. Wire inhomogeneity can make the result depend on where the <temperature gradient> lies, and reference connections at unequal or unknown <temperatures> introduce errors. An electronic reference correction uses a measured reference <temperature> in place of an ice bath. https://www.nist.gov/system/files/documents/director/planning/report97-1.pdf[Marx, Link and Scott (1997), §2.1: reference correction and wire homogeneity]{ref}. Burns and colleagues provide the standardized reference functions for common letter-designated <thermocouple> types; the toy line above is not a replacement for them. https://www.nist.gov/publications/temperature-electromotive-force-reference-functions-and-tables-letter-designated[Burns et al., Temperature-electromotive force reference functions and tables (1993), NIST Monograph 175]{ref}.
The historical experiment used a different detector. In his 1821–1823 investigations, Seebeck found that a closed <electrical circuit> of two dissimilar materials with junctions at different <temperatures> deflected a compass needle. The modern explanation is that thermoelectric <electromotive force> drives an <electric current>, whose <magnetic field> deflects the needle. The open-circuit experiment above detects voltage while drawing negligible <electric current>. The striking connection is that a thermal imbalance can become an electrical signal and then a magnetic response. https://thermoelectrics.matsci.northwestern.edu/thermoelectrics/history.html[Northwestern thermoelectrics research group: Seebeck's compass experiment]{ref}.
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