Thermoelectricity is a branch of physics that studies the direct conversion of temperature differences into electrical voltage (and vice versa). A thermocouple is a device made of two dissimilar metals that utilizes these effects.
When the two junctions of a thermocouple are held at different temperatures (a "hot" junction T and a "cold" reference junction T0, usually 0°C), a thermo-EMF (E) is produced.
For most metal pairs, the relationship between EMF (E) and the hot junction temperature (T) is a parabolic function:
Formula: E = aT + (1/2)bT2
Where 'a' and 'b' are thermoelectric constants specific to the pair of metals.
This is the temperature of the hot junction at which the thermo-EMF (E) is maximum.
To find it, we find where the rate of change of EMF is zero: dE/dT = 0.
dE/dT = a + bT. Setting to zero: a + bTn = 0
Formula: Tn = -a / b
This is the temperature of the hot junction (other than the reference T=0) at which the thermo-EMF becomes zero again. The EMF changes direction (inverts) beyond this temperature.
We set E = 0: aTi + (1/2)bTi2 = 0
Ti (a + (1/2)bTi) = 0. This gives T=0 or Ti = -2a / b.
Formula: Ti = -2a / b
Important Relation: Ti = 2Tn. The neutral temperature is exactly halfway between the cold junction (0°C) and the inversion temperature.
Explanation: The Seebeck effect is the production of a voltage (thermo-EMF) in a circuit made of two dissimilar conductors (e.g., Copper and Iron) when their two junctions are maintained at different temperatures (TH and TC).
This is the basic principle of a thermocouple, which is used as a thermometer.
The Seebeck Coefficient (S), or Thermoelectric Power, is the local voltage produced per unit temperature difference: S = dV/dT.
Explanation: The Peltier effect is the reverse of the Seebeck effect. When an electric current (I) is passed through a junction of two dissimilar metals, heat is either absorbed or liberated at that junction, causing it to heat up or cool down.
The direction of heating/cooling reverses if the current direction reverses. This is the basis for thermoelectric coolers (Peltier coolers).
The Peltier Coefficient (Π) relates the rate of heat (Q/t) absorbed or liberated to the current (I):
Formula: (Q/t) = Π * I
The unit of Π is Joules/Coulomb, which is Volts.
Explanation: The Thomson effect describes the absorption or liberation of heat when an electric current flows through a single conductor that has a temperature gradient along its length.
Prediction: William Thomson (Lord Kelvin) predicted this effect using thermodynamics. He argued that if current flows from a hot end to a cold end, the Seebeck and Peltier effects alone do not conserve energy, and a third effect must exist.
The Thomson Coefficient (σ) relates the heat generated (dQ) to the charge (dq) and the temperature difference (dT): dQ = σ dq dT.
The total EMF in a thermocouple circuit is the sum of all Seebeck and Thomson EMFs in the loop. Two fundamental laws govern this behavior:
Statement: If a thermocouple (A, B) produces an EMF E12 with its junctions at temperatures T1 and T2, and an EMF E23 with junctions at T2 and T3, then the EMF E13 it produces with junctions at T1 and T3 is:
Formula: E13 = E12 + E23
Use: This law allows us to calibrate a thermocouple at a reference temperature (like 0°C) and then calculate the EMF for any other pair of temperatures.
Statement: The introduction of a third, different metal (C) into a thermocouple circuit (A-B) will not change the total EMF, provided that the two new junctions created by inserting metal C are at the same temperature.
Use: This is a crucial practical law. It means we can insert a voltmeter (which is a "third metal") into a thermocouple circuit to measure its EMF without altering the reading, as long as both terminals of the voltmeter are at the same (room) temperature.
Thermoelectric Power (S) is another name for the Seebeck Coefficient. It is defined as the rate of change of the thermo-EMF (E) with respect to temperature (T).
Definition: S = dE / dT
If we use the parabolic relation E = aT + (1/2)bT2, then the thermoelectric power is:
Formula: S = a + bT
This shows that the thermoelectric power (S) is a linear function of temperature.
This is a graph of the Thermoelectric Power (S) versus Temperature (T). Since S = a + bT, this graph is a straight line.
Use: The area under this (S vs. T) graph between two temperatures (T1 and T2) gives the total thermo-EMF generated between those temperatures.
E = ∫T1T2 S dT = Area under S-T graph
Lord Kelvin applied the laws of thermodynamics (particularly the Second Law) to the thermoelectric circuit, treating it as a reversible heat engine. This led to two critical relationships (Kelvin's Relations) that link the three thermoelectric coefficients (S, Π, σ).
It connects the Peltier coefficient (Π) at a junction with the Seebeck coefficient (S) of the two metals at that temperature (T, in Kelvin).
Kelvin's 1st Relation: ΠAB = T * SAB
Where SAB = dE/dT. This shows that the Peltier effect (heating/cooling) and Seebeck effect (voltage) are fundamentally linked.
It connects the difference in the Thomson coefficients (σA, σB) of the two metals to the second derivative of the EMF (i.e., the slope of the S-T graph).
Kelvin's 2nd Relation: σA - σB = T * (d2E / dT2) = T * (dSAB / dT)
This relation proves that if the EMF-Temperature graph (E vs. T) is not a straight line (i.e., d2E/dT2 ≠ 0), then the Thomson effect *must* exist.
These equations are the theoretical foundation of thermoelectricity.