# Thermo-electricity

## Seebeck effect

If a thick iron wire is welded at its ends A and B to two pieces of a thick copper wire and the free ends of the copper wire are connected to a galvanometer, the galvanometer shows a deflection when one of the junction is kept cold and the other is heated. It is found that a current flows from copper to iron at the hot junction.

If the temperature of the hot junction is raised keeping the temperature of the cold junction constant, the emf increases and becomes maximum at a certain temperature of the hot junction. This temperature is called the neutral temperature. Its value depends upon the pair of metals chosen and is independent of the temperature of the cold junction. The neutral temperature for copper-iron is 270 deg C.

If the temperature of the hot junction is increased beyond the neutral temperature the emf begins to decrease and becomes zero at a certain temperature known as the temperature of inversion. The temperature of inversion is not constant for the given metals pair but depends upon the temperature of the cold junction. It is as much above the neutral temperature as the neutral temperature is above that of the cold junction.

The thermo-emf $e$ is related to the temperature of the hot junction $T$ by \begin{align} e=aT+\frac{1}{2}bT^2 \end{align} The thermo-electric power is defined as \begin{align} \mathrm{d}e/\mathrm{d}t=a+bT. \end{align} The thermo-electric power is zero at the neutral temperature $T_n$ given by \begin{align} T_n=-a/b. \end{align} The thermo-emf is zero at the inversion temperature given by \begin{align} T_i=-2a/b. \end{align}

A thromo-couple is based on the Seebeck effect. It is used to measure the temperature. The antimony-bismuth thermo-couple is one of the most sensitive thermo-couple. The direction of current is from antimony to bismuth at the cold junction (ABC).

Thermo-electric series.

## Peltier effect

If a current is passed through a circuit consisting of two dis-similar metals (thermo-couple), there is either an evolution or absorption of heat at the junctions. This effect is known a Peltier effect. The direction of current decides whether a junction gets heated or cooled. Reversing the direction of current reverses the heating/cooling of the junctions. It is a reversible effect.

The Peltier effect occurs due to contact potential difference at the junction of two dis-similar metals.

The thermoelectric cooler works on the Peltier effect.

The emf is given by \begin{align} e&=\frac{\Delta H}{\Delta Q} \\ &=\frac{\text{Peltier heat}}{\text{charge transferred}}. \end{align}

## Thomson effect

When a current is passed through a non-uniformly heated conductor (i.e., whose different parts are at different temperatures), there is an evolution or absorption of heat in the conductor. The direction of current decides whether the conductor gets heated or cooled. This effect is known as Thomson effect and is reversible.

In an unequally heated conductor, different parts are at the different potentials and hence there will be a potential gradient along the conductor.

The emf is given by \begin{align} e&=\frac{\Delta H}{\Delta Q} \\ &=\frac{\text{Thomson heat}}{\text{charge transferred}}=\sigma \Delta T. \end{align}

## Problems from IIT JEE

Problem (JEE Mains 2006): A thermocouple is made from two metals, Antimony and Bismuth. If one junction of the couple is kept hot and the other is kept cold then, an electric current will

1. flow from Antimony to Bismuth at the cold junction
2. flow from Antimony to Bismuth at the hot junction
3. flow from Bismuth to Antimony at the cold junction
4. not flow through the thermocouple

Solution: The current will flow from antimony to bismuth at the cold junction. At the cold junction, the current flows from metal with a higher Seebeck coefficient (47 $\mu$V/K for antimony) to a metal with a lower Seebeck coefficient (-72$\mu$V/K for bismuth). See the thermoelectric series for more details.

Problem (JEE Mains 2004): The thermo emf (in volts) of a thermocouple varies with the temperature $T$ of the hot junction as \begin{align} V=a(T-T_c)+b(T-T_c)^2, \nonumber \end{align} where $T_c$ is the temperature of the cold junction and the ratio $a/b$ is 700 deg C. If the cold junction is kept at 0 deg C, then the neutral temperature is

1. 700 deg C
2. 350 deg C
3. 1400 deg C
4. no neutral temperature is possible for this thermocouple

Solution: The thermo emf attains its maximum value at the neutral temperature ($T_n$). Thus, $\mathrm{d}V/\mathrm{d}T=0$ at $T=T_n$ i.e., \begin{align} a+2b(T_n-T_c)=0.\nonumber \end{align} Solve to get the neutral temperature \begin{align} T_n & =T_c-\frac{a}{2b} \\ &=0-\frac{700}{2}\\ &=-350\;\text{deg C}.\nonumber \end{align} But $T_n\nless T_c$, hence no neutral temperature is possible for this thermocouple.

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