If equation 4.18b instead of 4.18a is substituted in 4.16 we get for Ф, assuming B = 0, the following equation (instead of equation 4.21)

Note again that in the above equation we have written q= -e. Using the general expression for the current density, equation 4.28, and following the same steps as in equations 4.28-4.30, we find that for the current density in the presence of a temperature variation we can write



Equation 4.42a may be written after some standard manipulations involving differentiation of products of functions


is called the Seebeck coefficient. Now we consider an open circuited bar with a temperature difference at its two ends as in figure 4.5. No electric field is applied, but an electric field

A temperature difference Г — 7j creates an electric field in a bar, as shown

FIGURE 4.5 A temperature difference Г2 — 7j creates an electric field in a bar, as shown.

develops in the bar because electrons move in a transient condition from the hot end to the cold end. At steady state, however, J will be zero because we have an open circuit. Then

As a result of the movement of the carriers and the establishment of the electric field £ in the bar, a potential difference AV appears between the ends of the bar. This is the Seebeck effect. A slightly more complicated experimental configuration than that of figure 4.5 is needed to measure this difference as is shown in figure 4.6. The two junctions /1, /2 between the two metals A and В are kept at different temperatures Tj and T2. At some point anywhere in metal A away from the two junctions a small cut is made creating two end points, call them P3 and P4, between which the potential difference AV is measured. The points P3 and P4 are kept at the same temperature Tp. We will have

The line integrals can be taken along any path joining P3 to P4.

Experimental set-up for the measurement of the Seebeck coefficient.


Since the temperature at P3 and P4 is the same (Tp), the line integral in the RHS of equation 4.45 involving the gradient of E? is zero. For the line integral involving the Seebeck coefficient we will have

Equation 4.46 is actually used to measure the Seebeck or thermoelectric power coefficient of a metal. The material lead is used as metal A which has a negligible Seebeck coefficient and then ДV is measured so as to obtain Sg(T).

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