9780030839931, Chapter 13, Problem 3P

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Chapter 13, Problem 3P Problem Show that the equations describing the electric and thermal currents, (13.45) and (13.50) to (13.53), continue to hold in the presence of a uniform magnetic field, provided that Eq (13.48) for σ(ε) is generalized to include the effects of the magnetic field by replacing the second v(K) by v(k), defined in Eq (13.70). Step-by-step solution Step of 3 The electrical current density and thermal current density from the distribution function is defined as (1) Here, is the matrix element and is defined in terms of Step 2 of 3 The structure of these results is simplified by defining In terms of which To evaluate equation (2) for metals, the fact that is negligible except within B of can be exploited. Since the integrands in and have factors that vanish when to evaluate them one the first temperature correction in the Somerfield expansion must retain to evaluate the equations. Thus, it is found an accuracy of order (3) (4) (5) Here, (6) Step 3 of 3 Equation (1) and equation (3) to equation (6) are the basic results of the theory of electronic contributions to the thermoelectric effects. Instead, the zero-field result must be replaced by (k) Here, is a weighted average of the velocity passing through (7) In the low field limit, the orbit is traversed very slowly, only points in the immediate vicinity of k contribute appreciably to the average in equation (7), and zero field result is recovered. Therefore, the equations describing the electric and thermal currents, equation (1) and equation (3) to equation (6), continue to hold in the presence of a uniform magnetic field, provided that equation (6) for is generalized to include the effects of the magnetic field by replacing the second (v(k)), defined in equation (7).