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A cylindrical metal wire of cross sectional area A = 1 mm2 and length, L = 1 cm, is composed of a material with electron density N = 1029/m3. As a result of scattering, it has a mean free time of ƒä torque = 3 ¡Ñ 10-12 sec. Describe the motion of electrons and the resulting current flow in the classical free electron model under the simplifying assumption that at any moment the electron is equally likely to be moving along or opposite the axis of the wire with a speed is vRMS appropriate to the temperature of the wire. The wire is in a room at a temperature T = 300 K. Recall that the current density in this model can be expressed as j = -eNvd. The drift velocity was shown to be the average of the additional velocity resulting from the acceleration of an electron during the time, , such that vd = ½a, where , and E is the electric field due to the applied voltage, V, E = V/L. This gives, .

(a) For t = 100t, 500t; and 2500t;, plot the probability distribution of P(x), for the displacement x of the electron from its initial position at t = 0, in the case that no electric field is applied. Recall that since P(x) is a probability distribution, such that P(x)dx is equal to the probability that the electron has reached a value x between x and (x + dx). The probability of finding an electron at some value x is unity so You may obtain P(x) by following the history of 10,000 electrons starting from the time they arrive at the point x = 0. Break the range of x values into at least 100 bins of equal width dx and plot the number of electrons that arrived between the points x and (x + dx), N(x), divided by the total number of points and also divide by dx, to obtain P(x) = N(x)/(10000dx). Note that P(x)dx = N(x)/10000 is then equal to the fraction of electrons that have a displacement between x and (x + dx), which is nothing but the probability that this result will occur. If the displacement after i steps is xi, then the displacement after the next step is given by,

xi+1 = xi ¡Ó vRMS + ½a 2, where either the + or ¡V sign is chosen randomly.

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# Electron diffusion

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