How Does Electron Diffusion Affect Current Flow in Cylindrical Metal Wires?

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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.

 

[azntoon]

The initial displacement is x0 = 0 at t = 0.(b) Repeat the process for the case where an electric field E is applied. The drift velocity can be obtained as, vd = ½a + E. Plot P(x) for the same values of t as in part (a).Solution: (a) In the absence of an electric field, the motion of electrons in a cylindrical metal wire of cross sectional area A = 1 mm2 and length, L = 1 cm, composed of a material with electron density N = 1029/m3 can be described using the classical free electron model. In this model, the electrons have a mean free time of 3 ¡Ñ 10-12 sec and are equally likely to move along or opposite the axis of the wire with a speed vRMS appropriate to the temperature of the wire. At room temperature (T = 300 K), the average speed of the electrons is approximately vRMS = 8.9 ¡Ñ 105 m/s. The probability distribution of P(x), for the displacement x of the electron from its initial position at t = 0, can be obtained by following the history of 10,000 electrons starting from the time they arrive at the point x = 0. The displacement after i steps is given by xi+1 = xi ¡Ó vRMS + ½a 2, where either the + or ¡V sign is chosen randomly. The initial displacement is x0 = 0 at t = 0. For t = 100t, 500t; and 2500t;, the probability distribution P(x) can be plotted by breaking the range of x values into at least 100 bins of equal width dx and plotting 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). (b) When an electric field E is applied, the motion of electrons in a cylindrical metal wire is described using the classical free electron model. In this model, the electrons have a mean free time of 3 ¡Ñ

 

[R0man]

Electron diffusion is the process by which electrons move through a material due to scattering. In this scenario, we have a cylindrical metal wire with a cross-sectional area of 1 mm2 and length of 1 cm. The wire is made of a material with an electron density of 1029/m3 and a mean free time of 3x10-12 seconds. At room temperature (T = 300 K), the electrons in the wire have a root mean square speed, vRMS, which is appropriate for the wire's temperature.

In the classical free electron model, the current density can be expressed as j = -eNvd, where e is the charge of an electron, N is the electron density, and vd is the drift velocity. This drift velocity is the average additional velocity gained by an electron due to acceleration during the mean free time, τ. It can be calculated using the formula vd = 1/2aτ, where a is the acceleration and E is the electric field due to the applied voltage, V. In this case, the electric field is given by E = V/L, where L is the length of the wire.

To plot the probability distribution of P(x) for the displacement x of an electron from its initial position at t = 0, we need to follow the history of 10,000 electrons starting from x = 0. The range of x values should be divided into at least 100 bins of equal width, dx. The probability distribution is then given by P(x) = N(x)/(10,000dx), where N(x) is the number of electrons that arrived between x and (x + dx). This value is also equal to the fraction of electrons that have a displacement between x and (x + dx), which is the probability of this result occurring.

To calculate the displacement after i steps, we use the formula xi+1 = xi + vRMSτ + 1/2aτ2, where the + or - sign is chosen randomly. This process is repeated for t = 100τ, 500τ, and 2500τ, and the resulting probability distribution is plotted for each time interval. This will give us a better understanding of the motion of electrons and the resulting current flow in the wire without the application of an electric field.

In summary, electron diffusion is a complex process that can be studied using the classical free electron model. By following the history of electrons