Microscopic View Of Ohm’s Law
To find out why particular materials obey Ohm’s law, we must look into the details of the conduction process at the atomic level. Here we consider only conduction in metals, such as copper. We base our analysis on the free-electron model, in which we assume that the conduction electrons in the metal are free to move throughout the volume of a sample, like the molecules of a gas in a closed container. We also assume that the electrons collide not with one another but only with atoms of the metal. According to classical physics, he electrons should have a Maxwellian speed distribution somewhat like that of the molecules in a gas. In such a distribution (see Section 20-7), the average electron speed would be proportional to the square root of the absolute temperature. The motions of electrons are, however, governed not by the laws of classical physics but by those of quantum physics. As it turns out, an . assumption that is much closer to the quantum reality is that conduction electrons i a metal move with a single effective speed very, and this speed is essentially independent of the temperature. For copper, Vera = 1.6 X 106 m/s.
When we apply an electric field to a metal sample, the electrons modify their random motions slightly and drift very slowly-in a direction opposite that of the field-with an average drift speed Vd’ As we saw in Sample Problem 27-3, the drift speed in a typical metallic conductor is about 5 X 10-7 mIs, less than the effective speed (1.6 X 106 m/s) by many orders of magnitude. suggests the relation between these two speeds. The gray lines show a possible random path for an electron in the absence of an applied field; the electron proceeds from A to 8, making six collisions along the way. The green lines show how the same events might occur when an electric field E is applied. We see that the electron drifts steadily to the right, ending at B’ rather than at B. was drawn with the assumption that d = 0.021·eff’ However, because the actual value is more like the drift displayed in the figure is greatly exaggerated. The motion of the conduction electrons in an electric field E is thus a combination of the motion due to random collisions and that due to E. When we consider all the free electrons, their random motions average to zero and make no contribution to the drift speed. Thus, the drift speed is due only to the effect of the electric field on the electrons. If an electron of mass m is placed in an electric field of magnitude E, the electron will experience an acceleration given by Newton’s second law:
The nature of the collisions experienced by conduction electron,s is such that, after a typical collision, each electron will-so to speak-completely lose its memory of its previous drift velocity. Each electron will then start off fresh after every encounter, moving off in a random direction. In the average timer between collisions, the average electron will acquire a drift speed of ’d = a r. Moreover. if we measure the drift speeds of all the electrons at any instant, we will find that their average drift speed is also air. Thus, at any instant, on average, the electrons will have drift speed I’d = ar. Then
Equation 27-20 may be taken as a statement that metals obey Ohm’s law if we can show that, for metals, their resistivity p is a constant, independent of the strength of the applied electric field E. Because n, m, and e are constant, this reduces to convincing ourselves that ‘T. the average time (or mean free rime) between collisions, is a constant, independent of th~ strength of the applied electric field. Indeed, ‘T can be considered to be a constant because the drift speed Vd caused by the field is so much smaller than the effective speed I’eff that the electron spell-and thus ‘T-is hardly affected by the field.