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Why does resistivity increase with temperature?

The explanations I have heard so far are that increasing temperature increases vibrations in the lattice structure resulting in the number of collisions increasing.

But the vibrations from thermal energy are random, so around half of the vibrations would result in the atoms moving out of the electrons way and half would result in atoms moving into the way of the electrons.

And since half of the vibrations result in increasing the drift speed and half reduce it, shouldn't the average drift speed remain the same?

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  • $\begingroup$ Okay, but what about the other kind? o_O $\endgroup$ – Waffle's Crazy Peanut Feb 20 '14 at 12:51
  • $\begingroup$ There's no moving out of the way of electrons. Wherever the conductor exists, electrons flow. So increasing the vibrations, simply reduces the free space for electrons to move. See this. Do you think it'd be easier to pass through the gaps in this structure when it is moving, or when it is relatively stationary? $\endgroup$ – mikhailcazi Mar 21 '14 at 11:38
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The increase in resistance in metals is mainly due to the increasing velocity of the thermic electron motion (I think $v_{\rm therm}\sim \sqrt{kT}$). This shortens the time $\tau$ of free motion of the electron. If $l$ is the mean free path length we have $\tau=\frac{l}{v_{\rm therm}}\sim \frac{1}{\sqrt{kT}}$. If the electron bumps into the next atom/ion it looses all its directed velocity $v_{\rm drift}$. That means it has only the time $\tau$ to accelerate into the field-direction and we have a mean drift velocity $v_{\rm drift} = -\frac{eE}{2m_e}\tau\sim\frac1{\sqrt{kT}}$. This is surely simplified. The change in the mean free path length also must be taken into account.

See also Section 2.1.3 "Temperature Dependence of Conductivity" in http://www.newagepublishers.com/samplechapter/002014.pdf.

But, also see http://en.wikipedia.org/wiki/Classical_and_quantum_conductivity#Flaws_in_classical_conductivity.

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Yes, there will be increase in velocity of free electrons, when temperature increases. But the electrons will not be accelerated in a particular direction.

Consider a conductor, when potential difference is applied across the two ends of the conductor, an electric field is set up. Under the effect of electric field, the free electrons accelerate and acquire a velocity component in a direction opposite to the direction of electric field in addition to their thermal velocity. Due to electric field, electrons gain drift velocity in a direction opposite to it. If you increase the temperature of the conductor now, the electrons velocity increases in the direction other than due to electric field. Of course there will be some electrons which will be accelerated in the direction opposite to that of electric field, but there will be greater number of electrons which will be accelerated in other directions. As a consequence, current should decrease.

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  • $\begingroup$ Directly taken from [1]: "...an electrical field is applied to a solid the free electrons are accelerated. Their kinetic energy increases." | [1]: Section 2.1.3 "Temperature Dependence of Conductivity" in newagepublishers.com/samplechapter/002014.pdf $\endgroup$ – Tobias Feb 19 '14 at 11:53
  • $\begingroup$ Thank you for the comment. I haven't taken anything from the link you provided. Anyway, even if I would had, it doesn't matter. We always gain knowledge from one or other the source. Yes, the answer I have sent is directly extracted from the knowledge I have gained from my teachers and my learning source. $\endgroup$ – Immortal Player Feb 19 '14 at 12:57
  • $\begingroup$ (1) I missread your answer. The first time you speek about acceleration you meen the increase of velocity due to higher temperature == higher kinetic energy. (2) In the classical theory the mean reason for the higher resistance is the shorter time $\tau$ of free motion. See my answer or the corresponding wiki-article. $\endgroup$ – Tobias Feb 20 '14 at 8:10
  • $\begingroup$ I hope you are speaking about relaxation time. For the same question there will be different way of answering. My answer imply the same, that relaxation time becomes shorter, please read it once again. $\endgroup$ – Immortal Player Feb 20 '14 at 12:04
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I think you have some of the right intuition, but the orders of magnitude of the problem need to be considered. Look at Electricity and Magnetism by Purcell. http://www.amazon.com/Electricity-Magnetism-Vol-II-Berkeley/dp/0070049084 They have a chapter on these concepts.

The idea is that conductivity in a metal is primarily the motion of free electrons. However, remember they are small (scattering cross-section is small and therefore they don't hit each other often) and their mean free path can be quite long.

So, as the temperature increases, the kinetic energy increases and the velocity of electrons increases. This reduces the time between collisions (not necessarily the path length or mean free path). Therefore, it is less likely an electron will flow from the high electrical potential to the low electrical potential, increasing the electrical resistance.

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Let us begin by taking a real life example.

You are in your room and at the door, there is a large block that blocks almost the entire way( the space between the ions is very small ). Say it is motionless. Now, you would not find it hard to pass the block. All you have to do is to squeeze a bit to get through.

enter image description here

Now, let us say that the block is vibrating. It will be difficult for you to get through. You've got to go extremely fast in order to have a chance to get through. Remember, electrons drift very slowly, slower than you could imagine. They travel approximately 1 meter every hour.

enter image description here

You might say that as a result of thermal agitation, the electrons gain velocity and so the gain and loss balance each other. The question is, in what direction does it gain speed? Thermal agitation results in randomness. So, it is not right to say that the electrons gain speed in a particular direction. Electrons can gain speed in any random direction due to thermal agitation. There is a chance for an electron to gain speed in a particular direction but, there is also a chance for another electron to slow down( equal amount ) in that direction. So, the average velocity as a result of thermal agitation remains zero. Furthermore, the electric field created by the voltage source stabilizes everything and the electrons drift in a particular direction with an average velocity of 1 m/hr( approx ).

In summary, the number of collisions increases as the temperature increases. This is due to the lattice vibrations. The electric field tends to set the electrons in drift motion against thermal agitation. The vibrational kinetic energy of the ion increases which results in a greater number of collisions.

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