Resistance dependence on temperature for metals

[furryelephant]

I'm writing some high school physics teaching animations and I need to make sure I have the right idea for why increasing the temperature of a metal increases its resistance.

It's high school, so I need some classical analogy which isn't just blatantly wrong or stupid.

The explanations I have always heard involve increased movement of the metal lattice disrupting (scattering) the flow of electrons.

It is always presented as intuitively obvious that:

1. Increased ion vibration would obviously cause more frequent interaction with electrons.
2. Electron collision with vibrating ions would necessarily slow the progress of the electrons down the wire.

Neither of these two assumptions are remotely intuitive to me.

My feeling is that

1. is like running down a corridor with a sliding door that opens and shuts continuously. If you can't control your speed, would you rather a high frequency door or a low frequency one to avoid running into it? - seems to me like swings and roundabouts unless there is a second order effect to do with relative size of you and door.

2. we can ignore the component of collisions that cause motion perpendicular to the drift direction since the electrons are still being accelerated by the field - analogous to droppping a ball and launching it horizontally.

So the only interactions we need to worry about are the ones where the component of the ion's motion is in the same sense as drift direction.

Now these interactions could either

a. tend to reduce the drift rate
b. tend to increase it
c. have no effect

My model is that they tend to reduce it.

If the maximum velocity of the ion vibration is much higher than the typical electron velocity at interaction then the electron can only interact with the ion under these circumstances (let's imagine that drift direction is left to right):

a. the ion has just has just gone past it's left maximum and is moving slowly right (the electron would have a 'soft landing' and so this would tend to hasten its progress down the wire since it wouldn't be smacked back the way it came)

b. the ion is just about to reach it's right maximum (the electron would have a 'soft landing' and so this would tend to hasten its progress down the wire since it wouldn't be smacked back the way it came)

c. the ion is moving at any velocity to the left (the electron will be batted back up the wire by the approaching ion, which will reduce its progress down the wire)

Most of the time when the ion is moving from left to right the electron can't interact with it because it's not going fast enough.

This means the majority of the possible interactions tend to knock it back to where it came from rather than giving it a soft landing.

At high school level you don't want to get too involved (these explanations look a lot simpler when they're animated) but I want to avoid telling them a story that is just blatantly wrong.

Comments on my ideas, please or alternative suggestions.

Thanks

 

[Vern]

It's like trying to move down a corridor full of people. If all the people drift along the same direction things go better than if the people are all bouncing off the walls and colliding with each other.

Seems to me

 

[Graeme Robinson]

for your question and for wanting to make sure you have the right understanding before teaching this concept to your students. The relationship between resistance and temperature for metals is a fundamental concept in physics and it's important to have a clear understanding of it before teaching it to others.

Your analogy of running down a corridor with a sliding door is a good start, but there are a few key points that can be added to provide a more accurate explanation.

Firstly, it's important to understand that metals have a crystal lattice structure, meaning that the atoms are arranged in a regular pattern. When a metal is heated, the atoms in the lattice vibrate more vigorously, which can cause the lattice to expand. This increased vibration and expansion of the lattice can create more obstacles for the flow of electrons, which results in an increase in resistance.

Now, let's revisit your analogy. Instead of a sliding door, imagine a series of hurdles placed along the corridor. As you run, you have to jump over each hurdle, which slows you down. The higher the hurdles are, the more they impede your progress. In this analogy, the hurdles represent the vibrating atoms in the lattice, and the height of the hurdles represents the strength of the resistance.

Additionally, as the temperature increases, the atoms in the lattice also gain more kinetic energy, meaning they move around more and can collide with the electrons traveling through the metal. These collisions can also slow down the flow of electrons, contributing to an increase in resistance.

Your second point about electron collisions causing motion perpendicular to the drift direction is correct. However, it's important to note that these collisions can also affect the overall drift velocity of the electrons. When an electron collides with an atom, it loses some of its energy and momentum, which can reduce its overall drift velocity. This is similar to your "soft landing" scenario in your analogy.

In summary, the increase in resistance of a metal with increasing temperature is due to the increased vibration and expansion of the lattice, as well as the increased collisions between the vibrating atoms and the moving electrons. Your analogy of running down a corridor with hurdles can be a helpful way to visualize this concept, but it's important to also include the role of the crystal lattice structure and the collisions between atoms and electrons in your explanation. I hope this helps clarify the concept for you and your students.