in this video we discuss the physical reasons why resistance increases with temperature in
an ordinary conductor we state a simple linear formula for computing the increase in resistance
with temperature and we work an example to demonstrate how to use the formula so what we're
looking at in the first picture here it's a simple model for what's going on in a conductor when
there's a current flowing through it now in this conductor current is actually flowing to the right
but we know what's really going on is that electrons are moving to the left
at an average speed that's called the electron drift velocity and i'll post a link to where
i derived the electron drift velocity real quick now resistance arises in the conductor because of
collisions with the atoms in a material so like if this was a copper wire these big red things here
those are the copper atoms so when electrons collide with the atoms in a material energy
is dissipated and randomized and it turns into heat and it translates into an increase in the
temperature of the material so these electrons are losing potential energy but they always maintain
the same speed why aren't they speeding up that missing energy goes into the thermal energy
that's heating up the material as the current flows so how does higher temperature change the
resistance of a material that's what i'm trying to illustrate in the second picture temperature
is literally the average kinetic energy of the particles in a material and so what's
happening as this thing gets hotter is the atoms in the material are vibrating faster and faster
and with larger and larger amplitude well this means they effectively have a larger cross-section
as they're bouncing back and forth you're more likely to have a collision with an electron
the classic metaphor here is that it's like trying to run past somebody who keeps jumping side to
side you're more likely to run into them than if they were standing still so this is the physical
reason why we expect resistance to increase with temperature now we get into our simple linear
model for how to compute resistance changes as the temperature changes and the first thing i'll
do is phrase this in terms of the resistivity of the material so there's a couple assumptions
that go into this model we have to assume that delta t is not exceptionally large so this is a
very simple linear model that's only going to work over a fairly narrow range of temperature changes
also i need to be clear about what rho naught is so that's the resistivity of the material at some
reference temperature that you're going to look up in a chart or that would be given to you in a
problem finally i wanted to just look at this new thing alpha that's called the temperature
coefficient of resistivity and that just measures how sensitive a material is to temperature changes
like a material with a large alpha would have large resistivity changes when the temperature
changes and i want to look at the units of alpha real quick in this term alpha delta t
i've got to get just a constant out of that with no units on it so when i add a 1 to that basically
that's going to give me a percentage change on the original resistivity so for this to come
out unitless i need the units of alpha to be 1 over celsius degrees now we originally introduced
resistivity in a video about how resistance depends on the length and cross-sectional area
of a resistor and i'll just go ahead and post a link to that real quick and in that video we
deduced this formula the resistance is equal to the resistivity times the length over area the
main point being right now that the resistance of a resistor is proportional to the resistivity of
the material well if resistance is proportional to the resistivity of the material then i could write
my simple linear model like this and that might be more useful depending on how a problem is given to
you all right let's wrap things up with a simple example of how to apply this formula so we're told
that a nichrome resistor has a resistance of 5 ohms at a temperature of 20 degrees celsius
and we know the temperature coefficient of resistivity for nichrome is about
0.0004 per degree celsius now the chart that i used to find this
if i could be a little more specific said the alpha is equal to .0004 per degree celsius at
20 degrees celsius so it's important you pay attention to what the reference temperature is
when you do these calculations then we're asked to find the resistance after the resistor heats up to
100 degrees celsius so this is a plus 80 for delta t and in this case i'm using the second formula i
don't have to go back and worry about resistivity and the length and cross-sectional area of the
resistor i know the original resistance is 5 ohms i'll find my new resistance at the temperature t
by taking the original resistance 5 ohms and multiplying by 1 plus alpha times delta t
and again that alpha times delta t is unitless so i end up with my original units of ohms and
i'm just scaling up the resistance here when i run the numbers on this i get 5.16 ohms for the final
resistance so that gives you a sense for what the magnitude of these changes with temperature
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