in this video we're determining the dependence of resistance on the area and length of a conductor
so in this top picture i have a little section of a conductor and i'm trying to look at what's
happening microscopically there's a potential difference across this thing i see the current
is flowing to the right but we know what that really means is electrons are flowing to the left
and these electrons are responding to the presence of an electric field through the conductor that
puts an average force on them to the left and they move to the left with a velocity that's called
the drift velocity and i'll post a derivation to the drift velocity formula real quick now
resistance arises from collisions with atoms in the material so these electrons are slamming into
atoms and they get slowed down and then they get sped back up by the electric field and
then they collide again and so on and so on these collisions are what dissipates power in a resistor
and i'll post a link real quick to the video where i derive the power formula for resistors and this
energy loss is what results in the constant drift velocity of the electrons if there was no
energy loss here the electrons would be speeding up and speeding up and speeding up as they went
from low to high potential so let's think about a couple changes we could make to this conductor
and then we'll figure out what effect these changes have on the resistance
but first a reminder of ohm's law we can write ohm's law as current is voltage over resistance
you may be more used to seeing that as v equals ir which is just a simple algebraic manipulation
of the equation and then i'm going to think about what happens if i make my conductor wider so i'm
increasing the cross-sectional area which we'd normally call a well i have the same potential
difference across the conductor and having the same potential difference in the same length means
the strength of the electric field in there is going to be the same remember electric field could
be measured in volts per meter and that's staying the same in this picture if the strength of the
electric field is the same it means the drift velocity or the average velocity of the electrons
in this conductor should be the same but with a fatter wire and the same electron velocity that
means i have more electrons per second passing through the wire in other words the current has
just gone up if i then look at ohm's law and say oh the current just went up because of the changes
that i made well i didn't change v what happened here is r changed r must have gone down in order
to increase the current and we've obtained the first proportionality for the equation we're
trying to derive here the resistance is inversely proportional to the cross-sectional area of the
wire next we look at what happens if we take the original wire and just make it longer well in this
case we're taking the same potential difference and spreading it out over a larger distance but
we know that electric field can be measured in volts per meter we didn't change the volts but we
just made the length longer so this decreases the electric field that decreases the drift velocity
of the electrons because they're responding to a smaller force and now these electrons are moving
slower they're carrying less charge per second through the conductor then i realized looking at
ohm's law that if the current went down it means the resistance went up and we obtain
our second proportionality we can put these two proportionalities into an equation and there's a
constant of proportionality in here that constant of proportionality given by the greek letter rho
is called the resistivity of the material and this is just a property of the material
telling you how bad of a conductor the material really is the bigger the resistivity the more the
resistance so a good conductor would have a very small resistivity a bad conductor has a big one
notice real quick the units of resistivity if i want ohms to pop out of this new formula
resistivity better have ohms in it and then that second term length over area
that's a one over meters so resistivity better having meters in it to cancel that
so resistivity is measured in ohm meters finally we'll wrap things up with an example
so in the example we're told that nichrome has a resistivity of about 1.01 times 10 to the negative
6 ohm meters and i want to know what length of 30 gauge nichrome wire of course the units we
want are those millimeters for the diameter of the wire is necessary to make a 5 ohm resistor
so we'll start with our new formula r equals rho l over a and what i'm trying to solve
for here is length so i multiply by a on both sides and divide by rho and i get the length
is r times a over rho now just a quick reminder here we don't have to cut that diameter in half in
order to compute the cross-sectional area we can just change the area formula in terms of diameter
our classic formula is pi r squared for the cross sectional area but r is half the diameter
and when i square that that squares the 2 in the denominator giving me a 1
4 pi diameter squared going back to my length calculation the resistance i want
is five ohms the cross sectional area is one quarter times pi times the diameter squared to get
this in meters i need to shift the decimal point three times point zero zero zero two five four six
squared then i plug in my resistivity in the denominator one point zero one
times 10 to the negative 6 ohm meters i can see the units of ohms are going to cancel
one factor of meters cancels from the squared meters in the numerator and one survives leaving
me with meters for my final units and that's a good thing because we're trying to find a length
when i run the numbers on this i get .242 meters just keeping three sig figs here
and that looks nicer in centimeters as 24.2 centimeters of nichrome wire
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