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- [Voiceover] You
probably know that if you

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hook up a battery of voltage V,

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to a resistor of resistance R,

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then you'll get a certain
amount of current,

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and you can determine how
much current flows here

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by using Ohm's law.

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And remember, Ohm's law
says that the voltage

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across a resistor equals the
current through that resistor

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times the resistance of that resistor.

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So, this pretty much gives you
a way to define resistance.

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The resistance of the
resistor is defined to be

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the amount of voltage applied across it

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divided by the amount
of current through it.

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And this is good, we like
definitions, because we want

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to be sure that we know
what we're talking about.

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That's the definition of resistance.

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Remember, it has units of ohms.

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But be careful, don't fall
into the trap of thinking

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about this the way some people do.

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Some people think, "Oh okay,
if I want a bigger resistance,

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"I'll just increase the
voltage, 'cause that'll give me

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"a bigger number up top."

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Bigger resistance doesn't work that way.

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If you increase the voltage, you're gonna

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increase the current.

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And this ratio is gonna stay the same.

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The resistance is a constant.

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And this resistor, if you're
not changing the material

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makeup or size or
dimensions of this resistor,

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this number that is the
resistance is a constant,

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if it's truly an Ohmic material.

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So, Ohmic materials maintain

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a constant resistance,
regardless of what voltage

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or current you throw at them.

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It'll just be constant.

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Yeah, if you throw too
much current or voltage,

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the thing'll burn up.

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I don't suggest you do that.

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So, there's an operating
range here, but if you're

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within that range, this
resistance, this number,

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this number of ohms is a constant.

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It stays the same no matter
what voltage or current

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you put through it.

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So, we define it by talking
about voltage and current,

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but it doesn't even really depend on that.

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If you really want to change
this ratio, this number

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that comes out here for
the resistance, you need to

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change something about
the resistor itself.

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Its size, what it's made out
of, its length, its shape.

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So let's figure out how to do
that if we take this resistor.

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Imagine taking this resistor,
bringing it into the shop.

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What is it gonna look like?

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Well, for simplicity's
sake, let's say we just have

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a perfectly cylindrical resistor.

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So, this is the wire going into one end.

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This is your resistor.

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It's a cylinder, let's say.

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And this is the wire going
out of the other end,

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so this is the blown up
version of this resistor.

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One thing you could
depend on is the length.

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So, the length of this
resistor could affect

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the resistance of this resistor.

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Another thing it could
depend on is the area

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of this front part here,
this cross sectional area.

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It's called the cross sectional
area, because that's the

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direction that the current's heading into.

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This current's heading into
that area there, like a tunnel,

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and it comes out over here.

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Now this is full, this isn't hollow.

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This is made up of some material.

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Maybe it's a metal or some
sort of carbon compound

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or a semiconductor, but it's
a solid material right here

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that the current flows
into and then flows out of.

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So, what would happen if we
made this resistor longer?

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Let's say we start changing
some of these variables,

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and we increase the
length of this resistor.

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Well, now this current's gotta flow

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through a longer resistor.

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It's gotta flow through this
resistor for more of this path.

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And it makes sense to me to
think that the resistance

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is going to increase.

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If I increase the length of this resistor,

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then the resistance is gonna go up.

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How about the area, this
cross sectional area?

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Let's say I increase this
area, I make it a wider,

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larger diameter cylinder.

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Well, it makes sense to me to
think that now that current's

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got more room to flow
through, essentially.

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There's a bigger area through
which this current can flow.

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It's not as restricted.

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That means the resistance should go down.

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And if we try to put this
in a mathematical formula.

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What that means is, if
I increase the length,

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R should depend on the length.

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It turns out it's directly
proportional to the length.

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If I double the length of a resistor,

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I get twice the resistance.

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But area, if I increase the area,

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I should get less resistance.

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'Cause there's more room to flow.

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So, over here in this formula, my area has

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go to go on the bottom.

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The resistance of the resistor
is inversely proportional

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to this cross sectional area.

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But there's one more
quantity that this resistance

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could depend on, and
that's what the material is

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actually made of.

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So, the geometry determines the resistance

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as well as what the material is made of.

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Some materials just naturally offer more

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resistance than others.

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Metals offer very little resistance,

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and non-metals typically
offer more resistance.

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So, we need a way to quantify
the natural resistance

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a material offers, and that's
called the resistivity.

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And it's represented with
the greek letter rho.

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And the bigger the
resistivity of a material,

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the more it naturally resists the flow

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of current through it.

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To give you an idea of the numbers here,

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the resistivity of copper.

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Well, that's a metal,
it's going to be small.

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It's 1.68 times

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10 to the negative eighth.

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We'll talk more about
the units in a second.

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But the resistivity of
something like rubber,

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an insulator, is huge.

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It can be on the order of 10 to the 13th.

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So, there's a huge
range of possible values

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as you go from metal
conductor to semiconductor

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to insulator, huge range
of possible resistivities.

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And this is the last key here.

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This is the last element in this equation.

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The resistivity goes right here.

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So, the bigger the resistivity,
the bigger the resistance.

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That makes sense.

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And then it also depends
on these geometrical

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factors of length and area.

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So, here's a formula to
determine what factors actually

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change the resistance of a resistor.

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The resistivity, the length, and the area.

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So, what are the units of resistivity?

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Well, I can rearrange this
formula, and I can get

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that the resistivity equals

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the resistance times the area

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of the resistor divided by the length.

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And so that gives me units of

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ohms times meters squared,

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'cause that's area, divided by meters.

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And so I end up getting ohms.

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One of these meters cancels out.

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Ohms times meters.

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Those are the units of
these resistivities.

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Ohm meters.

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But how do you remember this formula?

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It's kind of complicated.

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I mean, is area on top,
is length on bottom?

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Hopefully you can remember why those

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factors affected it.

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But sometimes students have a hard time

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remembering this formula.

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One of my previous students
from a few years ago

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figured out a way to remember it.

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He thought this looked like "Replay".

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So, this R is like R,

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and the equal sign kind
of looks like an E.

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And the rho kinda looks like a P.

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And the L looks like an L,
and the A looks like an A.

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And it kinda looks like "Replay".

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There's a missing Y here,
but every time I think

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of this formula, I think of
it as the "Replay formula".

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'Cause my former student Mike
figured out this mnemonic.

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And it's handy, I like
it, so thank you, Mike.

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And since we're talking about resistivity,

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it makes sense for us to
talk about conductivity.

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Electrical conductivity.

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Now the resistivity gives
you an idea of how much

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something naturally resists current.

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And the conductivity tells
you how much something

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naturally allows current.

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So, they're inversely proportional.

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And if you're thinking it
might be this easy, it is.

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The resistivity is just equal to

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one over the electrical conductivity.

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And the symbol we use for
electrical conductivity is sigma.

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So, this Greek letter sigma is
the electrical conductivity.

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And rho, the resistivity,
is just one over sigma,

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the electrical conductivity.

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And vice versa, sigma is gonna equal

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one over the resistivity,
because if something's

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a great resistor, it's a bad conductor.

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And if something's a great conductor,

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it's a bad resistor.

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So, these things are
inversely proportional.

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They're like two peas in a pod.

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If you know one of these,
you know the other.

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All right if this all seems a
little bit too abstract still,

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there's a nice analogy
you can make to water.

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We saw that a resistor
depended on a few things,

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like the resistivity.

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The bigger the resistivity,
the bigger the resistance.

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And we saw that the bigger
the length of the resistor,

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the larger the resistance.

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And if you divide by the
area of the resistor,

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it shows that the resistance
is inversely proportional

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to the area of the resistor.

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So, let's make an analogy to water.

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Let's say you have, instead of electrons,

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flowing through a wire.

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Instead of the wire, let's
say you had a tube, a pipe

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that water could flow through.

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So, instead of electrons, you've got water

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flowing through a pipe.

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Different pipes are gonna
offer different amounts

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of resistance to the water
flowing through that pipe.

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What would affect it?

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Well, imagine you had a
constriction in this pipe.

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If this pipe got
constricted, it'd be harder

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for the water to flow.

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You'd find that it resists
the flow of water more

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because of this constriction.

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And what would it depend on?

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Well, the smaller this
area of the constriction,

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the larger the resistance.

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And that agrees with what we have up here.

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If you have a really small
area, you're dividing by

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a small number, and when you
divide by a small number,

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you get a big number.

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That'd be a big resistance.

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So, that makes sense.

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Also the length, if
you increase the length

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of this constriction, the water will have

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a harder time flowing.

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There's manuals for plumbers,
and you can look it up.

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There's a key to determine if
your pipe is a certain length,

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you're gonna need more pressure over here.

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So, the smaller the constriction
in terms of its area,

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and the longer it is, the more
pressure you need back here.

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The pressure is like the
source of the battery.

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So instead of a battery
providing the voltage

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to this circuit, you'd have
something offering pressure

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to get the water flowing.

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And just like a battery, what
matters is the difference

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in electric potential.

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What matters for the pressure
here is the change in pressure

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between one point in the system

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and another point in the system.

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So that makes sense.

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A longer constriction
means more resistance.

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A smaller area means more resistance.

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What would this resistivity
be analogous to?

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Well, it would be kind of like
what the pipe is made out of.

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If this pipe has a rough inner surface,

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the water wouldn't flow as smoothly.

255
00:09:41,452 --> 00:09:43,821
You would get a greater
resistance regardless

256
00:09:43,821 --> 00:09:46,491
of how long it is or what the area is.

257
00:09:46,491 --> 00:09:49,927
Just the natural built-in
affect of the pipe itself

258
00:09:49,927 --> 00:09:51,924
is what the resistivity would depend on,

259
00:09:51,924 --> 00:09:52,969
just like up here.

260
00:09:52,969 --> 00:09:55,964
The resistivity depends on what
the material is made out of.

261
00:09:55,964 --> 00:09:58,611
The resistivity of this pipe
depends on what this pipe

262
00:09:58,611 --> 00:10:00,747
is made out of, at least the inner wall.

263
00:10:00,747 --> 00:10:03,348
So hopefully this analogy
makes this formula seem

264
00:10:03,348 --> 00:10:05,275
a little more intuitive.

265
00:10:05,275 --> 00:10:07,202
But just in case, let's do an example.

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00:10:07,202 --> 00:10:09,129
Let's get rid of all this.

267
00:10:09,129 --> 00:10:10,267
And let's say you got this question.

268
00:10:10,267 --> 00:10:12,287
"How much resistance would be offered by

269
00:10:12,287 --> 00:10:15,120
"12 meters of copper wire with a diameter

270
00:10:15,120 --> 00:10:16,885
"of 0.01 meters?"

271
00:10:16,885 --> 00:10:19,601
If copper has a resistivity of 1.68

272
00:10:19,601 --> 00:10:21,250
times 10 to the negative 8th.

273
00:10:21,250 --> 00:10:23,456
Now, what units does resistivity have?

274
00:10:23,456 --> 00:10:26,915
Turns out resistivity
has units of ohm meters.

275
00:10:26,915 --> 00:10:28,842
So, ohms times meters.

276
00:10:28,842 --> 00:10:29,887
Well, let's try this out.

277
00:10:29,887 --> 00:10:30,909
We've gotta use our formula.

278
00:10:30,909 --> 00:10:33,765
Remember "Replay", R equals rho,

279
00:10:33,765 --> 00:10:34,949
L over A.

280
00:10:34,949 --> 00:10:36,644
The resistivity we have right here.

281
00:10:36,644 --> 00:10:40,313
1.69 times 10 to the negative 8th.

282
00:10:40,313 --> 00:10:41,775
Notice how small this is.

283
00:10:41,775 --> 00:10:43,308
This is hardly anything at all.

284
00:10:43,308 --> 00:10:45,305
Copper's a great conductor.

285
00:10:45,305 --> 00:10:47,092
It's a terrible resistor.

286
00:10:47,092 --> 00:10:49,461
It let's electrons flow
through it like a charm.

287
00:10:49,461 --> 00:10:51,574
All right, so the length,
that's pretty easy.

288
00:10:51,574 --> 00:10:52,990
The length is 12 meters.

289
00:10:52,990 --> 00:10:55,057
Notice, we're asking,
what's the resistance

290
00:10:55,057 --> 00:10:57,077
of the wire itself?

291
00:10:57,077 --> 00:11:00,118
Now there's not really a
quote-unquote resistor in here.

292
00:11:00,118 --> 00:11:02,951
But every piece of wire's
gonna offer some resistance.

293
00:11:02,951 --> 00:11:05,761
And this formula applies just
as well to a piece of wire

294
00:11:05,761 --> 00:11:07,293
as it does to a resistor.

295
00:11:07,293 --> 00:11:09,685
So, the length of the wire's 12 meters,

296
00:11:09,685 --> 00:11:11,380
and the diameter is 0.01.

297
00:11:11,380 --> 00:11:12,053
What do we do with that?

298
00:11:12,053 --> 00:11:13,144
Well, we need the area.

299
00:11:13,144 --> 00:11:15,048
Remember the cross sectional area.

300
00:11:15,048 --> 00:11:18,368
And the area of a circle is pi r squared,

301
00:11:18,368 --> 00:11:21,364
so the area down here is gonna be pi times

302
00:11:21,364 --> 00:11:22,943
not 0.01 squared.

303
00:11:22,943 --> 00:11:24,220
That's the diameter.

304
00:11:24,220 --> 00:11:26,565
We need the radius, we
need to take half of this.

305
00:11:26,565 --> 00:11:30,985
So 0.005 meters squared.

306
00:11:30,985 --> 00:11:34,380
And if you calculate all
this, you get a resistance of

307
00:11:34,380 --> 00:11:38,211
0.0026

308
00:11:38,211 --> 00:11:39,140
ohms.

309
00:11:39,140 --> 00:11:41,649
Hardly anything, but
there is some resistance.

310
00:11:41,649 --> 00:11:43,390
And if this is going
to have an affect on a

311
00:11:43,390 --> 00:11:45,225
very delicate experiment, you've gotta

312
00:11:45,225 --> 00:11:46,386
take that into account.

313
00:11:46,386 --> 00:11:48,661
If you get this really
long, the longer it is,

314
00:11:48,661 --> 00:11:50,008
the more resistance is has,

315
00:11:50,008 --> 00:11:51,447
that could affect your system.

316
00:11:51,447 --> 00:11:53,746
But typically, it doesn't matter too much.

317
00:11:53,746 --> 00:11:56,370
The copper wire, electron's
flow through that like water,

318
00:11:56,370 --> 00:11:57,600
like it's not even there,

319
00:11:57,600 --> 00:00:00,000
because the resistance is so small.