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- [Voiceover] Now that
we have our collection
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of components, our favorite
batteries and resistors,
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we can start to assemble
these into some circuits.
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And here's a circuit shown here.
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It has a battery and
it has three resistors,
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and a configuration that's called
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a series resistor configuration.
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Series resistors is a familiar pattern,
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and what you're looking for is resistors
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that are connected head
to tail, to head to tail.
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So these three resistors are in series
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because their succession of nodes
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are all connected, one after the other.
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So that's the pattern that tells you
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this is a series resistor connection.
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So we're gonna label
these our resistors here.
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We'll call this R1, R2, and R3.
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And we'll label this as v.
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And the unknown in this
is what is the current
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that's flowing here, that's
what we want to know.
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We know v, we want to know i.
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Now one thing we know about i
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is i flows down into resistor R1,
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all of the current goes out of
the other end of resistor R1
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because it has to, it
can't pile up inside there.
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All that goes into here,
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and all that comes out of R3.
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And i returns to the place it came from,
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which is the battery.
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So that's a characteristic
of series resistors,
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is in a series configuration
is they are head to tail,
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and that means that all the components,
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all the resistors share the same current.
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Current.
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That's the key thing.
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The thing that we don't know
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that's different between each resistors,
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is the voltage here, and the voltage here,
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let's call that v1,
this is v2, plus, minus,
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and this is v3, plus, minus.
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So in general, if these
resistors are different values
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because they have the same
current going through them,
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Ohm's Law tells us these
voltages will all be different.
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So the question I want to
answer with series resistors
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is could I replace all three of these
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with a single resistor that
cause the same current to flow?
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That's the question we have
on the table right now.
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So we make some observations,
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we have Ohm's Law, our friend, Ohm's Law.
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And we know that means v equals i times R,
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for any resistor.
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That sets the ratio of voltage to current.
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And this is another
thing we know about this,
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which is that v3, plus v2, plus v1,
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those are the voltages
across each resistor,
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those three voltages have
to add up to this voltage
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because of the way the
wires are connected.
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So the main voltage from the battery
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equals v1, plus v2, plus v3.
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We know that's for sure,
and now what we're gonna do
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is we're gonna write Ohm's Law
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for each of these individual resistors.
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v1 equals i, and i is
the same for everybody,
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times R1.
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v2, this voltage here, equals i times R2.
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And v3 equals i times R3.
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Now you can see, if I had four,
or five, or six resistors,
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I would have four, or five, or
six equations just like this
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for each resistor that was in series.
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So now what I'm gonna do is
substitute these voltages
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into here, and then we'll
make an observation.
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So let's do that substitution.
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I can say v equals i, R1,
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plus i, R2, plus i, R3.
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And because it's the
same i on every resistor,
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I can write v equals i,
I'm gonna factor out the i.
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R1, plus R2, plus R3.
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Now what I want to do
is take a moment here
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and compare this expression
to this one here,
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the original Ohm's Law.
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Alright, there's Ohm's Law.
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So we have v equals i, some current,
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times some resistor.
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I can come up with a resistor value,
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a single resistor that would
give me the same Ohm's Law.
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And that is gonna be called,
let's draw it over here.
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Here's our battery.
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And I'm gonna say there's a
resistor that I can draw here,
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R series, that's equivalent
to the three resistors here.
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And it's equivalent in the sense that
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it makes i flow here, that's
what we mean by equivalent.
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So in our case, to get the
same current to flow there
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I would say v equals i times R series,
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in which case, what I've done
is I've said that R series
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is what, is the sum of these three things,
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R1 plus R2, plus R3.
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This is how we think
about series resistors.
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We can replace a set of series resistors
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with a single resistor
that's equivalent to it
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if we add the resistors up.
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Let's just do a really fast
example to see how this works.
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I'm gonna move this screen.
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Here's an example with three resistors.
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I have labeled them 100
ohms, 50 ohms, and 150 ohms.
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And what I want to know
is the current here.
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And we'll put in a voltage,
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let's say it's 1.5 volts,
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just a single small battery.
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So what is the equivalent resistance here?
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One way to figure this out
and to simplify the circuit
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is to replace all three of those resistors
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with a series resistor, RS,
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and that is, as we said here, is the sum,
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so it's 100, plus 50, plus 150.
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And that adds up to 300 ohms.
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So that's the value of the equivalent
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series resistor right here.
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And if I want to calculate the current, i,
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i equals v over R, and
this case, it's R series,
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and that equals 1.5 divided by 300.
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And if I do my calculations right,
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that comes out to .005 amperes.
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Or an easier way to say it is
five milliamps, milliamperes.
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So that's i.
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And now that I know i, I can
go ahead and I can calculate
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the voltage at each point
across each resistor
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because I know i, I know R,
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I can calculate v.
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So v1, v1, which is the
voltage across that resistor,
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v1 equals i, R1, as we said before.
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So it's five milliamps
times 100 ohms, 0.5 volts.
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Let's do it for the other one, v2,
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equals i, same i, this time times R2,
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five milliamps times 50 ohms,
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and that equals 0.25 volts.
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And finally, we do v3.
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This is plus, minus v3.
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And that equals the same
current again times 150 ohms,
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which is equal to 0.75 volts.
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So we've solved the
voltage and the current
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on every resistor, so this
circuit is completely solved.
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And let's do one final check.
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Let's add this up.
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Five, five (mumbles) is zero.
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Carry the one, six, seven eight.
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15, 1.5 volts, and that's very handy
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because that is the same as that.
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So indeed, the voltages across
the resistors did add up
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to the full battery that was applied.
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There's one more thing
I want to point out.
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Here's an example of
some series resistors.
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And that's a familiar pattern.
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And you'll say, "Oh, those
are series resistors."
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Now, be careful because if
there's a wire here going off
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and there's, doing this,
or there's a wire here,
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connected to this node here,
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this still looks like they're in series,
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but there might be current flowing
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in these branches here.
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If there's current flowing out
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anywhere along a series branch,
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anywhere along what looks
like a series branch,
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then this i may or may
not be the same as this i.
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And it might not be the same as this.
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So you gotta be careful here.
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If you see branches going
off your series resistors,
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these are not in series
unless these are zero current.
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If that's zero current, and
if that is zero current,
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then you can consider these in series.
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So that's just something to be careful of
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when you are looking at a circuit
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and you see things that
look like they're in series,
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but they have little branches coming off.
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So a little warning there.
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So that's our series resistors.
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If you have resistors and series,
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you add them up to get
an equivalent resistance.