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# Resistors in series

Video transcript

Let's make our circle a little
bit more complicated now. So let's say I have a battery
again, and let me do it in a different color just
for variety. That's the positive terminal,
that's the negative terminal. Let's say I have this perfect
conductor, and let's say I have one resistor and I
have another resistor. I don't know, just
for fun, let's throw in a third resistor. And we know, of course, that
the convention is that the current flows from positive to
negative, that that's the flow of the current. And remember, current is just
the charge that flows per unit of time or the speed
of the charge flow. But we know, of course, that in
reality what is happening, if there's any such thing as
reality, is that we have a bunch of electrons here that,
because of this voltage across the battery terminals, these
electrons want to really badly get to the positive terminal. And the higher the voltage, the
more they really want to get to this positive terminal. So what's going to happen
in this circuit? Actually, let me label
everything. So let's call this R1,
let's call this R2, let's call this R3. The first thing I want you to
realize is that between elements that the voltage
is always constant. And why is that? Well, we assume that this is a
perfect conductor-- let's say this little segment
right here, right? And so it's a perfect
conductor. Well, let's look at
it at this end. So you have all these
electrons. This is a perfect conductor,
so there's nothing stopping these electrons from
just distributing themselves over this wire. Before you encounter an element
in the circuit or device or whatever you want to
call that, you can view this ideal conducting wire just from
a schematic point of view as an extension of the
negative terminal. And similarly, you can view
this wire right here, this part of the wire, as
an extension of the positive terminal. And the reason why I want to
say that is because it actually turns out that it
doesn't matter if you measure the voltage here. So let's say if I take a measure
of the voltage across those two terminals using what
we call a voltmeter. And I'll later do a whole video
on how voltmeters work, but remember, when we measure
voltage, we have to measure it at two points. And why is that? Because voltage is a potential
difference. It's not some kind of
absolute number. It's a difference between
essentially how bad do electrons want to get
from here to here. So if we measure the voltage
between those two points, it would be the exact same thing as
if we measured the voltage between these two points. Theoretically. As we know, no wires really
have no resistivity. All wires have a little bit,
but when we draw these schematics, we assume that the
wires are perfect conductors and all the resistance takes
place in the resistor. So that's the first thing I want
you to realize, and it makes things very-- so, for
example, everywhere along this wire, this part of the wire,
the voltage is constant. Everywhere along this wire,
the voltage is constant. Let me erase some of this,
because I don't want this to get too messy. That's a big important
realization when you later become an electrical engineer
and have much harder problems to solve. Let me erase all of this. Let me erase all of that. Let me redraw that, because we
can't have that gap there, because if there was that gap,
current wouldn't flow. That's actually-- well, I'll
draw later how you can draw a switch, but a switch is
essentially a gap. It looks like a gap in the
circuit that you can open or close, right? Because if you open it,
no current will flow. If you close it, current
will flow. OK, so you now know
that the voltage between devices is constant. The other thing I want to
convince you is that the current through this entire
circuit is constant, and that applies to any circuit
in series. Now, what do I mean by series? Series just means that
everything in the circuit is after one another, right? If we take the convention and
we say current flows in this direction, it'll hit this
resistor, then the next resistor, then the
next resistor. At no point does the circuit
branch off and have to choose whether I want to go down
path A or path B. So this circuit is completely
in series, and there's a couple ways I can convince you
that the current-- let's call the current here I1. Let's call this current
here I2. Let's call this current
here I3. I could draw another
one here, I3. So there's a couple of ways
I can convince you that I1 equals I2, I3. One is I could just say if you
experimentally tried it out using an ammeter, which measures
current, you would see that they are identical. But the other way to think about
it, and this time I'm going to actually talk about the
electrons, so let's talk about things going in this
direction, is-- so these electrons, through this wire,
they can go as fast as they want to go, right? The speed of light or close to
the speed of light since they have very, very,
very low mass. And we'll go into relativity
one day. But once they get to this
resistor, they start bumping into things, and
they slow down. This resistor is a bit of
a bottleneck, right? So as fast as they're traveling
here, they have to slow down here. And if they slow down here, they
have to slow down here, because if they kept going
superfast here and then they slowed down here, then they
would start building up here, and that just doesn't make
sense, because we know that they're evenly spread
out, et cetera. And similarly, they might exit
this resistor at a certain speed and then slow down even
further as they bump into resistors here, but if they're
going even slower at this point, then there would be
a bottleneck here, so essentially, they would have to
go at that rate throughout the whole thing. And another way to think about
it is the resistance is kind of a probabilistic thing. I know when you think on a macro
level, you say, oh, it has this resistance. It just slows it down. But the longer there's a
resistor, it increases the probability that some of the
electrons are going to bump into something and create
a little bit of heat, et cetera, et cetera. So when you put resistors in
series, what you're actually doing is increasing the
probability that more electrons will bump into
more things, right? Say there's an electron that
travels-- say, somehow through freak luck, it doesn't bump
into anything as it goes through here's because it's
going really fast, but then it bumps into something
here, right? It only increases the
probability that something bumps into it. So there's a bunch of ways you
can think about it, and I encourage you to let me
know if there's other ways that help you. But the current through
this entire series circuit is constant. Now if we say that, what
else can we say? Well, if the current here--
let's say the current through here is I1. If the current through here is
I1, what is going to be the voltage if I measured it
from here to here? What is this voltage here? I measured it with
a voltmeter. Well, V1 is going to be
equal to I1 times R1. I don't know why I put an R. That's a 1, not an I. I1 times R1, right? And similarly, if I measured the
voltage from here to here, that voltage is going to be
equal to I2 times R2. Let's say this is where I3 is. So the voltage, if I were to
measure it from here to here-- But anyway, if we look at the
voltage from here to here, it's going to be I3 times R3. So what we see is that the
voltage across the entire circuit, which I can write as
V-total, is going to be equal to the potential drops, the
total potential drop across each of these devices. So the way to think about it
is that-- well, let's think about the electrons. The electrons here, they really
want to get here. But after they've bumped around
a little bit and they get here, they've experienced
some potential drop. So the electrons here actually
are a little bit less eager to get here. And then once they've gone
through here, maybe they're just tired of bumping
around so much. And once they're here, they're
a little bit less eager to get here. So there's a voltage drop across
each device, right? So the total voltage is equal
to the voltage drop across each of the devices. And now let's go back to the
convention, and we'll say that the current is going
in that direction. The total voltage drop is equal
to V1 plus V2 plus V3, so the total voltage drop
is equal to I1 R1 plus I2 R2 plus I3 R3. And what's the total
voltage drop? Well, that's equal to the total
current through the whole system. I-total, or we just call it I,
times the total resistance is equal to I1 R1, plus
I2 R2 plus I3 R3. Well, we know that all
the I's are the same. Hopefully, you can take it as,
just conceptually it makes sense to you that the current
through the entire circuit will be the same. So all these I's are
the same, so we can just cancel them out. Divide both sides by that I. We assume it's non-zero, so I,
I, I, I, and then we have that the total resistance of the
circuit is equal to R1 plus R2 plus R3. So when you have resistors in
series like this, the total resistance, their combined
resistance, is just equal to their sum. And that was just a very
long-winded way of explaining something very simple, and
I'll do an example. Let's say that this voltage
is-- I don't know. Let's say it's 20 volts. Let's say resistor 1 is 2 ohms.
Let's say resistor 2 is 3 ohms, and let's say resistor
3 is 5 ohms. So what is the total resistance through
this circuit? Well, the total resistance is
2 ohms plus 3 ohms plus 5 ohms, so it's equal to 10 ohms.
So total resistance is equal to 10 ohms. So if I were to ask you what is
the current going through this circuit? Well, the total resistance is
10 ohms. We know Ohm's law: voltage is equal to current
times resistance. The voltage is just
equal to 20. 20 is equal to the current
times 10 ohms, right? We just added the resistances. Divide both sides by 10. You get the current is
equal to 2 amps or 2 coulombs per second. So what seemed like a very
long-winded explanation actually results in something
that's very, very, very easy to apply. When resistors are in series,
we just add them up. I will see you in
the next video.