Electricity and magnetism
Circuits (part 2) Resistors in series
Circuits (part 2)
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- Let's make our circuit 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.
- 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.
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