Video Transcript
In this video, we’re talking about
series circuits. These are electrical circuits where
charge has only one path to follow as it moves through the circuit. It never splits up or divides
across different branches, but rather it all moves within the same closed,
continuous loop.
As an example of this, consider
this circuit. This circuit consists of a cell,
one resistor here, and a second resistor here. Conventionally, charge flows from
the positive to the negative terminal of a cell in a circuit. So, we could draw that current in
like this, counterclockwise, also known as anticlockwise, in the circuit. And since there’s only one pathway
for the moving charge to follow, we know that this is a series circuit.
Now in this or any other series
circuit, potential difference 𝑉, current 𝐼, and resistance 𝑅 all follow certain
rules. If we start out by considering 𝑉,
the potential difference, we know that, in general, a cell in our circuit will
supply a potential difference that causes charge to move. As charge moves throughout the
circuit though, whatever increase in potential difference we experience thanks to
the cell must decrease as charge moves throughout the rest of the circuit. Whereas the cell is the place where
potential difference is increased, the remaining circuit components, in this case,
these two resistors, are the places where potential difference is decreased.
To see how potential difference
changes throughout this circuit, say that we start at this point right here and we
move in the direction of conventional current, counterclockwise. One way we could depict all this is
this way. We open up our circuit, starting at
that negative terminal of our cell, and then we stretch it out as though it’s in a
line, being sure to include all the other components in our circuit. In this case, these two
resistors. And let’s say that on a vertical
axis, we have potential difference Δ𝑉 plotted. So, we’ll start here at the
negative terminal of our cell, and then we’ll move in the direction of conventional
current.
If we did this, then we would see
that over the course of traveling across this cell, the potential difference would
increase to this value 𝑉 supplied by the cell. Then, as we follow the direction of
charge flow, our potential difference would be maintained at this value until we
reach the first resistor. As charge crosses that first
resistor, potential difference is lost. And in terms of the amount of how
much is lost, if we assume that this resistor is the same as this one, then the
potential difference lost across this first resistor will be equal to half the
potential difference supplied by the cell. More on that later, but for now,
let’s continue to follow charge as it moves anticlockwise through our circuit.
Once we cross over the first
resistor, potential difference stays constant until we reach the second
resistor. And moving across this resistor,
our potential difference again drops. In this time, it drops all the way
down to its original value at the negative terminal of our cell, zero. From that point on, until we reach
what we can call the start point of our circuit, the negative terminal of our cell,
the potential difference is zero. Notice that the potential
difference starts and ends at the same value; that’s required in order for this
circuit to be complete. And notice also that what we gain
in potential difference from the cell is dropped across the various components of
our circuit. This also is always true in an
electrical circuit.
Now, so far, we haven’t given any
labels to these two resistors. But let’s do that now. Let’s call this first resistor 𝑅
one and the second resistor 𝑅 two. Looking back down at our sketch, we
could say that this amount of potential difference, that was dropped over 𝑅 one, is
𝑉 one. And we can call this amount of
potential difference dropped over 𝑅 two 𝑉 two. And then with these values labeled,
we can see there’s a relationship between 𝑉 one, 𝑉 two, and 𝑉 which is the
potential difference supplied by the cell. Specifically, we can see that 𝑉 is
equal to 𝑉 one plus 𝑉 two.
Now, earlier, we talked about the
possibility that 𝑅 one is equal in value to 𝑅 two. That is, each resistor supplies the
same number of ohms of resistance. But if 𝑅 one is equal to 𝑅 two,
then that means the potential difference dropped over each resistor is the same. In other words, 𝑉 one is equal to
𝑉 two. And then, if 𝑉 one is equal to 𝑉
two and 𝑉 one plus 𝑉 two is equal to 𝑉. Then that means the potential
difference dropped over the first resistor, 𝑉 one, is equal to the potential
difference supplied by the cell divided by two. And that 𝑉 two, the potential
difference dropped over the second resistor, is also 𝑉 over two. And this conclusion, we recall,
requires that 𝑅 one be equal to 𝑅 two.
Now, in a series circuit with two
resistors, it’s a bit of a special case when those resistors have the same exact
value. When that happens, we were able to
make this conclusion about the potential difference that’s dropped over each
one. But even when that’s not the case,
even when the resistors have different values, it’s still true that the potential
difference supplied by the cell is equal to the sum of the potential differences
dropped over the two resistors. And this equation here is very
close to a general expression we can write for potential difference in a series
circuit.
The expression we can write will
look like this. We’ll say 𝑉 sub 𝑡, where 𝑉 sub
𝑡 is the potential difference supplied by the cell or battery in the circuit, is
equal to the potential difference dropped over the first resistor. Plus the potential difference
dropped over the second resistor, if there is one, plus the potential difference
dropped over the third resistor, if there is one of those, plus dot, dot, dot. Meaning the potential difference
dropped over any other resistors that might be in this series circuit.
If we happen to have a series
circuit with only one resistor, then our equation would simply look like this: 𝑉 𝑡
is equal to 𝑉 one. But if we had two resistors, it
would look like this, very much like the one we generated here. And then if we had three resistors,
we would add together the potential difference dropped over those three. And that would equal the total
supplied by the cell and so on, up to as many resistors as we might find in our
series circuit. So, that’s potential difference and
how it behaves in a series circuit.
Now, let’s move on to consider
current. Remembering our definition of a
series circuit, a closed loop where charge only has one path to follow. The implication of that is as
charge travels around this circuit, the value of the current at any given point is
always the same. That’s because any bit of charge
that reaches this point will also reach this point and this point and this point and
this point and this point and any other point in the series circuit. The charge has no choice, we could
say, but to follow that same path. Let’s say that in this series
circuit we’re considering, there’s some total value of the current we’ll call it
𝐼. And what we want to know is how
does the current at any given point in the circuit, say, at the resistor 𝑅 one or
at the resistor 𝑅 two, compare to this value of the total current.
From what we’ve seen so far,
because any individual charge that travels through the circuit must cross all of it,
that would mean that if 𝐼 one is the current through the first resistor, then
that’s equal to the total current. And it’s also equal to 𝐼 two, what
we could call the current through the second resistor. And it’s furthermore equal to the
current at any given point, whether in a component or not, in this circuit. In a series circuit then, the rule
for current is simple. It’s the same everywhere.
And we can write that as a general
expression this way. We can say that the total current
in the circuit is equal to the current in the first component, which is equal to the
current in the second component, if there is one, which is equal to the current in
the third component, if there is one of those, and so on. This fact that current is the same
everywhere in a series circuit is often very helpful to us when we’re solving
exercises. Okay, now, let’s move on to our
last property that of resistance.
How do resistors behave in a series
circuit? In particular, given a series
circuit, like this one here, that has more than one resistor in it. What we may wonder is the total
equivalent resistance of that circuit. For this circuit, the total
equivalent resistance it has is equal to the sum of its individual resistors 𝑅 one
and 𝑅 two. This may seem common sense. But there is another type of
circuit known as a parallel circuit, where this sort of rule does not hold. But in this case, it does. In series circuits, to find the
overall resistance of the circuit, and we can write this as a general equation where
our overall resistance we’ll call 𝑅 sub 𝑡. To find that value, we take all the
individual resistances that may appear in the circuit, and we add them together.
So if we have a series circuit with
just one resistor, then the total resistance is equal to that resistor’s value. But if we have a circuit with two
resistors like we do over here, then the total resistance is 𝑅 one plus 𝑅 two, and
so on and so forth when we have progressively more resistors in our series
circuit. These rules that we’ve discovered
for series circuits are worth memorizing because they come up over and over again
when we talk about these types of circuits.
Taking a step back now, let’s
consider this simplified circuit diagram that we’ve drawn, which has been stripped
of any labels of 𝑉 or 𝑅 values. When we’re first learning about
series circuits, it may seem that there is some significance in the position of
these different components, the resistors in the cell relative to one another. For example, we might think that
this circuit is essentially different from, say, this circuit.
And the reason we might think that
is because we can see that this resistor, which was originally here in the circuit,
has now been moved to this side of it. Or to continue the example, we
might think that this circuit is different in an important way from this one. And we could think that because the
cell is reversed in polarity in this third circuit we’ve sketched in compared to the
first one.
As it turns out though, the changes
that we’ve sketched in, as well as many that we haven’t, don’t essentially change
the properties of the circuit we’re considering. In all cases, we still have one
cell and two resistors in series with one another. And whether charge moves clockwise
or counterclockwise through these circuits doesn’t make a difference in terms of
their performance. This means that there are many
different ways of drawing the same circuit. So long as the various components
in the circuit are all the same, in other words that, say, the cell supplies the
same potential difference and the resistors all have the same value. Then these circuits are all what
are called equivalent.
Equivalent circuits perform the
same way and have the same characteristics. For this circuit and effectively it
is a single circuit, even though we’ve sketched it three different ways, it doesn’t
matter whether the resistors are on the bottom in the side versus the two sides, or
whether the cell’s positive terminal faces left or right. In all these cases, the circuit’s
performance is the same. So, these are all equivalent
expressions of one and the same circuit. And that’s why we say they are
equivalent.
Knowing all this about series
circuits, let’s now get a bit of practice through an example exercise.
A battery supplying 12 volts is
connected in series with two resistors. The potential difference across
the first resistor is four volts. What is the potential
difference across the second resistor?
Okay, so in this scenario, we
have a battery connected in series with two resistors. So, let’s begin by sketching
the circuit. Here is our battery, and we’re
told that it supplies a potential difference of 12 volts. And then here are the two
resistors it’s connected with in series. We’re not told about the values
of these resistors. But just to give them names so
we can refer to them, let’s call this one 𝑅 one, and this one we’ll call 𝑅
two. The problem goes on to tell us
that the potential difference across the first resistor, across what we’ve
called 𝑅 one, is four volts. So, let’s label that this
way. Let’s say that 𝑉 one is the
potential difference across this first resistor. So, 𝑉 one is four volts. And let’s call 𝑉 two the
potential difference across 𝑅 two.
Now, it’s that value 𝑉 two
that we want to solve for. And to do it, we can recall
something about electrical circuits. In any electrical circuit, if
we make one complete closed loop around the circuit, then the potential
difference across that entire loop must be zero. It’s like saying that we need
to start and end in the same place. And that’s true because we are
talking about a circuit. In terms of potential
difference, that means that whatever potential difference is supplied by our
cell or battery, in this case, 12 volts, must be dropped or diminished across
the rest of the circuit outside of the battery.
Now, in our case, the rest of
the circuit consists of our two resistor components, 𝑅 one and 𝑅 two. These are the only places in
the circuit where potential difference drops. This means we can now write an
equation in terms of the various potential differences in the circuit. We can say the potential
difference supplied by the battery 12 volts must be equal to the potential
difference dropped across 𝑅 one, that’s 𝑉 one, plus the potential difference
dropped across 𝑅 two, what we’ve called 𝑉 two. And by the way, this expression
we’ve written here is a specific case of a general equation. That equation says that the
total potential difference supplied by a battery or a cell in a series circuit
is equal to the sum of the potential difference drops across the various
components of the circuit.
In our case, we only have two
components, our two resistors 𝑅 one and 𝑅 two. So, our equation basically
looks like this. And that’s what we see here
with 12 volts in place of 𝑉 sub 𝑡. In this equation, it’s 𝑉 two
that we want to solve for. And we already know that 𝑉 one
is equal to four volts. So, when we substitute that in,
we just need to subtract four volts from each side of this equation. And then, positive four volts
minus four volts on the right cancels one another out. And we wind up with this
expression; 12 volts minus four volts is equal to 𝑉 two, and 12 minus four is
eight. And so, this is our value for
𝑉 two. The potential difference across
the second resistor is eight volts.
Let’s summarize now what we’ve
learned about series circuits. In this lesson, we’ve learned that
a series circuit is an electrical loop in which there is only one path for charge to
travel. We also learned a series of
equations describing potential difference, current, and resistance in series
circuits. These equations showed us that the
total equivalent resistance in a series circuit is equal to the sum of the
individual resistors. That current in a series circuit is
the same everywhere. And that the potential difference
supplied by the cell or battery is equal to the sum of the potential differences
dropped across the other components of the circuit. And lastly, we learned about
equivalent circuits, which are circuits that are arranged differently but that
perform in the same way. This is a summary of series
circuits.
