A student set up the electrical
circuit shown in figure one. The ammeter displays a reading of
0.12 amps. Calculate the potential difference
across the 72-ohm resistor.
Okay, so in this question, what we
need to do is to find out the potential difference across this resistor here, the
72-ohm resistor. And we’ve been told that the
ammeter — that’s this ammeter here in the circuit — displays a reading of 0.12
amps. Now that’s all good, but let’s
first start by considering the flow of current in this circuit, because it’s going
to give us some vital information for both this part of the question and later parts
of the question as well.
Now let’s consider conventional
current, in other words current that flows from the positive terminal of the cell to
the negative terminal. So let’s start at the positive
terminal of the cell.
We’ve got a conventional current
flowing anticlockwise in our circuit. The first thing it meets in the
circuit is the ammeter, which we’ve been told displays a reading of 0.12 amps. Now because this is all of the
current in the circuit — it’s not like some of it has been split off or anything —
this means that the total current in the circuit is 0.12 amps. In other words, the ammeter is
measuring all of the current in the circuit, not just, for example, the current
flowing through one component in the circuit or anything like that.
Anyway, so the conventional current
continues to flow through the circuit anticlockwise and flows through a resistor
with resistance 𝑅. Now we don’t know what this
resistance is. And I think we can safely assume
that, in a later part of the question, we’ll have to work it out.
So the current having flown through
the resistor with resistance 𝑅 continues to flow anticlockwise and now arrives at a
junction. Now at this junction, we’d expect
the current to split two ways, some going this way and some going this way. However, let’s take a closer look
at what’s actually happening in the two branches along which the current could
Now along the bottom branch, we’ve
got a 48-ohm resistor. And that’s about it; there’s
nothing else on that branch. And at the top branch, we’ve got
just a switch. Now this switch is labeled 𝑆. And importantly, the switch is
closed. That means that there is a path for
current to flow along this branch as well, because the switch is closed, so current
can flow through it.
But here’s the interesting bit. Along the branch with the switch 𝑆
on it, there is absolutely no form of resistance. First of all, we don’t have a
resistor on there. And secondly, because this is a
circuit diagram and this is an ideal situation, we assume that the wires firstly
don’t have any resistance and a switch can basically be as simple as just a piece of
wire that’s moved into position depending on whether the switch needs to be open or
closed. In other words then, this entire
branch is just a long piece of wire. And in an ideal situation such as a
circuit diagram, we assume that wire has no resistance.
Therefore, the current flowing from
this junction has two options. One is to flow along a branch which
has some resistance, and the other is to flow along a branch which has zero
resistance. In a situation like this, the
current will always flow entirely along the branch which has no resistance. In other words, no current flows
along the branch with the 48-ohm resistor. All of it flows along the branch
with the switch on it.
Now we’ve mentioned that this is
only true for an ideal situation such as a circuit diagram. But what happens in a real
circuit? Well, in a real circuit, wires of
course do have some resistance. So both branches have some
resistance. However, because the resistance of
the wire is very small — it’s basically negligible — this means that only a very
tiny current flows through the 48-ohm resistor. And even in a real circuit, we can
basically ignore this current. In other words, we can just assume
that there is absolutely no current flowing through the branch with the 48-ohm
resistor, again coming back to the same conclusion as before.
Now remember, this is only true
when switch 𝑆 is closed and there is a path for current to flow through. If switch 𝑆 was open instead,
well, we’ll worry about that later.
Anyway, so the current having
flowed through the switch 𝑆 arrives at the second junction and continues to travel
anticlockwise, through the 72-ohm resistor, at which point we’ve now completed the
circuit. It’s arrived at the negative
terminal of the cell. So now we have a pretty good idea
of how the current flows through the circuit. This is useful for solving the
first part of the question, because we can see that at no point along the circuit
does the current have to split. Even at this junction, where we
would have expected it to, it doesn’t. And more importantly, all of the
current in the circuit is flowing through the 72-ohm resistor because the 72-ohm
resistor is not on a branch or anything like that. So all of the current is flowing
But we know already that the total
current in the circuit is 0.12 amps. Therefore, we now know that 0.12
amps are flowing through the 72-ohm resistor as well. This means that we know the current
through the resistor and the resistance of the resistor. And we’re trying to find the
potential difference across that resistor.
To work this out then, we can
recall something known as Ohm’s law. Ohm’s law tells us that the
potential difference, 𝑉, across a component in a circuit is equal to the current
through that same component, 𝐼, multiplied by the resistance of that component,
Now in this case, our component is
the 72-ohm resistor. And we already know the current
through it, 0.12 amps, and the resistance of the resistor, 72 ohms. So we can say that the potential
difference, 𝑉, is equal to, well, we have to substitute in the values of the
current and the resistance. So we plug in 0.12 amps for the
current and 72 ohms for the resistance.
Now at this point, it’s important
to note that we’re working in standard units. The standard unit of current is the
amp, and the standard unit of resistance is the ohm. Therefore, when we evaluate the
right-hand side of this equation, we’ll find our answer to be in the standard unit
of potential difference, which is volts. And volt is exactly what we need
our answer to be in. So we’re on the right track, and we
don’t need to manipulate any units here. Instead, we just work out the
right-hand side of the equation.
When we do this, we find that the
potential difference is 8.64 volts, and so our final answer to this part of the
question is that the potential difference across the 72-ohm resistor is 8.64
Now earlier, we guessed that we’d,
at some point in the question, have to calculate the resistance, 𝑅. Yeah, guess what we’re about to do
now! Yep, that’s right, calculate the
resistance of the resistor labeled 𝑅.
Okay, so how are we gonna go about
doing this? Well, one way to do this is to
consider the total resistance in the circuit. Why we consider the total
resistance will become clear momentarily. But the way to work out the total
resistance of the circuit is to use Ohm’s law once again, except this time we modify
it slightly to represent the entire circuit.
Now earlier, we saw that Ohm’s law
said 𝑉 is equal to 𝐼𝑅, where 𝑉 represented the potential difference across a
component in the circuit, 𝐼 represented the current through that component, and 𝑅
was the resistance of that component. What we’ve done here is to modify
this slightly. Instead of talking about the
potential difference, current, and resistance of one component in the circuit, here
we’re talking about the total potential difference across the circuit, the total
current through the circuit, and the total resistance of the entire circuit. That’s why we’ve got the subscripts
“tot” for total in each case.
So now we’re trying to work out the
total resistance of the circuit. Firstly, we already know the total
current through the circuit. It’s 0.12 amps. And if we look carefully, we
actually know the total potential difference across the circuit as well.
Do we really? Yes we do. We know that the total potential
difference across the circuit is 24 volts. That’s the emf of the cell. And so as the current flows along
the circuit, 24 volts of potential difference must be dropped across all the
different components before the current reaches the negative terminal of the
cell. So yes, we do actually know the
potential difference across the entire circuit.
This means that we need to
rearrange this equation to solve for 𝑅 total. We do this by dividing both sides
of the equation by 𝐼 total, the current. This way, the current on the
right-hand side cancels out, and we’re just left with the total resistance of the
circuit on the right-hand side.
Now at this point, we substitute in
the values for 𝑉 total and 𝐼 total. So we have 24 volts divided by 0.12
amps. And taking a quick look at the
units once again, we see that we’re working at standard units. So our answer here is going to be
in the standard unit of resistance, which is the ohm. Now this is a good thing because
our final answer needs to be in ohms as well.
Anyway, so when we evaluate the
left-hand side of the equation, we find that 200 ohms is the total resistance of the
circuit. Now we’ve gone through all this
trouble to find the total resistance of the circuit. Why? Well, because there’s another way
to find the total resistance of the circuit, specifically by looking at which
resistors the current is flowing through in the circuit.
Now as we said earlier, the current
flows anticlockwise through the circuit and flows through the resistor, 𝑅, and
along this branch, entirely avoiding the 48-ohm resistor, and then it flows through
the 72-ohm resistor. This means that, in our circuit,
the current is flowing through this resistor, 𝑅, and the 72-ohm resistor. And that’s it.
Now we can use this information to
work out the total resistance in the circuit in a different way. To do this, we first needed to
realize that the two resistors are connected in series, because the current first
flows through one of them and then through the other. And then we can recall a way to
work out the total resistance of a circuit when the resistors in the circuit are
connected in series.
So let’s remember then that the
total resistance of a circuit when the resistors are in series is given by simply
adding up all of the resistances of the resistors. So you add up the first resistance
to the second resistance to however many there are to the final resistance.
Now in this case, we’ve only got
two, the resistance 𝑅 and the 72-ohm resistor. Once again, it’s important to
reiterate that the current does not flow through the 48-ohm resistor, so we do not
include this in the resistance of the circuit. And so we can say that the total
resistance of the circuit is equal to 𝑅 plus 72 ohms.
Now at this point, we can see that
we’ve got two different expressions for the total resistance of the circuit. One of them tells us that the total
resistance of the circuit is 200 ohms. The other one tells us that this
must be the same thing as 𝑅 plus 72 ohms. So let’s equate 200 ohms with 𝑅
plus 72 ohms.
When we do this, all we need to do
now is to rearrange the equation to find 𝑅. We can subtract 72 ohms from both
sides. This way, we’ve only got 𝑅 left on
the left-hand side. And on the right-hand side, we’re
left with 128 ohms. Hence, we’ve just calculated the
resistance of the resistor labeled 𝑅. It’s 128 ohms.
Now this entire time we’ve been
thinking about the circuit with the switch 𝑆 being closed. Now the switch 𝑆 is in the circuit
for a reason. If we weren’t going to open it,
then they would’ve just given us a piece of wire. So let’s think about what happens
when the switch 𝑆 is opened. So state what happens to the total
resistance of the circuit and the current through the circuit when switch 𝑆 is
Now what do we have here? We know that the switch 𝑆 is no
longer closed. We’ve just opened it. This means that there’s no longer a
path for current to flow along this branch. None of this is happening. And so when the current reaches the
junction, it now must flow through the 48-ohm resistor. And none of it flows through the
other branch because of course there’s no path for current to flow. This means that, now in our
circuit, we’ve got an additional 48 ohms of resistance. And this is important because one
of the things we need to consider is the total resistance of the circuit when the
switch is opened.
We can recall from the previous
part of the question that the total resistance of the circuit is given by adding all
of the resistances when they’re placed in series. And in this case, they still are in
series, because the current first flows through the resistor 𝑅, then the 48-ohm
resistor, and then through the 72-ohm resistor. So now we can say that the total
resistance of the circuit is equal to 𝑅 plus 48 ohms plus 72 ohms.
Now initially, when the switch 𝑆
was closed, we only had 𝑅 and 72 ohms. But after we opened the switch,
we’ve got this additional 48 ohms of resistance. Now we can actually work out what
this is, but that’s not what we need to do. The question only asks us to state
what happens to the total resistance. It doesn’t say calculate the total
resistance. So we know that the resistance has
increased by an extra 48 ohms, and hence all we need to say is that the total
resistance of the circuit, 𝑅 tot, increases.
Now the second thing that we’ve
been asked to do is to state what happens to the current through the circuit when
the switch is opened. To do this, we can once again think
about a modified version of Ohm’s law for the entire circuit. The total potential difference
across the circuit is equal to the total current through the circuit multiplied by
the total resistance of the circuit. In this case, we’re trying to work
out what happens with the total current, so let’s rearrange the equation first.
If we divide both sides of the
equation by the total resistance of the circuit, then we’re left with the potential
difference divided by the resistance is equal to the current. Now with the potential difference
in the circuit, we know that initially it was 24 volts, and now after opening the
switch it must still be 24 volts, because simply opening a switch is not going to
change the behavior of the cell. It still has an emf of 24
And so 𝑉 tot, the total potential
difference of the circuit, stays the same. And as we’ve seen already, the
total resistance of the circuit increases, so the value of 𝑅 tot is now larger. So we’re taking the same number, 𝑉
tot, and dividing it by a larger number, 𝑅 tot. If we take the same value and
divide it by a larger value than before, then the end result is going to be smaller
than before. In other words, the total current
in the circuit is now smaller than 0.12 amps. And this means that this reading is
no longer true.
The value of 𝐼 tot, the total
current through the circuit, is going to decrease. And so that is our final
answer. When the switch 𝑆 is opened, the
total resistance of the circuit increases and the current through the circuit