Video Transcript
The decrease in potential across the resistor in the circuit shown is 12 volts. The terminal voltage of one of the batteries powering the circuit is 5.5 volts. Find the terminal voltage 𝑉 of the other battery powering the circuit.
In this question, we’re shown a circuit, and this circuit contains three components,
a resistor and two batteries. The question tells us that the decrease in potential across the resistor is 12 volts,
and this is shown on the diagram. The question also tells us the terminal voltage of one of the batteries, which is 5.5
volts, which is also shown on our circuit diagram. The question wants us to find the terminal voltage of the other battery powering the
circuit. So that’s the potential difference across this other battery and this is labeled
𝑉.
To answer this question, we will use Kirchhoff‘s second law. And this tells us that the sum of the potential difference across each component in a
loop in a circuit is equal to zero. And what this means is if we have 𝑁 components in the circuit that the sum of the
potential difference across each of these 𝑁 components is equal to zero. In our question, we have three components. So we know that the sum of the potential differences across each of these three
components must add up to zero volts.
However, the direction we go around this loop matters. For example, if we’re going around the loop and we come across a battery where the
positive terminal is on the right and the negative terminal is on the left, if we go
across the battery from left to right, this is an increase in potential because we
have gone from the negative terminal to the positive terminal. However, if we go from right to left, this is a decrease in potential because we’ve
gone from the positive terminal to the negative terminal. Something similar happens with a resistor. If we have a resistor that has a decrease in potential across it of 𝑉 sub 𝑅, in one
direction, this will be a decrease in potential, and in the other direction, this
will be an increase. We just don’t know which direction this is.
Now, let’s take a look at how we can apply Kirchhoff’s second law to the circuit in
this question. Starting from this point here and moving clockwise around the circuit, the first
component we get to is the battery with the terminal voltage of 5.5 volts. And as we move across it, we’re going from positive to negative. So the potential difference in our calculation is negative 5.5 volts. Next, we continue around until we get to the battery with the unknown terminal
voltage. This time, we’re going across it from its negative terminal to its positive terminal,
so its potential difference in our calculation is positive 𝑉. Next, we continue around the loop until we get to the resistor.
As we go across the resistor, we don’t know whether this is an increase or decrease
in potential in this direction. So for now, we’ll write down both alternatives. Finally, we continue and complete the loop. And because we’ve completed the loop, we know that these potential differences must
sum to zero volts. So we’re left with two equations, but only one of them is correct. What we have to do now is work through each one of these to find out which one.
Clearing some space on the right, we’ll start with our first equation. Negative 5.5 volts plus 𝑉 plus 12 volts is equal to zero volts. Our first step is to note that negative 5.5 volts plus 12 volts is equal to 6.5
volts, so 6.5 volts plus 𝑉 is equal to zero volts. Subtracting 6.5 volts from both sides, we see that the two 6.5 volts on the left
cancel, and zero volts minus 6.5 volts is just negative 6.5 volts. So this first equation gives us a value of 𝑉 of negative 6.5 volts, and we’ll keep a
note of this here.
Next, we move on to the second equation. Negative 5.5 volts plus 𝑉 minus 12 volts is equal to zero volts. We can simplify the left-hand side by noting that negative 5.5 volts minus 12 volts
is equal to negative 17.5 volts. So negative 17.5 volts plus 𝑉 is equal to zero volts. This time adding 17.5 volts to both sides, we see that the 17.5 volts on the left
cancel, and zero volts plus 17.5 volts just equals 17.5 volts. So equation two gives us a value of 𝑉 is equal to 17.5 volts, and we’ll keep a note
of it over here.
So we have two possible answers for the value of 𝑉. But which one should we accept as our correct answer? Well, we should note that 𝑉 is the terminal voltage of a battery. And the terminal voltage of a battery is positive, and therefore 𝑉 itself is
positive. The value of 𝑉 that we got from our first equation of negative 6.5 volts is not
positive, so we should reject this answer. However, the value of 𝑉 that we got from our second equation of 17.5 volts is
positive, so we can accept this as our correct answer. So the terminal voltage 𝑉 of the other battery powering the circuit is 17.5
volts.
