### Video Transcript

The decrease in potential across the resistor in the circuit shown is 15 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.

Okay, in this question, we’re shown a circuit, which consists of a resistor and two batteries. The question tells us that the decrease in potential across the resistor is 15 volts, and this is shown on our diagram. The question also tells us that the terminal voltage of one of the batteries powering the circuit is 5.5 volts. And this is also shown on our diagram. What the question wants us to do is find the terminal voltage across the other battery in the circuit. And to do that, we will use Kirchhoff’s second law, which tells us that the sum of the potential differences across each component in a loop in a circuit is zero. So, mathematically, if we had a loop with 𝑛 components, if we add together the potential difference across each of those components, it will sum to zero.

When we look at our circuit, we can see that it is made up of one closed loop. If we follow this loop around from the positive terminal of the 5.5-volt battery, the first component we come to is a resistor. And because this is a resistor, this is a decrease in potential, so this is negative 15 volts. Continuing to follow the loop around, the next component we get to is our battery with unknown terminal voltage. This is just a capital 𝑉 in our calculation. Finally, we get back around to the 5.5-volt battery, which we write as positive 5.5 volts.

We have now completed the loop. So this means that the sum of these three potential differences is equal to zero volts. The only unknown in this equation is the terminal voltage of our other battery, capital 𝑉. So all we have to do is rearrange the equation to make 𝑉 the subject and we’ll have our answer.

First, we’ll note that negative 15 volts plus 5.5 volts is equal to negative 9.5 volts. So negative 9.5 volts plus 𝑉 is equal to zero volts. If we add 9.5 volts to both sides of the equation, we see that the 9.5 volts on the left cancels with the negative 9.5 volts. And on the right, zero volts plus 9.5 volts is just equal to 9.5 volts. And this gives us our value for 𝑉. The terminal voltage 𝑉 of the other battery powering the circuit is 9.5 volts.