Name one factor that can increase the potential difference at the terminals of an electric cell in a closed circuit.
To get started, let’s make a sketch of an electric cell in a closed circuit so we know what we’re working with. Okay, so here in our circuit, we have our electric cell. We’ve called the emf supplied by the cell 𝑣 sub 𝑐. And as well, that cell has an internal resistance what we’ve called lowercase 𝑟. Since the emf 𝑣 sub 𝑐 and the internal resistance lowercase 𝑟 are really all part of the same unit, we can draw a little box around them to show that. Besides our cell, the only other element in the circuit is this external variable resistor. Our goal is to name one factor that can increase the potential difference at the terminals of our cell.
Based on the way we’ve sketched it, the terminals of our cell are here and here. But in order to explicitly include the internal resistance of our cell, let’s say that the terminals of the cell are here and here, just outside of the box we’ve drawn. In other words, if we were to put a voltmeter across these two points in our circuit, then it’s the reading of that voltmeter measured across the real terminals of our electric cell that we want to increase.
Interestingly, there are a couple of different ways to think about what it is that this voltmeter measures. In a very literal sense, it measures the potential difference between one point here and another point here in our circuit. One way for electrical current to move between these two points is starting at the point on the right to move from right to left across the cell and then across the internal resistor. This is the most direct way to get from one of these points to the other. But as we can see, because we’re working with a closed circuit, there is another path. Beginning instead at the point on the left that we’re measuring, we can move around our circuit in a clockwise direction until we reach the point on the right.
Whether we think of our voltmeter as measuring the potential difference across our cell or the potential difference across the rest of the circuit, the magnitude of that reading will be the same, but the signs will be opposite. The reason for this is that when we move right to left across our cell, we’re adding potential difference into the circuit. After all, the cell is what powers the circuit. But then outside the cell, the function of the other elements in our circuit in our case just as variable resistor are to reduce potential difference. That is potential is lost or dropped across this element.
As a side note, note that the voltage drop across our external resistor because it’s the only element in our circuit outside of the cell is equal in magnitude to the potential difference the cell supplies. There is a mathematical way that we can express these potential magnitudes. That is the measurement across the terminals of our cell as well as the measurement corresponding to the rest of the circuit. Ohm’s law tells us that the potential difference across a circuit is equal to the current in the circuit multiplied by its resistance.
Speaking of current, since we know that in our closed loop, the current is the same everywhere even though we don’t have a value for it, let’s give it a name. Let’s call the current in this circuit capital 𝐼. And using this name, let’s apply Ohm’s law to figure out what our potential difference measurement would be when we’re considering the pathway across our cell. That is, including 𝑣 sub 𝑐 and its internal resistance. The reading of the voltmeter — in this case what we can call capital 𝑉 — will be equal to the emf or electromotive force supplied by the cell — what we’ve called 𝑣 sub 𝑐 — minus the current circuit 𝐼 times the internal resistance of the cell.
This equation tells us that unless the current in the circuit is zero, 𝑣 sub 𝑐, the emf supplied by our cell, is not equal to the potential difference across the terminals of the cell. We see that when current is flowing, 𝑉 the potential difference across those terminals is less than the emf. It’s this potential difference 𝑉 that our question is asking about. How do we increase that value? To figure it out, let’s take a look at the right side of this expression.
𝑣 sub 𝑐, the cell emf, is a fixed quantity. It’s like when you buy a battery and you look in the back and it says it supplies a certain number of volts. That value won’t change so we can’t adjust it to increase or decrease 𝑉. And likewise, the internal resistance of the cell, lowercase 𝑟, is also fixed. That value can’t get bigger or smaller. So the only adjustable parameter on the right-hand side of this equation is the current 𝐼. We see that if we were to decrease 𝐼, that is decrease the current flowing through our circuit, then because we’re subtracting this term 𝐼 times 𝑟 from a term which stays the same, that means that subtracting a smaller number will give us an overall larger result. In other words, this is one way to increase the potential difference of the terminals of the cell.
So let’s write that down as part of our answer. We found that decreasing the circuit current is one way to increase the potential difference of the cell terminals. But actually, there is a second equivalent way of having the same effect. And it has to do with the rest of the circuit, the circuit outside of the cell. Remember we said that our voltmeter would give the same magnitude result whether we were reading across our cell or whether we’re reading across the rest of the circuit. That outcome by the way is required by Kirchhoff’s voltage rule and ultimately by the conservation of energy.
This means that if we think purely of the magnitude of our reading, that is, the magnitude of the potential difference across the cell terminals, then there’s a second way we can write this potential. Referring to Ohm’s law, we can say that the potential drop across our external circuit — that is the circuit outside of our cell — is equal in magnitude to the potential supplied by the cell that is capital 𝑉. And that this magnitude of potential is equal to the product of the current in the circuit capital 𝐼 multiplied by the total resistance in the circuit outside of our cell. That is the circuit resistance starting at the left side of our cell and moving clockwise until we reach the right side.
As we can see, this total external resistance is simply the resistance of our adjustable resistor capital 𝑅. Notice now that our top equation as well as our bottom equation both have a magnitude of 𝑉 on one side. And that means we can equate the right-hand sides of these two expressions. So 𝐼 times capital 𝑅 is equal to 𝑣 sub 𝑐 minus 𝐼 times lowercase 𝑟. And if we add 𝐼 times lowercase 𝑟 to both sides of the equation and then on the left-hand side factor out our common 𝐼 term and then finally divide both sides of this equation by the sum of capital 𝑅 plus lowercase 𝑟, we find that that term in parentheses cancels out on the left. And we now have an expression for the current in our circuit in terms of the cell emf and our total resistance.
Notice by the way that this equation is a form of Ohm’s law for our circuit. Our emf, 𝑣 sub 𝑐, divided by the total resistance is equal to the current in our circuit. Anyway, we’ve developed this equation to show that there is an equivalent way to increase the potential difference across our cell terminals. As we saw, decreasing circuit current has this effect. Based on this equation we’ve just developed though, we can see just how that might happen. Recalling that 𝑣 sub 𝑐 and lowercase 𝑟 are fixed constants, we see that capital 𝑅 is the only adjustable factor on the right-hand side of this expression.
Now if we wanted to decrease the overall value of this fraction 𝑣 sub 𝑐 over capital 𝑅 plus lowercase 𝑟, what can we do to capital 𝑅? Well, if we increase the denominator of a fraction, that has the overall effect of decreasing the fraction itself. This realisation shows us a second equivalent way of increasing potential difference across the cell terminals. We could write it as decreasing the circuit current or we could also say by increasing the equivalent external resistance in the circuit. Either one of these statements explains one factor that increases the potential difference across the cell terminals.