To confirm Ohm’s law, an experiment is performed with a battery, an ammeter, a voltmeter, a fixed-value resistor, and a variable resistor. If 𝑅 is resistance, 𝑉 is voltage, and 𝐼 is current, which of the following shows a relationship between the graphed quantities that is consistent with Ohm’s law? In each case, assume the third quantity is held constant.
We’re asked to choose among five graphs. Each graph has three evenly spaced points labeled one to three on each axis, on the horizontal and on the vertical. Where the graphs differ is in the labeling of their axes and in their shapes. For example, choice (d) has resistance on the horizontal axis and current on the vertical axis, and its shape is a straight line. So our task is to select the correct graph such that the labeling of the axes and the shape of the graph are consistent with Ohm’s law.
Let’s recall Ohm’s law. Stated symbolically, Ohm’s law has that 𝑉 is equal to 𝐼 times 𝑅, where we’ve used the variables as defined in the question. 𝑉 is voltage, 𝐼 is current, and 𝑅 is resistance. Remember that the answer choices each have two quantities from Ohm’s law, and we’re told that, in each case, the third quantity is held constant. So let’s take two quantities from Ohm’s law, let them vary, and see what happens when we hold the third constant. Let’s start by letting current and resistance vary and holding voltage constant. When we do this, we can write Ohm’s law as 𝐼 times 𝑅 equals constant. In a situation like this, where two quantities multiply together to make a third constant quantity, we call those two quantities inversely proportional.
To understand why we refer to these quantities as inversely proportional and also figure out the relationship that we’re looking for, let’s divide both sides of 𝐼 times 𝑅 equals constant by 𝑅. On the left-hand side, 𝑅 divided by 𝑅 is just one and we’re left with 𝐼. And on the right-hand side, we have constant divided by 𝑅. So if we can vary 𝐼 and 𝑅 and hold 𝑉 constant, then we can write current is equal to some constant divided by resistance. We can now see why 𝐼 and 𝑅 are inversely proportional. 𝑅 is in the denominator of the fraction on the right-hand side, which means that as 𝑅 increases, the size of that fraction must decrease since the denominator is getting larger. But the fraction itself is equal to 𝐼. So if the fraction is getting smaller, 𝐼 is also getting smaller.
Conversely, if 𝑅 gets smaller, then the denominator of the fraction gets smaller, the whole fraction gets larger, and 𝐼 gets larger. We could’ve stated the same in terms of current. As current gets larger, resistance must get smaller. As current gets smaller, resistance must get larger. So current and resistance are inversely proportional since as one gets larger, the other must get smaller. As one gets smaller, the other must get larger.
Let’s now look at choices (d) and (e), both of which have 𝑅 and 𝐼 on the axes of the graph. Choice (d) shows a straight line with positive slope such that when 𝑅 is one, 𝐼 is one. And when 𝑅 is three, 𝐼 is three. But this violates the condition for inverse proportionality since a larger 𝑅 is giving a larger 𝐼 instead of a smaller 𝐼, as it ought to. So (d) is not the right answer. (e) is also not the right answer, since in this graph, as 𝐼 increases, 𝑅 also increases. When 𝐼 is one, 𝑅 is a little bit more than one. When 𝐼 is two, 𝑅 is close to 2.5. So clearly, this graph does not show inverse proportionality, and (e) is not the right answer.
It’s worth noting the general shape of a graph of two quantities that are inversely proportional. If 𝑥 and 𝑦 are such that 𝑥𝑦 equal 𝑐, a positive constant, then the graph of 𝑦 versus 𝑥 looks something like this. The important thing to note, which is true whether 𝑐 is positive or negative, is that the graph goes to very large magnitudes of 𝑦 when the magnitude of 𝑥 is very small, and goes to very small magnitudes of 𝑦 when the magnitude of 𝑥 is very large. So just by generally knowing what the graph of inversely proportional quantities should look like, we could’ve eliminated (d) and (e) since clearly neither of them have the right shape.
Let’s now turn to (a), (b), and (c). All three of these are graphs of voltage versus current. So let’s see what happens to Ohm’s law when we let voltage and current vary and hold resistance constant. In this case, Ohm’s law reduces to voltage is equal to constant times the current. Stated this way, voltage and current are what we call directly proportional. This is because two quantities that are directly proportional either both increase or both decrease in magnitude at the same time since one is a constant times the other. Although their signs could vary depending on the sign of the constant. But regardless of the sign of the constant, two quantities getting larger or smaller in magnitude together just has the shape of a straight line. And the slope of the line is the constant.
Furthermore, if one quantity is zero, the other quantity, being a constant times the first quantity, must also be zero. So the graph of directly proportional quantities would be a straight line passing through the origin whose slope is the constant. Note this fundamental difference between directly and inversely proportional quantities. Directly proportional quantities have a graph that must pass through the origin, whereas inversely proportional quantities have a graph that cannot pass through the origin.
At any rate, choice (b) can’t be the right answer because it isn’t a straight line. And of choices (a) and (c), only choice (a) is a straight line that, if extended, would pass through the origin. So only choice (a) meets the conditions for being the graph of directly proportional quantities. So choice (a) is the correct answer. And remember, this is because Ohm’s law dictates that if resistance is held constant, voltage and current must be directly proportional.
As one final note, let’s just imagine that the graph in choice (c) had also passed through the origin. How would we distinguish between (a) and (c)? The answer is that the constant of proportionality between the two directly proportional quantities is the slope of the graph. In this case, that constant is resistance, and resistance is always positive or zero. Choice (a) has a positive slope. Choice (c) has a negative slope. But since the slope is the resistance and resistance must be greater than or equal to zero, choice (c) does not represent a possible value for the resistance. Which is another way we could’ve eliminated it as an incorrect answer.