A resistor is placed in a circuit with an adjustable voltage source. Measurements of the voltage across and the current through the resistor are shown in the accompanying graph. Find the resistance of the resistor.
We can call that resistance capital 𝑅. The measurements that are shown on this graph are very much like the measurements George Ohm himself made in order to derive Ohm’s law from data. Each one of the data points on this graph gives us two pieces of information about the resistor in the circuit.
See me look more closely at this point where the resistor has 400 volts of potential difference across it and four amps of current moving through it. So for each of the points on the graph, we know voltage and current, 𝑉 and 𝐴.
Looking at the graph overall, we see there’s a trend. That trend is made more visible if we connect all the points with a line of best fit that moves through them. The slope of this line overall equals the change in voltage divided by the change in current. One of Ohm’s great observations was that, for a given resistor, this ratio is a constant value.
As we can see by looking at the different data points on this graph, Ohm found that this constant slope ratio was in fact equal to the resistance of the resistor 𝑅. Based on the data, we can write that the resistance 𝑅 of the resistor in the circuit equals the change in voltage over the change in current in the circuit.
If we slightly rearrange this expression, it may look more familiar. 𝑉 equals 𝐼 times 𝑅. This is what we more commonly recognize as Ohm’s law. But this law is based on measurements that Ohm made that look when they’re plotted much more like the graph we have shown here. So what is the resistance 𝑅 of the resistor in our circuit?
To find out, we look at the slope of our best fit line. Since the slope is constant, we can choose any two points to determine it. Let’s choose the point at the origin zero, zero and the point we chose earlier at 400 volts and four amps.
So we see that Δ𝑉 is 400 minus zero volts and Δ𝐼 is four minus zero amps. This equals 400 volts over four amps or 100 ohms. That’s the value of the resistor whose measurements are shown in this graph.