A student has a resistor of unknown
resistance. She places the resistor in series
with a source of variable potential difference. Using an ammeter, she measures the
current through the resistor at different potential differences and plots her
results on the graph as shown in the diagram. What is the resistance of the
Looking at our graph, we see it’s a
plot of the current, in amperes, running through this resistor plotted against the
voltage, in volts, running across it. And based on the description in the
problem statement, we can make a little sketch of the circuit that generated the
data plotted here.
Let’s say that this is our resistor
of unknown value. We’re told that this resistor is
connected to a variable potential difference supply and that also in this circuit is
an ammeter for measuring current. The idea then is that we use this
variable supply of potential difference to apply two, four, six, and eight volts
across this resistor. And then using our ammeter, we read
out corresponding current values of 0.4, 0.8, 1.2, and 1.6 amperes.
With these values plotted on the
graph, we see they’ve been fit with a line of best fit that runs directly through
all four points and also passes through the origin. Now this line is indeed a line that
has a constant slope. And it’s that slope that will help
us answer this question of what is the resistance of our unknown resistor.
To see how, let’s recall Ohm’s
law. This law tells us that, for a
resistor of constant value, that resistance multiplied by the current running
through the resistor is equal to the voltage across it. In our case, we want to rearrange
this equation to solve for 𝑅. And we see that that’s equal to the
potential difference divided by the current. We aren’t given explicit values for
the potential difference or the current. But we can get those from the data
plotted in our graph.
Recall that those data points are
the basis for the line of best fit that passes through all of them. This means that, in order to supply
the voltage and current we need to solve for the resistance, 𝑅, we can choose from
among any of our four data points plotted in this graph. In fact, we could choose from any
point along this line of best fit line because it so happens to pass perfectly
through all these data points. But just to make things easier, we
may as well constrain our choice to these four. It doesn’t matter which of the four
we choose. Any of them will give the same
ratio and therefore the same overall result for the resistance of the resistor.
And just to pick one of the points
then, let’s choose the one at four volts. That voltage corresponds to a
current running through the resistor of 0.8 amps. So then to solve for the resistance
of the resistor, we’ll divide four volts by 0.8 amps. When we do this, we find a result
of five ohms, where ohm is the unit of resistance. Based on our graph and Ohm’s law,
we find the resistance of the resistor to be five ohms.