A sensitive galvanometer can measure current up to 𝐼 g. A number of multiplier resistances are connected to its coil one at a time to convert it into a voltmeter. The table below records the maximum potential difference 𝑉 measured by the voltmeter, in volts, and the total resistance 𝑅 of the voltmeter, in ohms. Plot a graphical relationship between 𝑉, on the 𝑦-axis, and 𝑅, on the 𝑥-axis.
Okay, so in this question, first of all we’ve been told that we’ve got a galvanometer. And the maximum current that this galvanometer can measure is 𝐼 g. In other words, the galvanometer cannot measure any current larger than 𝐼 g. If a current larger than 𝐼 g does pass through the galvanometer, then it will just register as 𝐼 g. Now, we’ve also been told that a number of multiplier resistances are connected to the coil of the galvanometer one at a time to convert it into a voltmeter.
Now, multiplier resistances are basically resistors placed in series. So we connect a resistor to the coil of the galvanometer in series with it. And that’s the first multiplier resistance we connect to the galvanometer. Then, we can connect another one, also in series. And we do this one at a time in order to convert this galvanometer set-up into a voltmeter. In other words then, all of these components together — the galvanometer and the multiplier resistances, however many there are — are equivalent to just one voltmeter. We’ll see how this is possible later on. But for now, let’s take a look at the table that we’ve been given in the question.
The table gives the maximum potential difference measured by this voltmeter as well as the total resistance of the voltmeter, where, of course, that total resistance includes the multiplier resistances and whatever the internal resistance of the galvanometer is. And we’ve been given a value of 𝑉 and 𝑅 every time we connect a new multiplier resistance. What we’ve been asked to do in this question is to plot the graphical relationship between 𝑉 on the 𝑦-axis, or the vertical axis, and 𝑅 on the 𝑥-axis, or horizontal axis. So let’s go about doing that. But first of all, let’s clear some space on the screen.
Okay, so what we’ve got here is the table from the question as well as two axes ready for producing a graph, where the vertical axis is labelled 𝑉, in volts, and the horizontal axis is labelled 𝑅, in ohms. So to start plotting our graph, we first need to realise that our range on the vertical axis is between 100 volts and 300 volts. Therefore, this range needs to comfortably fit on our vertical axis. So we can divide our axis up into equal sections. And starting at zero, we can go up in intervals of 100 volts. Therefore, the way we’ve drawn this axis now, it can comfortably fit our range from 100 volts to 300 volts.
Similarly, for the horizontal axis, we can see that the 𝑅-values are between 500 and 1500. So our horizontal axis needs to be labelled such that all values between 500 and 1500 will comfortably fit on our graph. So, once again, starting at zero and dividing our axis up into equal segments, we can now go up in steps of 200. So 200 ohms, 400 ohms, and so on, until we get to our final label which is 1600 ohms. Therefore, the range 500 to 1500, which is between here and here, is going to comfortably fit on our graph. So what are we waiting for? Let’s get plotting!
Let’s start with this pair of values here. We need to plot 100 volts against 500 ohms. Now, one way to do this is to start on the horizontal axis and find this value here, 500 ohms. To do that, we start at zero and go across 200 ohms, 400 ohms, and 500 ohms. 500 ohms is exactly halfway between 400 ohms and 600 ohms. So we can put a dot on the axis at 500 ohms. Then, we can draw a dotted line going upwards from that point. Now, this doesn’t have to be drawn on the graph. It can just be done in our mind. But it often helps to draw it out on the graph in pencil so that it can be erased later.
And then, we do a similar thing with the vertical-axis value. We need to plot 100 volts. So we start at zero on the vertical axis and move upwards 100 volts. Then, we place a dot on the axis at that point. And then, we draw a dotted line either in our minds or on the graph itself across. Now, at this point, we’ve got two dotted lines, this vertical one and this horizontal one. Where these two dotted lines intersect is where we plot our point. So now that we’ve plotted our first point on the graph, we can erase the dotted lines that we drew earlier on. And then, we can move on to plotting our next point.
This one has a 𝑉-value of 150 and an 𝑅-value of 750. So let’s plot this point. Starting with the horizontal value, we’re looking for 750 ohms. So what we do is we start at zero and then go across 200, 400, 600. Now, we’re looking for 750. Well, 700 is exactly halfway between 600 and 800. And 750 is exactly halfway between that and 800. So we can put a dot on the axis over here because that’s where 750 ohms is. Now this time, when we’re plotting the 𝑉-value of 150 volts, instead of going up on the vertical axis, we could just choose to go straight up from the point that we’ve drawn. So this is 100 volts up, and 150 is here. Because that point is exactly halfway between 100 volts and 200 volts. And so, we can just place our cross here. And at that point, we’ve plotted our second point on the graph. Then, we can repeat this process for the third point, the fourth point, and the fifth point.
Now, at this point, we’ve plotted all of the values on our table. We can even choose to draw a line of best fit. So here’s the line of best fit that we draw. And by some happy accident, all of the points on our graph happen to sit on the line of best fit. As well as this, the line of best fit goes through the origin, which is even neater. This will become important really soon. But for now, let’s realise that we’ve completed what the question asks us to do. We’ve plotted a graph of 𝑉 against 𝑅. So let’s look at the next part of the question.
From the graph, find the measuring range of the galvanometer, 𝐼 g.
Okay, so to answer this, let’s first recall what we said earlier. We said that we had a galvanometer. And to it, we were adding lots of multiplier resistances in series to turn it into a voltmeter. Now, adding these multiplier resistances would not change the behaviour of the galvanometer itself. In other words, the maximum current that the galvanometer itself can measure is still 𝐼 g. That doesn’t change. However, the reason that this whole thing works as a voltmeter is that as we add more and more multiplier resistances, there are more and more components in the circuit for there to be a potential difference across. In other words, when the circuit consisted of just the galvanometer and none of the multiplier resistances, then whatever potential difference was being measured had to all be across the galvanometer. And at this point, we can recall something known as Ohm’s law, to take us a little bit further.
We can recall that the potential difference across a component in a circuit is equal to the current through that component multiplied by the resistance of that component. Therefore, when our voltmeter circuit consisted of just the galvanometer, then the maximum voltage that could be measured by the galvanometer, let’s call that 𝑉 subscript g, was equal to the maximum current that could be measured by the galvanometer multiplied by the internal resistance of the galvanometer. However, when we add a multiplier resistance, so let’s ignore this one for now, then the maximum potential difference that can be measured by this whole set-up is now 𝑉 sub g across the galvanometer. But there can also be a potential difference, let’s call it 𝑉 subscript 𝑅, across the resistor. And let’s say that the resistor has a resistance 𝑅, by the way.
Well, in that case, the potential difference across the resistor is equal to the current through the circuit. But that must be 𝐼 sub g because we’re talking about the maximum current of the galvanometer. And the galvanometer and the resistor are in series. And so, we multiply 𝐼 sub g by the resistance of the resistor, 𝑅. So now that we have an extra resistor in the circuit, even though the potential difference across the galvanometer itself is still 𝑉 sub g, that hasn’t changed. But the fact that there’s an extra voltage across the resistor means that we can calculate the total potential difference across this entire set-up.
So even though the galvanometer doesn’t directly measure this potential difference, by measuring the current through it, we can calculate what this new maximum potential difference is based on the fact that we know the value of 𝑅. And, of course, the more resistors we add, the more of them can take up potential differences. And so, the maximum potential difference that can be measured via calculation, by this entire set-up, gets larger and larger. And that’s exactly what we’ve plotted here. We’ve plotted the entire potential difference across the entire set-up on the vertical axis against the total resistance of this entire set-up, so that’s these resistances plus the internal resistance of the galvanometer, on the horizontal axis.
Now, based on that, we’ve been asked to find the measuring range of the galvanometer, 𝐼 g. And that’s the key to this whole thing. The reason that the resistors are placed in series with the galvanometer is because the current through all of them is the same. And hence, the maximum current that can go through them is 𝐼 g because that’s the maximum current that can be registered on the galvanometer.
Now, since we know that we’ve plotted the maximum possible voltage measured by the voltmeter for every time we add a multiplier resistor, we know that the current through the voltmeter must have been 𝐼 g in each case. Because if the current was any less than 𝐼 g, then that wouldn’t be the maximum possible voltage that could be measured by the voltmeter. And if the current was greater than 𝐼 g, the galvanometer wouldn’t register it. And so, we just measured the voltage to be whatever those values were, even if the actual potential difference was larger. Now, the whole point of this is that we can use this graph, the 𝑉-against-𝑅 graph, to work out the current through the voltmeter.
Now, let’s go back, once again, to assuming that the entire set-up is equivalent to a voltmeter and that for a particular number of multiplier resistances, let’s say two, the maximum voltage measured is 𝑉. And the resistance is the sum of all of these, which is 𝑅. Well, in that case, we can just treat the voltmeter as a component in the circuit and use Ohm’s law once again. Ohm’s law tells us that the potential difference across the voltmeter is equal to the current through the voltmeter, which we said must be 𝐼 g, multiplied by the resistance, 𝑅. And therefore, if we want to find the value of 𝐼 g, we can simply rearrange by dividing both sides of the equation by the resistance 𝑅 so that it cancels on the right-hand side. And what we have is 𝑉 over 𝑅 is equal to 𝐼 g.
This means that we can take any one of the points on our graph and divide the voltage value across the voltmeter by the total resistance of the voltmeter set-up to find the maximum possible current measured by the galvanometer. So let’s do this; let’s just pick a point on the graph. We’ve already put an arrow next to this one; so that makes it convenient. And the good thing is we don’t actually have to read it off from the graph itself. We know that that point corresponds to this one in the data table. Therefore, we know that the value of 𝑉 at that point is 150 volts. And we divide this by the value of 𝑅, which is 750 ohms. And this will give us 𝐼 g. Then, we can see that because we’re working in standard units, so that’s volts for potential difference and ohms for resistance, our answer for 𝐼 g is going to be in its own standard unit. That’s the ampere. And so, when we evaluate 150 volts divided by 750 ohms, we get 0.2 amps as the measuring range of the galvanometer.
Now, let’s notice a couple of things. Firstly, we could’ve done this with any point, and we still would’ve got the same answer. Secondly, our line of best fit is a straight line through the origin. This means that our voltmeter is an ohmic conductor; it follows Ohm’s law. And so, in a special case like this, where we’ve got a straight line going through the origin, it turns out that instead of just dividing 𝑉 by 𝑅 for one of the points on the graph, we could’ve actually found the slope of the graph. In other words, that’s the change in 𝑉 divided by the change in 𝑅. And when we do this, we’d get exactly the same answer.
However, it’s important to know that this method only works for an ohmic conductor or, in other words, when we’ve got a straight line through the origin. Because this indicates direct proportionality. But anyway, coming back to our final answer then, we’ve just found that the measuring range of the galvanometer is 0.2 amps.