Video Transcript
Which of the following is the most correct description of how the range of values of voltages that a galvanometer being used as a voltmeter can produce can be extended when a multiplier resistor is used. (A) A multiplier resistor with a resistance much greater than that of the galvanometer is connected in parallel with the galvanometer. (B) A multiplier resistor with a resistance much smaller than that of the galvanometer is connected in parallel with the galvanometer. (C) A multiplier resistor with a resistance equal to that of the galvanometer is connected in series with the galvanometer. (D) A multiplier resistor with a resistance much greater than that of the galvanometer is connected in series with the galvanometer. And lastly, (E) a multiplier resistor with a resistance much smaller than that of the galvanometer is connected in series with the galvanometer.
In this situation, we’re thinking about a galvanometer, a device for measuring current, being used actually as a voltmeter. We can imagine that our galvanometer is part of a circuit that includes a variable voltage supply. If we think about the measurement scale that displays the galvanometer’s current reading, we know that there is some voltage that our source could theoretically supply that would lead to a current in our circuit that would maximally deflect the galvanometer arm.
If 𝐼 sub 𝐺 is this maximum current measurable by the galvanometer and 𝑅 sub 𝐺 is the total resistance of the circuit as it is, then 𝐼 sub 𝐺 times 𝑅 sub 𝐺 equals a potential difference. We’ll call it 𝑉 sub 𝐺. This is the voltage being supplied by our variable voltage source. Note that this equation here is an application of Ohm’s law. That law says that the potential difference in a circuit equals the current in the circuit multiplied by the circuit’s resistance. In the case of our galvanometer, we’re measuring this current 𝐼 sub 𝐺. We know the overall resistance of the circuit as it is, 𝑅 sub 𝐺. And by multiplying these values together, we can discover the potential difference or the voltage 𝑉 sub 𝐺 across our circuit. This is how we use a galvanometer to indirectly serve as a voltmeter.
With a circuit setup as we have it shown here, even a relatively low voltage will create a current 𝐼 sub 𝐺 that maxes out the measurement scale of our galvanometer. That is, in this circuit, the maximum measurable voltage is a fairly low value. Our question though talks about extending the maximum measurable voltage in our circuit. We do this by adding a resistor. We’ll call it 𝑅 sub 𝑚, since it’s a multiplier resistor to the circuit.
We can see that answer choices (D) and (E) talk about adding this multiplier resistor in series with the galvanometer. If we recall the descriptions of answer choices (A) and (B) on the other hand, we can note that these options are identical to options (D) and (E), except that they describe connecting the multiplier resistor in parallel with the galvanometer. And lastly, we can also use answer choice (E) as a template for answer choice (C). This answer option described a multiplier resistor with a resistance equal to that of the galvanometer, being connected in series with the galvanometer.
We have at least summary versions then of all of our answer options now. And we see that some of them involve connecting our multiplier resistor in parallel with the galvanometer and some in series. Let’s remember that the whole purpose of adding this multiplier resistor is to increase the measurement range of our voltmeter. As we think about doing this, we recognize that 𝐼 sub 𝐺, due to the limits of the scale of the galvanometer, is the maximum measurable current in the circuit. We can increase the current in the circuit anymore and still be able to measure the voltage across it. That means the only way to increase the maximum measurable voltage is to increase the overall resistance of the circuit. By Ohm’s law, if current 𝐼 stays constant and resistance 𝑅 increases, then the voltage 𝑉 will have to increase with 𝑅.
The question then becomes, how can we add 𝑅 sub 𝑚 into our circuit to maximally increase the circuit’s resistance? Say that we add our multiplier resistor in parallel with a galvanometer like this. Note that the galvanometer itself has some resistance. And so, what we basically have here is two resistors in parallel. Let’s imagine a general scenario where these two resistors are called 𝑅 one and 𝑅 two. The equivalent or total resistance of these two branches, which we’ll call 𝑅 sub 𝑡, is equal to the product of 𝑅 one and 𝑅 two divided by their sum.
Interestingly, this fraction will always be less than the value of the individual resistor 𝑅 one. And it will also always be less than the resistance of the individual resistor 𝑅 two. It’s a general rule that whenever two resistors are connected in parallel, the total or effective resistance is always less than the resistance of either of the individual resistors. This tells us that adding our multiplier resistor in parallel with a galvanometer is not an effective way to increase the overall resistance of our circuit. In fact, doing that decreases the circuit’s resistance. That will then decrease the maximum voltage measurable. So we know that we won’t put our multiplier resistor in parallel with the galvanometer. Therefore, we can eliminate answer choices (A) and (B) from consideration.
We know that really we’ll be putting our multiplier resistor in series with a galvanometer. As we do this, let’s say that 𝑅 sub 𝑡 is the total resistance in our circuit. With our multiplier resistor arranged this way in series, we can say that the total resistance in our circuit equals the circuit’s resistance when just the galvanometer was present with the voltage supply plus the resistance of our multiplier resistor. We’ve seen that since current 𝐼 has a maximum value of 𝐼 sub 𝐺, in this scenario, in order to increase the voltage measurement range, we want to increase the resistance of our circuit as much as possible.
The best way to increase 𝑅 sub 𝑡 is to add a resistor 𝑅 sub 𝑚 with a large resistance. If contrary to this, as answer option (C) says, we inserted a multiplier resistor with a resistance much smaller than that of the galvanometer, then we would only marginally increase the overall resistance of our circuit. That would mean we would only marginally increase the maximum voltage we can measure. Because we want to extend our measurement range to greater values, we won’t choose answer choice (E).
Similar to this, answer choice (C) says that we would choose a multiplier resistance equal to the resistance of our galvanometer. We can see that this would effectively double the overall circuit resistance. But considering that we could add a multiplier resistor with a resistance much greater than that of the galvanometer, even answer option (C) is not the best way of increasing the measurable voltage in our circuit.
To maximize that measurable voltage, we will maximize the circuit’s resistance. And we do that by adding a multiplier resistor with a resistance much greater than that of the galvanometer, in series with the galvanometer. For our answer, we choose option (D).
