Question Video: Understanding the Design of the Ohmmeter Physics

Video Transcript

A circuit that can be used as an ohmmeter is shown. The circuit uses a galvanometer, a direct current source with a known voltage, a fixed resistor, and a variable resistor. The angle 𝜃 is the full-scale deflection of the galvanometer. Two resistors, 𝑅 one and 𝑅 two, are connected to the ohmmeter so that their resistances can be measured by the ohmmeter. The galvanometer’s angle of deflection is reduced by the angle 𝜙 when 𝑅 one is connected and its angle is reduced by 𝛼 when 𝑅 two is connected. Which of the following correctly relates the resistances of 𝑅 one and 𝑅 two? (A) 𝑅 one equals 𝑅 two, (B) 𝑅 one is less than 𝑅 two, or (C) 𝑅 one is greater than 𝑅 two.

So in this question, we’ve been given a circuit diagram of an ohmmeter. And we’re also shown the same circuit diagram, but this time with a resistor 𝑅 one connected in series and then the same circuit again, but this time with a resistor 𝑅 two in place of 𝑅 one. So let’s start just by reminding ourselves that an ohmmeter is a device which measures the resistance of a component such as 𝑅 one or 𝑅 two. In order to measure the resistance of a component, we connect it in series with the ohmmeter. The deflection of the needle on a galvanometer, which is represented in our circuit diagrams as a capital 𝐺 in a circle, indicates the value of the resistance.

Now, at this point, it’s useful to remember that the needle in a galvanometer actually responds to current. The idea behind an ohmmeter is that by applying a known voltage to a circuit containing a test resistor and a galvanometer, the needle on the galvanometer will respond to the amount of current in the circuit. We then know that if the test resistor has a really large resistance, then only a small current will exist in the circuit. Conversely, if the resistor has a very low resistance, then we’ll end up with a larger current in the circuit.

This relationship is summed up by Ohm’s law, 𝐼 equals 𝑉 over 𝑅. If we consider 𝐼 to be the current in the circuit, 𝑉 to be the voltage that we’re applying to the circuit, and 𝑅 to be the total resistance of the circuit, then we can see that by increasing 𝑅, the resistance, by a certain amount, we’ll decrease 𝐼, the current, by a proportional amount. In other words, the current in the circuit and the total resistance of the circuit are inversely proportional to one another. Now, if we look at the diagram on the left, we can see that the needle on the galvanometer is deflected fully. And incidentally, the angle of this deflection has been called 𝜃.

Now, a given galvanometer will have some current which causes maximum deflection of the needle. And we generally find this is in the milliamp or microamp range. Any current smaller than this will only cause a partial deflection of the needle, enabling the galvanometer to effectively measure that current. But any current greater than the full deflection current will just cause the needle to be fully deflected. In other words, a galvanometer on its own is only useful for measuring current within a small given range. And this is where these resistors come into play.

The function of the variable and the fixed resistors are to ensure that the ohmmeter on its own has just enough resistance such that the current is just big enough to cause maximum deflection of the needle. And once this is achieved, we can say that the ohmmeter has been calibrated. Once this has been done, then adding a resistor in series to the ohmmeter will increase the total resistance of the circuit and therefore decrease the current such that it’s now less than the current which would cause full-scale deflection of the galvanometer.

And at this point, it might be useful to remind ourselves that when we connect resistors in series, the total resistance is simply the sum of the resistances of the individual resistors in the circuit. So we know that connecting a resistor in series with the ohmmeter increases the overall resistance of the circuit and therefore causes the galvanometer’s needle to back away from full deflection due to a drop in current. The bigger the value of the resistor that we connect in series with the ohmmeter, the more the deflection of the needle of the galvanometer will decrease by.

In this question, we’re told that connecting a resistor 𝑅 one in series with the ohmmeter will cause the needle’s deflection to decrease by an angle of 𝜙. And we’re also told that connecting a resistor 𝑅 two to the ohmmeter will cause the needle’s deflection to decrease by an angle of 𝛼. Crucially, we’ve been told that 𝛼 is greater than 𝜙. In other words, connecting 𝑅 two to the ohmmeter causes the needle’s deflection to decrease by a greater amount. This means that the resistor 𝑅 two must be decreasing the total current in the circuit by a bigger amount than 𝑅 one does. Therefore, we can conclude that 𝑅 two is greater than 𝑅 one or equivalently 𝑅 one is less than 𝑅 two.

If a galvanometer’s angle of deflection is reduced by an angle 𝜙 when 𝑅 one is connected and its angle is reduced by 𝛼 when 𝑅 two is connected and 𝛼 is greater than 𝜙, then we can conclude that the resistance of 𝑅 one is less than the resistance of 𝑅 two.