Lesson Video: Design of the Ohmmeter Physics

Video Transcript

In this video, we’re going to be looking at the design of the ohmmeter.

An ohmmeter is a device which is used to measure the electrical resistance of a component. In circuit diagrams, we can represent an ohmmeter with an uppercase letter Ω in a circle. And in this circuit diagram, the ohmmeter is being used to measure the resistance, 𝑅 𝑥, of a test resistor. In this video, we’ll see how we can construct an ohmmeter using a cell, a galvanometer, a variable resistor, and a fixed resistor connected in series.

So to start things off, let’s consider a resistor. Let’s say that this resistor has some resistance, 𝑅 𝑥, and we want to build an ohmmeter so we can measure the value of 𝑅 𝑥. Well, Ohm’s law tells us that the resistance of a component is given by the voltage across that component divided by the current in that component. So as a general idea, if we could apply some known voltage 𝑉 to a resistor and then measure the current 𝐼, then we could calculate its resistance by dividing the voltage by the current.

So with this in mind, let’s connect a cell to our resistor. Now that we’re applying a potential difference, let’s call this 𝑉, to our resistor, we’ll find that there’s a current in the circuit, and we can call this current 𝐼. If our voltage is known, then all we need to do is measure the current 𝐼 and we can calculate the resistance using Ohm’s law. To help us measure the current, let’s introduce a galvanometer to our circuit. And let’s quickly recall that a galvanometer is a device which can measure the magnitude and direction of a current using a needle and a dial.

As a quick side note, because most galvanometers measure current in both directions, the zero tends to be in the middle of the scale. Since the ohmmeter we’re trying to build is a direct current circuit, that is, a circuit where current only goes in one direction, we can modify our galvanometer so that the zero is at one end of the scale. So now things are set up so that the needle will deflect this way in response to a current in this direction in our circuit. With this slightly modified galvanometer in place, we now have some way of getting information about the current in the circuit and therefore the resistance of our test resistor.

Because we’re trying to build an ohmmeter, we want to change the dial on our galvanometer so that it shows resistance rather than current. Ohm’s law tells us that if the current in our circuit is really small, then the resistance must be really big. And similarly, if we’re getting a really big current, then it must mean that the resistance is small. Taking this idea to its extreme, we could say that if a circuit has zero current, then it must have effectively infinite resistance. This would be equivalent to putting a break in our circuit, which would make it impossible for a current to exist.

So since zero current means that resistance is infinite, then we could change the dial on our galvanometer so that when the dial is in this position, instead of reading zero current, it reads infinite resistance. So now if this end of our scale corresponds to infinite resistance, that means, ideally, we want the other end of the scale of our ohmmeter to correspond to zero resistance.

Now we know that galvanometers are very sensitive instruments. So they generally reach maximum deflection in this direction for a pretty small current, usually in the region of microamps or milliamps. We can call this current 𝐼 𝑔. If we want to set up our ohmmeter so that the needle reaches this position when the test resistance is zero, that means we need to make some modifications to our circuit so that when the value of the test resistance is zero, the value of the current in the circuit is 𝐼 𝑔, the maximum deflection current of the galvanometer.

To achieve this, we need to wire in a couple of resistors to our circuit, a variable resistor represented by a resistor sign with a diagonal arrow through it and an ordinary fixed resistor. And we can say that the resistance of the variable resistor is 𝑅 𝑉 and the resistance of the fixed resistor is 𝑅 𝐹. While we’re on the topic of resistors, it’s important to remember that the galvanometer has its own internal resistance, 𝑅 𝐺. The function of these additional resistors is to ensure that when the value of this test resistance is zero, the current in the galvanometer is just enough to cause maximum deflection of the needle.

We can now do some calculations to determine what these values of resistance need to be in order to make this the case. Since we’re now considering the situation where the test resistance is zero, this is equivalent to replacing the test resistor with just a wire. So since all of this stuff is effectively our ohmmeter, we’re effectively short-circuiting the ohmmeter. To work out the values of resistors that we need to use, we can use Ohm’s law. We want to construct our ohmmeter, so that the total effective resistance of this circuit when the test resistance is equal to zero is just sufficient to limit the current to 𝐼 𝑔. In other words, we’re looking to make the total resistance of our ohmmeter, which we can call 𝑅 Ω, satisfy this equation.

Let’s also recall that when we have several resistors connected in series, the total effective resistance, 𝑅 𝑇, is equal to the sum of the individual resistances. This means that the effective total resistance of all of our ohmmeter components connected in series is equal to 𝑅 𝑉, the resistance of the variable resistor, plus 𝑅 𝐹, the resistance of the fixed resistor, plus 𝑅 𝐺, the resistance of the galvanometer. So we can replace 𝑅 Ω in our expression with 𝑅 𝑉 plus 𝑅 𝐹 plus 𝑅 𝐺. If we now subtract 𝑅 𝐹 and 𝑅 𝐺 from both sides of this equation, we’re left with this expression, which enables us to calculate the value that we need to set our variable resistor to in order to correctly calibrate the position of the zero on our ohmmeter.

Practically speaking though because a variable resistor is by its nature easily adjusted, we could correctly calibrate our variable resistor simply by decreasing it from its maximum resistance until the needle on the ohmmeter reaches the maximum deflection. And at this point, we’ll know that the maximum deflection on the dial corresponds to a test resistance value of zero. So we can confidently write a zero at this end of the scale. So we now have a completely assembled and correctly calibrated ohmmeter. If we reintroduce a test resistor, then the needle on the dial will change to indicate its resistance. However, here we run into a problem. Our ohmmeter measures values of resistance from ∞ to zero. However, we don’t know what any of these values in the middle of the dial actually correspond to.

Luckily, we can figure out the scale on our ohmmeter by using the fact that the deflection of the needle on a galvanometer is proportional to the current. This means that if a current 𝐼 𝑔 is sufficient to cause full deflection of a needle, then a current half this size will cause half deflection of the needle, putting it in exactly the middle of the dial. Similarly, a current of a quarter 𝐼 𝑔 would cause the needle to deflect a quarter of the way round and so on.

Now, if we rearrange Ohm’s law to make 𝐼 the subject, we obtain the expression 𝐼 equals 𝑉 over 𝑅. Since 𝑉 in our circuit is a constant, this means that 𝐼, the current in our circuit, is inversely proportional to 𝑅 𝑇, the total resistance of our circuit. This means, for example, if we multiply the total resistance by two, then the current will halve or if we were to multiply the total resistance by four, then we would divide the total current by four.

Now, we’ve already shown that to go from full deflection of the needle which occurs when the test resistance is zero to a half deflection of the needle, we would need to halve the current. And Ohm’s law tells us that in order to halve the current, we would need to double the total resistance of the circuit. This means that if we add a test resistor and the needle moves to half deflection, then adding this test resistor must have doubled the resistance of the entire circuit.

This would mean that the resistance of the test resistor is exactly equal to the resistance of the ohmmeter, which means the resistance measured by the halfway position on the scale is equal to 𝑅 Ω, the resistance of the ohmmeter itself which, as we’ve shown, is equal to the resistance of the variable resistor plus the resistance of the fixed resistor plus the resistance of the galvanometer.

Following the same reasoning, the resistance indicated by this position on the dial would be half the resistance of the ohmmeter. And the resistance indicated by this position on the dial would be twice the resistance of the ohmmeter. So we can see that because the deflection of the galvanometer’s needle is proportional to current but current is inversely proportional to resistance that the scale on our ohmmeter is nonlinear. That is, the deflection is not proportional to the resistance that we’re measuring.

Okay, now that we’ve seen how to assemble and calibrate an ohmmeter and how to interpret the reading on the dial, let’s have a go at answering a question.

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.

Let’s now review the key points that we’ve learned in this video. Firstly, we’ve seen that an ohmmeter can be made by connecting a direct current source, a fixed resistor, a variable resistor, and a galvanometer in series with each other. We’ve also showed that to calibrate the resistor, the resistances of the fixed and variable resistors must be chosen such that when the ohmmeter is short-circuited, the current is equal to the full-scale deflection current of the galvanometer. And finally, we’ve seen that ohmmeters have a nonlinear scale, which varies from infinite resistance indicated by zero deflection of the galvanometer’s needle to zero resistance, which is indicated by full deflection. This is a summary of the design of the ohmmeter.