Video Transcript
In which of the following images of a hot-wire ammeter scale must it be true that a different increase in current is indicated when the arm of the ammeter changes from pointing at 𝐴 to pointing at 𝐵 than when the ammeter arm changes from pointing at 𝐶 to pointing at 𝐷? (A), (B), (C) both of these, or (D) neither of these.
Whenever we deal with hot-wire ammeter scales, it’s important to recall how they’re different from other ammeter scales, specifically that they are nonlinear. By this, we mean that a typical linear scale starting at zero and going all the way up to 𝐼 max, the maximum measurable current by the scale, will indicate that the same difference in current, in this case one ampere, will always result in the same deflection of the needle arm. But if the scale is nonlinear, like in the case of a hot-wire ammeter, then the same differences in current will not be reflected by the same increase or decrease in the deflection of the needle arm.
So, knowing that the scale must be nonlinear, when we look at the deflection difference in image (A), we see that it’s about the same from 𝐴 to 𝐵 and 𝐶 to 𝐷. But because the scale is nonlinear, the same increase in distance should indicate a different increase in current. So it therefore must be true that a different increase in current is indicated when the arm of the ammeter changes from 𝐴 to 𝐵 then from 𝐶 to 𝐷 in image (A). So we know that it’s true for image (A). But what about image (B)?
The distance from 𝐴 to 𝐵 is certainly different from the distance from 𝐶 to 𝐷, indicating a nonlinear relationship. So image (B) must be showing us the same increase in current because it’s a nonlinear scale, right? Well, not actually, we’re not looking for nonlinearity in general. We’re looking for a specific type that hot-wire ammeter scales have. The type of nonlinearity that this scale possesses is due to the hot wire of the hot-wire ammeter.
When there is a current through the hot wire of the hot-wire ammeter, resistive dissipation in the wire causes it to heat up, which in turn causes the metal to slightly expand and pull the string towards the spring on the opposite side. And since the string is connected to a pulley, it causes the pulley to rotate and the needle arm that is attached to it to rotate as well.
But there is a special relationship between the current in the wire and the heat that is produced because of it. The heat generated 𝑄 is proportional to the square of the current 𝐼. This means if you double the current, say from one ampere to two amperes, you will quadruple the heat that is generated. This is important to note because the rotation of this pulley and thus deflection of the needle arm depends on the expansion due to the heat of the wire, which means that each same current increase in the wire will produce a larger and larger deflection as we move along the scale.
So, with this in mind, when we look back at image (B), with a large initial difference between 𝐴 and 𝐵 at the beginning of the scale and a smaller difference between 𝐶 and 𝐷 near the end of the scale, we know that there must be a different increase in current same as image (A). If we’re using the arbitrary scale that we have here in the diagram, we might say that the distance between 𝐴 and 𝐵 would indicate a large increase in current of about two amperes.
Using the same arbitrary scale, we can see that the distance from 𝐶 to 𝐷 wouldn’t even be a full ampere. So it must be true that there is a different increase in current indicated when the arm of the ammeter changes from pointing at 𝐴 to pointing at 𝐵 than when it goes from 𝐶 to 𝐷 in both images (A) and (B), which means that the correct answer is (C), both of these.
