Video Transcript
Which of the following diagrams most correctly shows the divisions of the scale of a hot-wire ammeter corresponding to equal changes in current?
Looking at each of these diagrams, we see that there are four labeled lines: zero, 𝐼 one, 𝐼 two, 𝐼 three, and 𝐼 max. The distance between these lines indicates a particular range of current. And the question tells us that the divisions of these scales indicate equal changes in current, which means that the change in current from zero to 𝐼 one should be the same as the change in current from 𝐼 one to 𝐼 two.
So at first, (B) seems to be the obvious answer, since equal divisions should show an equal change in current, right? Well, for some ammeters, this may be true, but not for a hot-wire ammeter. This is because hot-wire ammeters have a hot wire, a wire that gets hot. And when it gets hot, it expands, causing a pulley with the dial arm on it to move. For the most part, this expansion of the wire linearly corresponds with the deflection of the needle arm on the pulley. This is to say that the expansion caused, and thus deflection caused, by the first 10-degree change should look the same as the second 10-degree change.
So this still seems linear. What’s the problem with answer (B)? Well, it has to do with how the heat in the wire is generated. It comes from current. The heat of a hot wire, which we’ll call 𝑄, is proportional to the square of current. And because we’re concerned about the current and not measuring the temperature, the scale will actually be nonlinear.
Looking at some example numbers, if we were to go from a magnitude of zero amperes, no current, to one ampere, that may correspond with an increase of 10 degrees. But if we were to go from zero amperes to two amperes, doubling the current, we should expect a quadrupling of the heat, since two squared is four, while one squared is only one. Similarly, three amperes would show a ninefold increase and so on.
As a reminder, these numbers aren’t meant to correspond with any real-life hot-wire ammeters. They’re just for this example. But the proportions are correct. For a hot-wire ammeter, if we’re looking at equal changes in current, we should be looking at an exponential increase in the divisions between those currents.
Answer (A) corresponds with an exponential decrease with the relationship between heat and current. Answer (B) corresponds with a linear relationship, which means the diagram that most correctly shows the divisions of the scale of a hot-wire ammeter corresponding to equal changes in current is answer (C).
