Why do hot-wire instruments have a non-linear scale? Hot-wire instruments are used to measure rms voltage or rms current in AC circuits. The construction of a hot-wire instrument is shown in the following diagram:

Image source: Hitzdrahtmesswerk (Hot wire measuring mechanism) - German Language Wikipedia | Annotations translated from German to English.
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When current is passed through the red wire (labelled "hot wire"), it gets heated up, due to which it expands. This expansion drives the pointer which moves on the graduated scale. It can be seen that the scale from which we would take the readings is non-linear or in other words, the distance between markings is not equal unlike ordinary ammeters, voltmeters, etc. 
However, I don't understand the reason for using a non-linear scale in hot wire instruments. Is it because the thermal expansion of the hot-wire is non-linear whereas the expansion/contraction of the spring is linear (as per Hooke's law)? Or is that because we're measure the rms current/voltage when the expansion is usually proportional to the square of the corresponding rms quantities? It would be helpful if you could explain which of the factors leads to the non-linear nature of the scale and in what manner in an intuitive way? 
 A: This happens not so much due to the fact that thermal expansion is not linear in temperature (you'd have to be either working with a very cold wire or else operating over very large temperature differences); it has more to do with the fact that when you apply a voltage across a resistance that object's equilibrium temperature is not linear in the applied voltage. For instance, assuming the wire's primary mode of heat transfer is radiative, the magnitude of power loss is given by
$$P_\text{loss}(T)\propto(T^4-T_0^4)$$
whereas the power supplied by a potential difference is
$$P_\text{gain}(T)\propto\frac{V^2}{1+\alpha(T-T_0)}$$
where $\alpha$ is the wire's resistivity coefficient. Equilibrium is achieved when the two powers are balanced, which sets the final temperature $T_\text{eq}$ as generally a non-linear function of $V$. We can make even more rough approximation to set $P_\text{loss}(T)\sim T^4$ and $P_\text{gain}(T)\sim V^2/T$ to see that $T_\text{eq}\sim V^{2/5}$, so that your change in length of the string is a linear function of $V^{2/5}$. This approximation should hold for very hot wires, such as the filament in an incandescent lightbulb. Here's some data I took off a miniature 3-Watt of resistance vs voltage fit to a linear function of $V^{2/5}$ (resistance, like length, increases roughly linearly in temperature even over somewhat large changes):

