# Problem 2 – Hint 4

In order to make a full use of the Hint 3, the following considerations might be useful.

First, suppose that the wire leftwards from a point $Q$ is in the low-resistivity state, and rightwards – in the high-resistivity state. This means that far from $Q$ (where we can neglect the heat transfer along the wire), the temperature of the wire $T(x)$ is almost constant, i.e. independent of the coordinate $x$ along the wire; let the temperatures to the left and to the right be respectively $T_1$, and $T_2$. A more complicated temperature profile will be formed in the immediate neighbourhood of $Q$. However, we can use the fact that the heat transfer equation determining the temperature profile $T(x)$ is a linear inhomogeneous differential to show that $T(x)-T(0)$, is an odd function (assuming that $x=0$ at $Q$). Hence, we can immediately express $T(0)$ in terms of $T_1$ and $T_2$.

Second, notice that at the moment when the hottest point of the wire $P$ is undergoing a phase transition, two fast process are triggered: widening of the high-resistivity region around $P$, and decreasing of the current strength.

Please submit the solution to this problem via e-mail to physcs.cup@gmail.com. The next and final hint for the Problem 1 will be published at 13:00 GMT, 29th January 2023, together with the next updates of the intermediate results.