EuPhO 2026 — Задача 2. Hysteresis (10 pts)

Theoretical Problems · 10 т.

Условие

Consider a solenoid with $N \gg 1$ turns, radius $R$ and length $\ell \gg R$. The solenoid has a ferromagnetic core with hysteresis curve as shown in the picture. $M$ denotes the magnetization of the core and would be zero without core material. It is related to $H$, the magnetic field strength generated by the current in the solenoid coil, and $B$ by $B = \mu_0 (H + M)$. $M_0$ and $H_0$ are of the same order of magnitude. Assume that $M$ can only change if $|H| \geq H_0$.

An ideal capacitor with capacitance $C$ is now connected to the solenoid forming a closed circuit. Assume that all wires have negligible resistance.

a) Initially there is a current $I_\mathrm{i}$ flowing through the circuit and the capacitor is uncharged. $I_\mathrm{i}$ is large enough that $H(I_\mathrm{i}) \gg H_0$. After one oscillation the current again reaches a maximum value. Determine the difference between $I_\mathrm{i}$ and this value. [3 т.]

b) Find the maximum current that can be attained after many oscillations. [2,3 т.]

c) The current $I(t)$ exhibits two qualitatively distinct phases of behaviour, A and B. The system can remain in phase A indefinitely, whereas any single interval spent in phase B has bounded duration. Find the maximal possible duration of a phase-B interval, if $I_\mathrm{i}$ is chosen optimally. [4,7 т.]

Решение

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Оригинал в Архива: eupho2026_theory_problems.pdf · официални решения: eupho2026_theory_solutions.pdf