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IPhO 1994, Теория — Задача 2. Superconducting Magnet

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XXV International Physics Olympiad — Beijing, People’s Republic of China — Theoretical competition, July13,1994 · 13 юли 1994 г. · 10 т.

Внимание: Бележка към темата

The complete archived booklet contains three theory tasks and two experiments. Theory is shown here separately using original pages1–21. Points are taken from the native marking schemes; the round total30 is their sum. Native source errors are retained: inconsistent S/Sprime/Sdoubleprime labels, the first quark-table header beta*x/e, Fig1 L=p0/f withoutc, inconsistent rubric subtotals, signed-current/rate wording in the magnet key, Vprime_tn in collision Eq2 and missing multiplicativeV in Eq9. These are printed source defects, not silently corrected solutions. The leading 0 of the upper0.5A label in native Fig7c is already clipped in the original page; the complete available source graphic is retained.

Theoretical competition — July13,1994 — Time available:5hours

Use only the ball pen provided.
Your graphs should be drawn on the answer sheets attached to the problem.
Your solutions should be written on the sheets of paper attached to the problems.
Write at the top of the first page of each problem:
- The total number of pages in your solution to the problem
- Your name and code number

Условие

Super conducting magnets are widely used in laboratories. The most common form of super conducting magnets is a solenoid made of super conducting wire. The wonderful thing about a superconducting magnet is that it produces high magnetic fields without any energy dissipation due to Joule heating, since the electrical resistance of the superconducting wire becomes zero when the magnet is immersed in liquid helium at a temperature of $4.2\ \mathrm{K}$ . Usually, the magnet is provided with a specially designed superconducting switch, as shown in Fig. 1. The resistance $r$ of the switch can be controlled: either $r=0$ in the superconducting state, or $r=r_n$ in the normal state. When the persistent mode, with a current circulating through the magnet and superconducting switch indefinitely. The persistent mode allows a steady magnetic field to be maintained for long periods with the external source cut off.

The details of the superconducting switch are not given in Fig. 1. It is usually a small length of superconducting wire wrapped with a heater wire and suitably thermally insulated from the liquid helium bath. On being heated, the temperature of the superconducting wire increases and it reverts to the resistive normal state. The typical value of $r_n$ is a few ohms. Here we assume it to be $5\ \Omega$ . The inductance of a superconducting magnet depends on its size; assume it be $10\ \mathrm{H}$ for the magnet in Fig. 1. The total current $I$ can be changed by adjusting the resistance $R$ .

This problem will be graded by the plots only!

The arrows denote the positive direction of $I$ , $I_1$ and $I_2$ .

Original Fig1 complete superconducting magnet circuit, heater-switch annotation, current arrows and liquid-helium note.

1) If the total current $I$ and the resistance $r$ of the superconducting switch are controlled to vary with time in the way shown in Figs. 2a and 2b respectively, and assuming the currents $I_1$ and $I_2$ flowing through the magnet and the switch respectively are equal at the beginning (Fig. 2c and Fig. 2d), how do they vary with time from $t_1$ to $t_4$ ? Plot your answer in Fig. 2c and Fig. 2d.

Original all four Fig2 graph panels, complete axes, prescribed I/r and initial I1/I2 segments.

[2 т.]

2) Suppose the power switch $K$ is turned on at time $t=0$ when $r=0$ , $I_1=0$ and $R=7.5\ \Omega$ , and the total current $I$ is $0.5\ \mathrm{A}$ . With $K$ kept closed, the resistance $r$ of the superconducting switch is varied in he way shown in Fig. 3b. Plot the corresponding time dependences of $I$ , $I_1$ and $I_2$ in Figs. 3a, 3c and 3d respectively.

Original complete four Fig3 blank/current-resistance graph panels and all minute tick labels.

[3 т.]

3) Only small currents, less than $0.5\ \mathrm{A}$ , are allowed to flow through the superconducting switch when it is in the normal state, with larger currents the switch will be burnt out. Suppose the superconducting magnet is operated in a persistent mode, i.e. $I=0$ , and $I_1=i_1$ (e.g. $20\ \mathrm{A}$ ), $I_2=-i_1$ , as shown in Fig. 4, from $t=0$ to $t=3\ \mathrm{min}$ . If the experiment is to be stopped by reducting the current through the magnet to zero, how would you do it? This has to be done in several operation steps. Plot the corresponding changes of $I$ , $r$ , $I_1$ and $I_2$ in Fig. 4.

Original four Fig4 panels including negative20A I2 level, complete axes and time scales.

[2 т.]

4) Suppose the magnet is operated in a persistent mode with a persistent current of $20\ \mathrm{A}$ ( $t=0$ to $t=3\ \mathrm{min}$ . See Fig. 5). How would you change it to a persistent mode with a current of 30a? plot your answer in Fig. 5.

Original Fig5a–c graph panels, complete20A/30A levels and minute ticks.

Original Fig5d I2 panel continued on next native page, including negative20A/30A levels.

[3 т.]

Решение

1)

For $t=t_1$ to $t_3$ :

Since $r=0$ , the voltage across the magnet $V_M=L\,dI_1/dt=0$ , therefore,

I_1=I_1(t_1)=\frac12I_0;\qquad I_2=I-I_1=I-\frac12I_0.

For $t=t_3$ to $t_4$ :

Since $I_2=0$ at $t=t_3$ , and $I$ is kept at $\tfrac12I_0$ after $t=t_3$ , $V_M=I_2r_n=0$ , therefore, $I_1$ and $I_2$ will not change.

I_1=\frac12I_0;\qquad I_2=0.

These results are shown in Fig. 6.

Original complete solution Fig6a–d including I2 negative segment.

2)

For $t=0$ to $t=1\ \mathrm{min}$ :

Since $r=0$ , $V_M=L\,dI_1/dt=0$ ,

I_1=I_1(0)=0,\qquad I_2=I-I_1=0.5\ \mathrm{A}.

At $t=1\ \mathrm{min}$ , $r$ suddenly jumps from $0$ to $r_n$ , $I$ will drop from $E/R$ to $E/(R+r_n)$ instantaneously, because $I_1$ can not change abruptly due to the inductance of the magnet coil. For $E/R=0.5\ \mathrm{A}$ , $R=7.5\ \Omega$ and $R_n=5\ \Omega$ , $I$ will drop to $0.3\ \mathrm{A}$ .

For $t=1\ \mathrm{min}$ to $2\ \mathrm{min}$ :

$I$ , $I_1$ and $I_2$ gradually approach their steady values:

I=\frac ER=0.5\ \mathrm{A},\qquad I_1=I=0.5\ \mathrm{A},\qquad I_2=0.

The time constant

\tau=\frac{L(R+r_n)}{Rr_n}.

When $L=10\ \mathrm{H}$ , $R=7.5\ \Omega$ and $r_n=5\ \Omega$ , $\tau=3\ \mathrm{sec}$ .

For $t=2\ \mathrm{min}$ to $3\ \mathrm{min}$ :

Since $r=0$ , $I_1$ and $I_2$ will not change, that is

I_1=0.5\ \mathrm{A}\quad\text{and}\quad I_2=0.

Original solution Fig7a/b with current jump/transient and resistance pulse.

Original solution Fig7c/d continued on native next page, complete rise/decay curves.

3)

The operation steps are:

First step

Turn on power switch $K$ , and increase the total current $I$ to $20\ \mathrm{A}$ , i.e. equal to $I_1$ .

Since the superconducting switch is in the state $r=0$ , so that $V_M=L\,dI_1/dt=0$ , that is, $I_1$ can not change and $I_2$ increases by $20\ \mathrm{A}$ , in other words, $I_2$ changes from $-20\ \mathrm{A}$ to zero.

Second step

Switch $r$ from $0$ to $r_n$ .

Third step

Gradually reduce $I$ to zero while keeping $I_2<0.5\ \mathrm{A}$ ; since $I_2=V_M/r_n$ and $V_m=L\,dI_1/dt$ , when $L=10\ \mathrm{H}$ , $r_n=5\ \Omega$ , the requirement $I_2<0.5\ \mathrm{A}$ corresponds to $dI_1/dt<0.25\ \mathrm{A/sec}$ , that is, a drop of $<15\ \mathrm{A}$ in $1\ \mathrm{min}$ . In Fig. 8 $dI/dt\sim0.1\ \mathrm{A/sec}$ and $dI_1/dt$ is around this value too, so the requirement has been fulfilled.

Final step

Switch $r$ to zero when $V_M=0$ and turn off the power switch $K$ . These results are shown in Fig. 8.

Original solution Fig8a–c, all operation stages and levels.

Original solution Fig8d continued on next page, full I2 rise from negative20A.

4)

First step and second step are the same as that in part 3, resulting in $I_2=0$ .

Third step Increase $I$ by $10\ \mathrm{A}$ to $30\ \mathrm{A}$ with a rate subject to the requirement $I_2<0.5\ \mathrm{A}$ .

Fourth step Switch $r$ to zero when $V_M=0$ .

Fifth step Reduce $I$ to zero, $I_1=30\ \mathrm{A}$ will not change because $V_M$ is zero. $I_2=I-I_1$ will change to $-30\ \mathrm{A}$ . The current flowing through the magnet is thus closed by the superconducting switch.

Final step Turn off the power switch $K$ . The magnet is operating in the persistent mode.

These results are shown in Fig. 9.

Original solution Fig9a–d with every graph panel, operation times and negative30A level.

Grading Scheme

Part 1 — 2 points:

0.5 point for each of $I_1$ , $I_2$ from $t=t_1$ to $t_3$ and $I_1$ , $I_2$ from $t=t_3$ to $t_4$ .

Part 2 — 3 points:

0.3 point for each of $I_1$ , $I_2$ from $t=0$ to $1\ \mathrm{min}$ , $I$ , $I_1$ , $I_2$ at $t=1\ \mathrm{min}$ , and $I_0$ , $I_1$ , $I_2$ from $t=1$ to $2\ \mathrm{min}$ ;

0.2 point for each of $I$ , $I_1$ , and $I_2$ from $t=2$ to $3\ \mathrm{min}$ .

Part 3 — 2 points:

0.25 point for each section in Fig. 8 from $t=3$ to $9\ \mathrm{min}$ , 8 sections in total.

Part 4 — 3 points:

0.25 point for each section in Fig. 9 from $t=3$ to $12\ \mathrm{min}$ , 12 sections in total.

Оригинал в Архива: ipho94.pdf · решение от архива: ipho94.pdf