Съдържание

IPhO 1985, Теория и експеримент — Задача 5. Exercise B

Оригинал на условието (PDF)Съобщи за грешка

Всички материали от IPhO 1985

Problems and solutions of the 16th IPhO Portorož, Slovenia, (Former Yugoslavia), 1985 · 10 т.

Условие

Locate the position of the centers and determine the orientations of a number of identical permanent magnets hidden in the black painted block. A diagram of one such magnet is given in Figure 1. The coordinates $x$ , $y$ and $z$ should be measured from the red corner point, as indicated in Figure 2.

Determine the $z$ component of the magnetic induction vector $\vec B$ in the $(x, y)$ plane at $z = 0$ by calibrating the measuring system beforehand.

Find the greatest magnetic induction $B$ obtainable from the magnet supplied.

Instrumentation

Permanent magnet given is identical to the hidden magnets in the block.
Induction coil; 1400 turns, $R = 230\ \Omega$
Field generating coils, 8800 turns, $R = 990\ \Omega$ , 2 pieces
Black painted block with hidden magnets
Voltmeter (ranges 1 V, 3 V and 10 V recommended)
Electronic circuit (recommended supply voltage 24 V)
Ammeter
Variable resistor $3.3\ \mathrm{k\Omega}$
Variable stabilized power supply 0 – 25 V, with current limiter
Four connecting wires
Supporting plate with fixing holes
Rubber bands, multipurpose (e. g. for coil fixing)
Tooth picks
Ruler
Thread

Side view of a flat rectangular magnet; an arrow B points up from the top face, labelled north-blue, and an arrow B enters the bottom face from below, labelled south-red. — Fig. 1

Perspective drawing of the flat block with the coordinate axes drawn from the marked (blackened) front corner: x along the front edge, y along the side edge into the depth, and a vertical axis marked z upward and -z downward. — Fig. 2

Instructions

For the magnet-search only nondestructive methods are acceptable. The final report should include results, formulae, graphs and diagrams. The diagrams should be used instead of comments on the methods used wherever possible.

The proper use of the induced voltage measuring system is shown in Figure 3.

This device is capable of responding to the magnetic field. The peak voltage is proportional to the change of the magnetic flux through the coil.

The variable stabilized power supply is switched ON (1) or OFF (0) by the lower left pushbutton. By the (U) knob the output voltage is increased through the clockwise rotation. The recommended voltage is 24 V. Therefore switch the corresponding toggle switch to the 12 V – 25 V position. With this instrument either the output voltage $U$ or the output current $I$ is measured, with respect to the position of the corresponding toggle switch (V,A). However, to get the output voltage the upper right switch should be in the ’Vklop’ position. By the knob (I) the output current is limited bellow the preset value. When rotated clockwise the power supply can provide 1.5 A at most.

Wiring of the measuring system: the induction coil L is connected to the left terminals of the electronic unit, which has a push button marked 1 and a dial marked 0; the unit's 24V (+/−) supply terminals are on the right, and the right-hand output terminals are wired to a voltmeter V. — Fig. 3 ’0’ zero adjust dial, ’1’ push reset button

Note: permeability of empty space $\mu_0 = 1.2\cdot 10^{-6}\ \mathrm{Vs/Am}$ .

Отговори

Решение

Two permanent magnets having the shape of rectangular parallelepipeds with sides 50 mm, 20 mm and 8 mm are hidden in a block of polystyrene foam with dimension 50 cm, 31 cm and 4.0 cm. Their sides are parallel to the sides of the block. One of the hidden magnets (A) is positioned so that its $\vec B$ (Fig. 15) points in $z$ direction and the other (B) with its $\vec B$ in $x$ or $y$ direction (Fig. 15).

Cross-section of the block between two horizontal lines (its surfaces): on the left magnet A lies flat with its field lines passing vertically through both surfaces (magnetization along z); on the right magnet B stands upright with a horizontal arrow, its field lines bulging out through the upper and lower surfaces on both sides (magnetization along x or y). — Fig. 15 A typical implementation of the magnets in the block

The positions and the orientations of the magnets should be determined on the basis of observations of forces acting on the extra magnet. The best way to do this is to hang the extra magnet on the thread and move it above the surface to be explored. Three areas of strong forces are revealed when the extra magnet is in the horizontal position i. e. its $\vec B$ is parallel to $z$ axis, suggesting that three magnets are hidden. Two of these areas producing an attractive force in position P (Fig. 16) and a repulsive force in position R are closely together.

Cross-section of the block with two flat magnets side by side; field lines of the left one emerge upward at P and enter from below at P’, those of the right one enter from above at R and leave downward at R’. — Fig: 16 Two ’ghost’ magnets appearing in the place of magnet B

However, by inspecting the situation on the other side of the block, again an attractive force in area P’ is found, and a repulsive one in area R’. This is in the contradiction with the supposed magnets layout in Fig. 16 but corresponds to the force distribution of magnet B in Fig. 15.

To determine the $z$ position of the hidden magnets one has to measure the $z$ component of $\vec B$ on the surface of the block and compare it to the measurement of $B_z$ of the extra magnet as a function of distance from its center (Fig. 18). To achieve this the induction coil of the measuring system is removed from the point in which the magnetic field is measured to a distance in which the magnetic field is practically zero, and the peak voltage is measured.

In order to make the absolute calibration of the measuring system, the response of the system to the known magnetic field should be measured. The best defined magnetic field is produced in the gap between two field generating coils. The experimental layout is displayed in Fig. 17.

Calibration circuit: a 24V DC source feeds, through a variable resistor and an ammeter A carrying current I, two field-generating coils in parallel (each with current I/2) facing each other across a gap with field B; the induction coil connected to the electronic unit, whose output goes to a voltmeter V. — Fig. 17 Calibration of the measuring system

The magnetic induction in the gap between the field generating coils is calculated using the formula:

B = \frac{\mu_O NI}{(2l + d)}\,.

Here $N$ is the number of the turns of one of the coils, $l$ its length , $d$ the width of the gap, and $I$ the current through the ammeter. The peak voltage, $U$ , is measured when the induction coil is removed from the gap.

Plotting the magnetic induction $B$ as a function of peak voltage, we can determine the sensitivity of our measuring system:

\frac{B}{U} = 0.020\ \mathrm{T/V}\,.

(More precise calculation of the magnetic field in the gap, which is beyond the scope of the exercise, shows that the true value is only 60 % of the value calculated above.)

The greatest value of $B$ is 0.21 T.

Graph of B_z (axis marks 0,1 and 0,2 T, with B_max marked at the top) against the distance z (10 to 50 mm): points with error bars on a decaying curve from about 0.17 T at 2 mm to nearly zero at 50 mm, extrapolated by a dashed line to B_max at z = 0. — Fig. 18 Magnetic induction vs. distance

Marking scheme:

a) determination of $x, y$ position of magnets (±1 cm) 1 p.

b) determination of the orientations 1 p.

c) depth of magnets (±4 mm) 2 p.

d) calibration (±50 %) 3 p.

e) mapping of the magnetic field 2 p.

f) determination of $B_{\max}$ (±50 %) 1 p.

Оригинал в Архива: ipho85.pdf