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APhO 2004, Експеримент — Задача 1. Hall effect and magnetoresistivity effect

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Experimental competition- Problem No.1. Hall effect and magnetoresistivity effect

Условие

APPARATUS AND MATERIALS

Three digital multimeters.
A Hall sensor with four pins MNPQ (M in black wire, N yellow wire, P red wire, Q green wire), fixed on a printed circuit, a pair of conductors leading to M, N; another pair of conductors leading to P, Q.
A permanent magnet in the shape of a disk, of radius $r = 14\ \mathrm{mm}$ , of thickness $t = 4\ \mathrm{mm}$ . The magnetization is perpendicular to the surface of the disk. The value $B_0$ (in Tesla) of the magnetic field at the surface of the magnet is written on the magnet's surface.

During the experiment, keep the magnet far away from the Hall sensor whenever you do not use it.
A coil of $N$ turns is wound on a core having the shape of a toroid, made of a ferromagnetic material. The average diameter of the core is $\rho = 25\ \mathrm{mm}$ . The toroid has a gap of width $d = 3\ \mathrm{mm}$ .
A box with two independent 1.5 V dry cells. The cell connected in series to a 10 k $\Omega$ variable resistor, called battery 1, is used to supply the current to the Hall sensor. The second cell, called battery 2, is used to supply the current to the coil only during the measurement.
A protractor with a small hole at its center.
A piece of plexiglass with a small needle fixed on it.
A holder for the printed circuit with the Hall sensor.
A small piece of plastic used to fix the sensor on the needle.
Conductors with negligible resistance.
Graph papers.

Photograph of the apparatus on a table with numbered call-outs: 1 – three digital multimeters; 5 – the battery box with the 10 kΩ variable resistor (labelled box “10 kΩ variable resistor”); 3 – the disk magnet on the protractor; 6 – the protractor; 2 – the Hall sensor at the end of the holder; 4 – the toroidal coil; 8 – the holder for the printed circuit; 7 – the plexiglass plate; 9 – the plastic piece fixing the sensor on the needle. — Figure 1

EXPERIMENT

I. Introduction

1. The magnetoresistivity effect and the Hall effect.

Consider a conductor sample in the shape of a parallelopiped of length $a$ , width $b$ and thickness $c$ (see Figure 2). The current $I$ flows along the direction of $a$ . If the sample is placed in a magnetic field $\vec{B}$ , the magnetic field affects the resistance $R$ of the sample. This effect is called magneto-resistivity effect (MRE). Let $\Delta R$ be the increase of the resistance $R$ of the sample, $R_0$ - the value of the resistance in the absence of a magnetic field, then the magnitude of the MRE is defined by the ratio $\Delta R/R_0$ .

A rectangular parallelopiped sample with edges a (length), b (width) and c (thickness); electrode M on the top face and N on the bottom face on a vertical line, P and Q at the ends along the horizontal current I (arrow to the right); a vector B drawn obliquely in the plane, and an angle θ marked between a rotated arrow and the current direction. — Figure 2

Assume that the applied magnetic field is uniform and the magnetic induction vector $\vec{B}$ is parallel to the top face of the sample as shown in Figure 2. If the charge carriers in the sample are electrons, the Lorentz force will bend them upward, and the top face of the sample will be charged negatively. This effect is called the Hall effect. The voltage appearing between electrodes M (on the top face) and N (on the bottom face) is called the Hall voltage. This can be measured by use of a voltmeter.

The potential difference measured between the electrodes M and N is given by

U_{\mathrm{MN}} = U_{\mathrm{H}} + V_{\mathrm{MN}} \qquad (1)

where, $U_{\mathrm{H}}$ is the Hall voltage, $V_{\mathrm{MN}}$ is the potential difference in the absence of a magnetic field due to some undesired effects (the electrodes M and N being not exactly opposite to each other, etc...).

Normally, the Hall voltage $U_{\mathrm{H}}$ is proportional to $IB.\sin\theta$ , and the magnitude of the MRE is proportional to $B^2\sin^2\theta$ , where $\theta$ is the angle between vector $\vec{B}$ and the current direction. But when the sample has a non regular shape, the dependence of $U_H$ and $\Delta R/R$ on $B\sin\theta$ may be more complicated.

The Hall effect is used to fabricate a device for measuring the magnetic field. This device is called the Hall sensor. For Hall sensor, the expression of $U_{\mathrm{H}}$ is given by:

U_{\mathrm{H}} = \alpha.I.B.\sin\theta \qquad (2)

where $\alpha$ is, by definition, the sensitivity of the Hall sensor.

II. The measuring sample:

The measuring sample in this experiment is a commercial Hall sensor. It consists of a small thin semiconductor plate covered with plastic, with 4 ohmic electrodes, leading to the pins M, N, P, Q (see Figure 3). It is used in this experiment to study both the MRE and the Hall effect.

Hall sensor drawn as a rectangle with four pins: M and Q on the left, P and N on the right. — Figure 3

Place the sensor in the magnetic field and use an ohmmeter to measure the resistance between pins M and N, we can deduce the magnitude of the MRE. Set a current ( $I \sim 1\ \mathrm{mA}$ ) flowing from P to Q, we can study the Hall effect by measuring the voltage between M and N with a millivoltmeter.

III. Experiment

1. Determination of the sensitivity $\alpha$ of the Hall sensor

Set the current through the sensor $I \sim 1\ \mathrm{mA}$ . Keep the distance between the sensor and the centre of the surface of the magnet $y = 2\mathrm{cm}$ . Adjust the orientation of the magnet to obtain maximal value of the Hall voltage. Measure the Hall voltage with some values of $I$ and determine the sensitivity $\alpha$ of the Hall sensor.

For a magnet having the shape of a disk of radius $r$ , thickness $t$ , the magnetic field at a point situated on its axis at a distance $y$ from the center of the disk surface with $y \gg t$ is given by the expression

B(y)=\frac{1}{2}B_0\left[\frac{y+t}{\sqrt{(y+t)^2+r^2}}-\frac{y}{\sqrt{y^2+r^2}}\right] \qquad (1)

where $B_0$ is the magnetic induction at the surface of the magnet. The value of $B_0$ is given on the surface of the magnet. [2.0 pts]

2. Study of the dependence of $U_{\mathrm{H}}$ on angle $\theta$ between $\vec{B}$ and the current direction.

Set the current through the sensor $I \sim 1\ \mathrm{mA}$ . Keep the distance between the sensor and the centre of the surface of the magnet $y = 2\ \mathrm{cm}$ . Put the magnet on the protractor so that the plane of the magnet is perpendicular to the line connecting the sensor and centre of the magnet.

a. Draw a sketch of the experimental arrangement.

b. Tabulate the values of $U_{\mathrm{H}}$ for $\theta$ in the range of $-90^{\circ} \le \theta \le 90^{\circ}$ .

c. Verify the proportionality between $U_{\mathrm{H}}$ and $\sin\theta$ by using a graph plotted in an appropriate way. [2.5 pts]

3. Study of the dependence of $\Delta R/R$ on $B$ , for $\vec{B}$ perpendicular to the sample plane.

The MRE is significant only at sufficiently strong magnetic field. So it is recommended to use a magnetic field as strong as possible.

a. Draw a sketch of the experimental arrangement and explain the principle of the measurements.

b. Perform measurements and tabulate the data.

c. Assume that $\Delta R/R \sim B^k$ , determine the value of $k$ by using a graph plotted in an appropriate way. Estimate the maximal deviation of the obtained value of $k$ . [4.0 pts]

4. Determination of the relative permeability $\mu$ of the ferromagnetic materials of the core of the toroidal coil

Determine the relative permeability $\mu$ of the core material at the measured current intensity $I$ by following this guidance step by step:

Put the Hall sensor into the gap on the core.
Connect the coil and an ammeter to battery 2. Use only the inputs COM and 20A of the ammeter in this case.
Measure the current $I$ in the coil and the magnetic field $B$ in the gap.
Calculate the value of $\mu$

You can use the following relation:

\frac{B.(\rho - d)}{\mu} + B.d = 4\pi.10^{-7}.N.I

[1.5 pts]

Решение

1. Determination of the sensitivity $\alpha$ of the Hall sensor:

The current $I_{\mathrm{H}}$ supplied by battery 1 and flowing through the milliammeter and the sensor via the pins P and Q is maintained unchanged. Use the variable resistance to maintain $I_{\mathrm{H}} \sim 1\ \mathrm{mA}$ . The Hall voltage $U_{\mathrm{H}}$ is measured across the pins M and N with a millivoltmeter.

In the absence of a magnetic field, we obtain $-1{,}3\ \mathrm{mV} < V_{\mathrm{MN}} < 1{,}3\ \mathrm{mV}$ .

Put the magnet on the protractor at a distance $y = 2\ \mathrm{cm} = 2.10^{-2}\ \mathrm{m}$ from the sensor.

Rotate the protractor to obtain the maximum value of $U_{\mathrm{H}}$ . We find the position of the magnet, in which the line joining the magnet and the sensor is perpendicular to the surface of the sensor. In this position of the magnet the axis is perpendicular to the surface of the sensor and $\vec{B}$ is perpendicular to the current $I_{\mathrm{H}}$ , and $\theta = 90^{0}$ .

Measure $U_{\mathrm{MN}}$ and calculate the value of $U_{\mathrm{H}}$ : $U_{\mathrm{H}} = U_{\mathrm{MN}} - V_{\mathrm{MN}}$ .

Calculate the value of $B$ by using equation (1).

Use the relation $\alpha = U_{\mathrm{H}}/IB$ to calculate the value of $\alpha$ .

Repeat the measurement to obtain 4-5 different values of $\alpha$ , and calculate the mean value of $\alpha$ and the deviation $\Delta\alpha$ .

2. Study the dependence of $U_{\mathrm{H}}$ on $\theta$ .

a. Keeping $y$ unchanged, $y = 2\ \mathrm{cm}$ . By rotating the protractor, we change the angle $\theta$ between $\vec{B}$ and the current direction.

b. At each angle $\theta$ , measure $U_{\mathrm{H}}$ (always with the same $I_{\mathrm{H}}$ ). Vary the angle in the range $-90^{\circ} \le \theta \le 90^{\circ}$ .

Tabulate $U_{\mathrm{H}}$ versus $\theta$ .

For $U_{\mathrm{H}} = \gamma \sin^m\theta$ , then:

\ln U_{\mathrm{H}} = \ln\gamma + m.\ln(\sin\theta) \qquad (2)

Plot $\ln U_{\mathrm{H}}$ versus $\ln(\sin\theta)$ , we get $m$ and the error.

We find that the graph of $\ln U_{\mathrm{H}}$ versus $\ln(\sin\theta)$ is a straight line. From the graph we can deduce $m = 1$ .

Graph of ln(U_H) (vertical axis, 0 to 3) versus ln(sin(θ)) (horizontal axis, −3 to 0): data points lie on a straight line; fit label “y = +1.09x^1 +3.35, max dev:0.0462, r^2=0.999”.

3. Study the dependence of $\Delta R/R$ on $B$ .

The Hall sensor, with its sensitivity $\alpha$ obtained from part 1., is used to measure the magnetic induction $B$ . It serves to study the magnetoresistivity as well. Because the resistance between N and M does not change during the experiment (and equals about 350 $\Omega$ ) we can study the dependence $\Delta R$ instead of $\Delta R/R$ on $B$ .

a. Keep the axis of the magnet perpendicular to the surface of the sensor by the procedure described in part 1. Provide a current $I = 1\ \mathrm{mA}$ to the Hall sensor and measure the intensity of the magnetic field (at the sensor).

Turn off the current, and switch the multimeter from the millivoltmeter regime to the ohmmeter regime.

Measure the resistance between N and M and calculate the value of $\Delta R$ .

Vary the distance $y$ from the sensor to the magnet from 2 cm to 0.6 cm and repeat the above measurement. Tabulate the values of $\Delta R/R$ or $\Delta R$ versus $B$ .

b. Assuming $\dfrac{\Delta R}{R} = \beta.B^k$ , we have:

\ln\left(\frac{\Delta R}{R}\right) = \ln\beta + k\ln B \qquad (1)

Plot $\ln\left(\dfrac{\Delta R}{R}\right)$ versus $\ln B$ .

Draw a straight line passing as nearly the experimental points as possible. The slope of this line gives the value of $k$ . One can find the value $k \sim 2$ . The deviation of $k$ can be estimated from the graph by using eye-balling method.

The student may use the less square method to determine $k$ and the error.

Graph of ln (ΔR) (vertical axis, about 0.7 to 1.8, ticks 1.0 and 1.5) versus ln(U_H) (horizontal axis, ticks 1.6, 1.8, 2.0): five data points on a straight line with a fitted red line; fit label “y = +1.74x^1 -1.87, max dev:0.0156, r^2=0.999”.

4. Determination of the permeability $\mu$ of a ferromagnetic core in a toroidal coil.

Put the Hall sensor into the gap. Keep the current through the sensor constant, $I = 1\mathrm{mA}$ . The sensor is used to measure $B$ in the gap.

Connect the coil to battery 2 via an ammeter. Note the value $I_C$ of the current flowing in the coil. $I_C$ varies from 3 to 4 amperes. Turn off immediately the current $I_C$ after reading its value, to avoid the variation of the temperature of the sensor.

Count the number $N$ of turns of the coil. From the values of $U_{\mathrm{H}}$ , , $\alpha$ , determine $B$ . From the values of $B$ , $\rho$ , $d$ and $N$ , using the relation:

\frac{B(\rho - d)}{\mu} + Bd = 4\pi.10^{-7}.N.I,

we obtain the value of $\mu$ .

Оригинал в Архива: exp1.pdf · официални решения: exp1_sol.pdf