APhO 2010, Експеримент — Задача 1. Experiment I. Magnetic force probe

Exp. I-A、Measuring the resonance frequency

Carefully take out the experimental components from Set-C and Set-I, and set up the experimental apparatus as shown in Fig. I-A-1. The schematic plot is shown in Fig. I-A-2. Connect the 5V-DC voltage source to the laser box (C-C-#). Connect the oscillating output of the sine wave generator to the driving box of the brass reed (I-I-#). Turn on the power and fix the output voltage of the sine wave generator. Direct the laser beam into the mirror at the free end of the brass reed so that the reflected beam spot on the screen (I-J) can be used to determine the vibrating amplitude of the reed.

Caution: 1) Carefully remove the paper protected cover before using the brass reed (I-I-#) for experimental measurements. The resonance frequency of the brass reed is very sensitive to its shape, and any deformation of reed during the experiment may give an inaccurate result.

2. Do not directly look into the laser beam, which can damage your eyes.

(1) Measure the amplitude $A$ of the oscillating laser beam spot by changing the frequency of the sine wave generator. Record the measured amplitude as a function of frequency in the data table on the answer sheet. (0.8 points) [0,8 т.]

(2) Make a proper plot on one of the supplied graph papers to determine the resonance frequency $f_{RO}$ and quality factor $Q$. Also record the obtained $f_{RO}$ and $Q$ in the proper blank spaces on the answer sheet. (1.2 points) [1,2 т.]

Exp. I-B、Resonance frequency versus the external force.

In this part of experiment, the resonance frequency under the influence of a non-uniform force is investigated. The non-uniform force is provided by a small 3-mm cylindrical metallic calibration magnet M$_\mathrm{C}$ fixed on a vertical slider (I-K-#) with its N pole pointing upward. The tip magnet M$_\mathrm{T,}$ adhered at the free end of the oscillating reed, has its N pole pointing downward. The pole axes of both magnets should be aligned along the same vertical line.

Set up the experiment as shown in Fig. I-B-1. The schematic plot is shown in Fig. I-B-2.

(1) On the scale of the vertical slider, read out the position $z_0$ of the bottom plane for the tip magnet M$_\mathrm{T}$ without the interaction of M$_\mathrm{C}$ by properly moving M$_\mathrm{C}$ away from M$_\mathrm{T}$. Record the measured $z_0$ in the data table. (0.2 points) [0,2 т.]

(2) Adjust the position of the magnet M$_\mathrm{C}$ to be right underneath M$_\mathrm{T}$. The pole axes of both magnets should be aligned along the same vertical line. Determine the position $z$ of the top plane of the N-pole of M$_\mathrm{C}$. Calculate the nominal distance $d$ by defining $d=z_0-z$. Record $z$ and $d$ in the data table. (Note: The equilibrium separation between the two magnets is not the same as $d$ because the two magnets repel each other.)

(3) Determine the resonance frequency $f_R$ for the distance $d$ by tuning the frequency of the sine wave generator until the maximum amplitude is reached, plotting amplitude versus frequency is not necessary for determining $f_\mathrm{R}$ of each distance $d$. Record the determined resonance frequency $f_R$ in the data table.

(4) Change the vertical position of the magnet M$_\mathrm{C}$ and repeat the steps (2) and (3) for a number of measurements of different distance $d$ and the corresponding resonance frequency $f_R$. (1.2 points) [1,2 т.]

(5) Plot a graph of $f_R$ as a function of distance $d$ using a graph paper. Guiding by eyes, draw the best line through the data points. (1.2 points) [1,2 т.]

(6) Define $\Delta f_R = f_R - f_{RO}$, and plot $\ln(\Delta f_R)$ as a function of $d$ using another graph paper. Guiding by eyes, draw the best line through the data points. (1.0 points) [1 т.]

Exp. I-C、Finding the positions and depths of the magnets inside a black box.

There are two magnets M$_\mathrm{A}$ and M$_\mathrm{B}$ buried in the black box (I-H-#) which is fixed on a 1D translational stage. The N poles of both magnets are pointed upward. Magnets M$_\mathrm{A}$ and M$_\mathrm{B}$, and M$_\mathrm{C}$ used in Exp. I-B are very close in size, shape, and magnetic properties. The depths of the magnets M$_\mathrm{A}$ and M$_\mathrm{B}$ may be different. Magnet M$_\mathrm{A}$ is located at the intersection of the two lines marked on the top surface of the black box. Magnet M$_\mathrm{B}$ is located somewhere along the longer line as shown in Fig. I-C-1. The horizontal distance between magnets M$_\mathrm{A}$ and M$_\mathrm{B}$ is denoted by $\overline{AB}$.

(1) On the scale of the vertical slider, read out the position $z_0$ (in this part, $z_0$ may be different from the $z_0$ in Exp. I-B) of the bottom plane for the tip magnet M$_\mathrm{T}$ without the interaction of the magnets inside the black box. On the scale of the vertical slider, read out the position $z_{box}$ of the top plane of black box. Record $z_0$ and $z_{box}$ on the answer sheet. (0.2 points) [0,2 т.]

(2) Move the black box along the longer line and observe the variation in resonance frequency $f_R$ of the reed to find the position of M$_\mathrm{B}$. Record the measured distances $y$ and their corresponding resonance frequencies $f_R$ in the data table. (1.4 points) [1,4 т.]

(3) Plot $f_R$ as a function of $y$ on a graph paper to determine the position of magnet M$_\mathrm{B}$. Mark the positions of magnets M$_\mathrm{A}$ and M$_\mathrm{B}$ on the $y$-axis of your graph, and write down the value of $\overline{AB}$ on the answer sheet. (1.2 points) [1,2 т.]

(4) Determine the depths $d_A$ and $d_B$ of the magnets M$_\mathrm{A}$ and M$_\mathrm{B}$ from the top surface of the black box using the results in Exp. I-B. Write down the values of $d_A$ and $d_B$ on the answer sheet. (1.6 points) [1,6 т.]