College Physics

Is it necessary to improve themost probable approximation by considering the finite size effect?

HOU Ji-xuan

College Physics. 2022, 41(12): 1. doi:10.16854/j.cnki.1000-0712.220272

Abstract ( 838 )

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The derivations of the most probable distribution are not self-consistent in most statistical physics textbooks since the Stirling’s approximation adopted in the derivations is not valid for systems with few particles. Thus, many attempts are devoted by adding higher Stirling terms to improve the prediction. However, by calculating the ground-state fraction of the ideal Bose system in one-dimensional harmonic trap, we show that the predictions considering the finite size effect are much worse than the predicitons given by the original version without modification. Hence, the finite-size modification to the most probable approximation is not necessary at all.

On the Huygens principle and spatial dimensions

YANG Shi-jie

College Physics. 2022, 41(12): 4. doi:10.16854/j.cnki.1000-0712.220107

Abstract ( 720 )

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In this paper we address the Huygens principle proposed in optics from the analytical solutions of the wave equation. It is revealed that the Huygens principle is correct only in one or three dimensional space while failed in two dimensional space. It is not certain that we can receive acoustic signals through air, it depends on the effectiveness of Huygens principle in the three dimensional space.

Description and analysis of Plane Kinematics by utilizing Euler formula

ZHENG Zheng-yu

College Physics. 2022, 41(12): 8. doi:10.16854/j.cnki.1000-0712.220001

Abstract ( 590 )

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In this paper，the complex function plane is used to replace the real plane of the
traditional Oxy co- ordinate system，and Euler＇s formula expressed in the form of complex
exponential is also introduced here to de- scribe and analyze the plane kinematics problem．
Thus，the confusion and the tedious vector analysis caused by the operation rules of point product
and cross product in the traditional real plane are avoided． The instantaneous coor- dinate system
is introduced，and the conclusion is completely consistent with the classical theory．

Strict proof of the commutation relations between orbital angular momentum operator and vector operator as well as scalar operator

QIN Xu-ming, KANG Dong-biao, HAO Hong-jun, ZHAO Dong-qiu, ZHANG Xi-wei

College Physics. 2022, 41(12): 12. doi:10.16854/j.cnki.1000-0712.220022

Abstract ( 1762 )

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Using the transformation characteristics of vector operator and scalar operator when space is rotated, this paper strictly proves the commutation relations between orbital angular momentum operator and vector operator as well as scalar operator, and constructs several specific scalar operators and vector operators to verify the commutation relations. Finally, the definitions of vector operator and scalar operator are discussed.

Chaotic pendulum analysis based on DDA

YU Hao, YU Yong

College Physics. 2022, 41(12): 17. doi:10.16854/j.cnki.1000-0712.220233

Abstract ( 464 )

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The method of discontinuous deformation analysis (DDA) is used to simulate the chaotic pendulum. Firstly, a two-degree-of-freedom model and a four-degree-of-freedom chaotic pendulum model with undamped nonlinear free vibration are established. The motion differential equations of the two models are derived and the theoretical solutions of the pendulum angles under two degrees of freedom are obtained by Matlab. The original DDA equation is simplified, the rigid DDA equation is derived, and the DDA program is written by Matlab. The motion of two kinds of pendulum is simulated numerically. The time history diagrams of different pendulum bars under the two models are obtained respectively, and the applicability of the program is illustrated by comparing with the theoretical results.

Analysis of third-order susceptibility tensor in isotropic media

YANG Guo-xia, GONG Wen-ping, LIU Da-he, SHI Jin-wei

College Physics. 2022, 41(12): 22. doi:10.16854/j.cnki.1000-0712.220226

Abstract ( 1082 )

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In textbooks, the third-order susceptibility tensor elements in isotropic media are not detailed and intuitive enough to be explained, especially in isotropic chiral media. A analysis method based on cascaded second-order effect is proposed in this paper, to determine whether third-order susceptibility tensor elements in isotropic media are zero. Compared with the symmetry operation method based on group theory, this method can make judgement more quickly, and give intuitive understanding. The limiting conditions of propagation directions of input fields required by the generation of third-order effects in isotropic media are analyzed, and a comparison with the second-order effects is made, where the key is to unify all non-zero third-order polarizations into a vector expression.

Momentum density in complex electromagnetic waves

DONG Zheng-gao

College Physics. 2022, 41(12): 26. doi:10.16854 /j.cnki.1000-0712.220085

Abstract ( 491 )

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Poynting vector clearly, as well as simply, describes the longitudinal propagation momentum in homogeneous plane waves. However, for complex electromagnetic waves with vortex phase or inhomogeneous polarizations, Poynting vector can exhibit more characteristics about the momentum density. Based on the phasor representation, we decompose the Poynting vector into orbital and spin momenta, in addition to the usual longitudinal momentum, and paraxially discuss the characterisitcs of transverse orbital momentum and spin momentum, by listing some cases of typical complex electromagnetic waves with physically different transverse flows, which are also explained by corresponding phase gradient pictures.

Crystal field splitting in local coordinate system analyzed by using maximally localized Wannier functions

LI Shu-zong, SI Jun-shan, WU Xu-cai, LI Hong-xing, ZHANG Wei-bing

College Physics. 2022, 41(12): 31. doi:10.16854/j.cnki.1000-0712.220192

Abstract ( 743 )

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Plane wave and Wannier function are two commonly-used basis sets in electronic structure of solid. The transformation between different basis sets and coordinate systems is helpful for understanding electronic structure. Taking 2D single-layer ferromagnetic semiconductor CrI3 as an illustration example, the present work performs a basis set transformation from plane-wave and Wannier function in localized coordinate system. And then, crystal field splitting of single-layer CrI3 is discussed in localized coordinate system. Moreover, the underlying mechanism of electronic structure and magnetism are also discussed clearly. Our work not only lets student understand the difference between two basis sets, but also increases the ability to solve physical problems.

A method for characterizing the performance failure of insulating materials

WANG Chun-chang, ZHAO Han-yu, BO De-fu

College Physics. 2022, 41(12): 36. doi:10.16854/j.cnki.1000-0712.220146

Abstract ( 364 )

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In this paper a method to characterize the performance failure temperatures of insulating materials is proposed. By measuring the AC impedance frequency spectrum of the parallel plate capacitor based on the tested insulating material, the relevant properties of the failure temperature of insulation performance, the upper limit of insulation performance holding temperature, heat resistance grade, and other parameters are quantitatively evaluated.

Mathematical model of K space signal in magnetic resonance imaging: a case study of spin echo sequence

FU Hong-bo, XIAO Wei-chen, GUO Xi-xie

College Physics. 2022, 41(12): 39. doi:10.16854/j.cnki.1000-0712.220120

Abstract ( 676 )

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K-space is the basis of signal spatially encoding in magnetic resonance imaging. Accurately understanding signal expression of K-space is the key step to understand the principles behind magnetic resonance imaging. In this work, we take conventional spin echo pulse sequence as an example to comprehensively derive the signal expression, by combining the signal spatial encoding process. In the signal acquisition step, a physical magnetization model was used to simplify the signal expression under the Faraday^,s law of electromagnetic induction. In the mathematically processing step, two-dimensional integral expression of K-space data is derived based on the micro-element analysis. The Details and the corresponding theory in each derivation step are clearly demonstrated. The model connects the key concepts and theories of magnetic resonance imaging, which can help to systematically understand the K-space concepts and the principles behind magnetic resonance imaging.

The elastic analysis of Wilberforce pendulum: characteristic derivation and verification of cylindrical spiral spring model

LIU Tian-heng, CAO Bo-xing, ZHANG Fu-lin

College Physics. 2022, 41(12): 43. doi:10.16854/j.cnki.1000-0712.210589

Abstract ( 1272 )

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The Wilberforce pendulum is a special coupled system,which is a combination of a vertically suspended cylindrical helical spring and an object fixed at the end of it. The block has two degrees of freedom: the displacement stretching along the central axis of the spring the angular displacement twisting around the central axis of the spring. Its translational kinetic energy and rotational kinetic energy can transform periodically during the viberation at a special set of parameters. The Wilberforce pendulum in question 12 of iypt2021 is studied. In this work，the Lagrange function of the system is given, and the motion equation describing the coupling phenomenon is obtained. Using the natural coordinate system of cylindrical helix, the model of cylindrical helix spring is established. From the perspective of theory of elasticity, the spring model is analyzed by microelement. The geometric meaning of the curvature and torsion of the cylindrical helix is connected with the physical meaning of the spring material strain, and the analytical solution of the spring elastic potential energy is obtained. The mathematical calculation of this method is strict and universal, and can be extended to all kinds of precision calculation related to homogeneous spring mechanics. The static experiment is also carried out, and the theoretical value of the elastic coefficient falls within the range of the synthetic standard uncertainty of the experimental value, which verifies the correctness of the cylindrical coil spring model.

Measurement method of liquid diffusion coefficient based on right-angle triangle cell

SU Jin-wei, YU Cheng-he, HU Tai-ran, ZHAO rui, YOU Hua-jie, WANG Jia-hui

College Physics. 2022, 41(12): 50. doi:10.16854/j.cnki.1000-0712.220147

Abstract ( 502 )

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Traditional measurement methods of diffusion coefficient have some defects, such as low accuracy, non-visual measurement, and critical environmental requirements. Therefore, a visualized measurement method of liquid diffusion coefficient based on a hollow right-angle triangle cell (RATC) has been provided. For this method, a linear laser is injected into the RATC filled by two kinds of liquid with different refractive indexes. The refractive index gradient generated by the liquid diffusion region can be visually reflected by the refractive spot after the laser passes through the RATC. For the refractive spot indicates the diffusion affecting on the refractive index, the equivalent interdiffusion coefficient of the two liquids is able to be calculated by combining the linear relationship between refractive index and concentration and the theoretical formula derived from Fick′s law. The experimental results show that the diffusion coefficient of 5.5mol/L glycerin solution is 7.47×10-10 m2/s, and the standard deviation of 125 groups of data is 7.12×10-12 m2/s. Meanwhile, the coefficient of variation is less than 1%, which shows that this method has strong consistency and high reliability.

Development of an experimental demonstration instrument for phased array technology

ZHANG Kang-kang, SHI Hua, YANG Xiao-wei, LIU Huan-ying

College Physics. 2022, 41(12): 56. doi:10.16854/j.cnki.1000-0712.220082

Abstract ( 476 )

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Radio waves and ultrasound cannot be directly seen by eyes, so the traditional phased array technology demonstrations are mostly animation demonstrations or Matlab simulations. In this paper, a 51 MCU and 3D printing technology are employed to design and produce an experimental instrument for phased array technology demonstration by water waves. The 51 MCU is used to control the time interval which determined the phase difference of the adjacent wave sources. The experimental instrument can be used to visually demonstrate the principle of phased array technology and interference phenomenon, which solves the problem that there is no experimental instrument to demonstrate the phased array principle in classroom. It can also be used to measure wavelength, wave speed, and the deflection angle of a line wave.

features of electrodes current of Franck-Hertz tube

JIANG Xiong-fei, LU Chang-hong, LI Zhen, LI Cheng-kai, LI Yu-han

College Physics. 2022, 41(12): 60. doi:10.16854/j.cnki.1000-0712.220170

Abstract ( 1148 )

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The measurement of the electrode current of argon-filled Frank-Hertz (F-H) tube shows that most of the electrons emitted by cathode are absorbed by grid1 and grid2, and only a small fraction forms the plate current. The grid1 and grid2 voltages have a significant influence on the number of electrons emitted by the cathode. Grid1 can not only absorb electrons, but also emit electrons under certain conditions. The direct reason why the F-H curve first increases and then decreases with grid1 voltage is given. The explanation of electron energy distribution with non-zero valley current of F-H curve is also given.

Design and manufacture of experimental device for #br# measuring metal linear expansion coefficient#br#

GUO Lei, WANG Mei-di, YAN Yan, YAO Ping

College Physics. 2022, 41(12): 66. doi:10.16854/j.cnki.1000-0712.220174

Abstract ( 610 )

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The structure, principle and characteristics of a selfdesigned and manufactured experimental device for measuring metal linear expansion coefficient are introduced. On this basis, the experiment for measuring metal linear expansion coefficient is carried out. The instrument adopts portable design, the combination of physical lever and optical lever, water circulation heating, so that the metal rod to be measured is heated evenly. Three common materials, aluminum, copper and iron are measured and the experimental error was less than 3%. The design and manufacture of the instrument are mainly used for teaching demonstration, which can help students learn and understand the knowledge of linear expansion coefficient, optical lever and water circulation heating. It provides a good experimental equipment for the experiment of measuring metal linear expansion coefficient.

Electrostatic resonance measurement of Young’s modulus for microfibers

ZAI Min-ming, HOU Li-zhen, WANG Shi-liang

College Physics. 2022, 41(12): 70. doi:10.16854/j.cnki.1000-0712.220122

Abstract ( 425 )

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A method based on electrostatic resonance for measuring Young’s modulus of microfibers is developed in this paper. The resonance frequencies of cantilevered microfibers are directly determined by using an optical microscope, rather than an expensive high-precision microsensors or laser Doppler vibrators used in previous studies. The Young’s moduli of microfibers are derived from Euler-Bernoulli beam theory by using measured resonant frequencies. This method is beneficial to improve the comprehensive experimental ability of undergraduates, stimulate their interest in learning, enhance their awareness of innovation, and help them to consciously apply physical principles and measurement methods to the new areas of science and technology.

Application of Green's reciprocation theorem in electrostatics and verification of mathematical identities

SONG Yu-fei, LIANG Zhao-xin

College Physics. 2022, 41(12): 75. doi:10.16854/j.cnki.1000-0712.220197

Abstract ( 679 )

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Green’s reciprocation theorem is refereed as to a mathematical relationship between the potentials of any two groups of conductors with electrostatic balance under different charge distributions. Such a relationship stands under any conductor shapes and any other conductor distributions. In this work, we discuss the application of Green’s reciprocation theorem in electrostatics and checking priori mathematical identities. By comparing with the method of images and the method of eigenfunction expansion, we show the advantages of Green’s reciprocation theorem in dealing with some electrostatic problems. In addition, this work lists the infinite parallel plate capacitor problems which can be easily solved by Green’s reciprocation theorem, but hardly be solved by the traditional methods. In particular, this work designs an ideal experimental protocol to test the famous Ramanujan mathematical formula and some other mathematical identities based on Green’s reciprocation theorem.

Calculation of equivalent resistance on infinite lattice point of Kagome plane

SHI Lu-jie, YIN Jian-kai, LU Wen, LI Dan, LI Xi-bin

College Physics. 2022, 41(12): 81. doi:10.16854/j.cnki.1000-0712.220228

Abstract ( 682 )

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Using the method of two-dimensional Fourier transform, the equivalent resistance between any two nodes on the regularly connected Kagome resistance grid is calculated, and the numerical calculation is carried out by using Mathematica software.

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