RESEARCH ON THE APPLICATION OF SYMMETRY METHODS IN SOLVING UNIVERSITY PHYSICS PROBLEMS — TAKING MECHANICS, ELECTROSTATIC FIELDS AND STEADY MAGNETIC FIELDS AS EXAMPLES

Danish Scientific Journal No100, 2025 55 RESEARCH ON THE APPLICATION OF SYMMETRY METHODS IN SOLVING UNIVERSITY PHYSICS PROBLEMS — TAKING MECHANICS, ELECTROSTATIC FIELDS AND STEADY MAGNETIC FIELDS AS EXAMPLES Dong Yongsheng JiNing Normal University, Science and Technology Department, Ulanqab, Inner Mongolia, 012000 ,P. R. China Fund Project: Key Project of Natural Science Research of Jining Normal University (jsky202205) Zhang Hongzhi JiNing Normal University, School of Mathematics and Statistic, Ulanqab, Inner Mongolia, 012000 ,P. R. China https://doi.org/10.5281/zenodo.17249788 Abstract In the knowledge system of university physics, symmetry is widely present in branches such as mechanics and electromagnetism. It is not only a conceptual tool that connects different disciplines, but also demonstrates significant value in solving problems. This paper conducts an analysis through specific examples: In particle kinematics of mechanics, the symmetry of motion is used to equate the trajectory of a small ball’s zigzag elastic collisions inside a well to projectile motion, simplifying multi-process problems into a single model; In electrostatic fields, taking a uniformly charged thin disk as an example, Gauss's theorem is applied in combination with the planar symmetry of the electric field to efficiently solve for the electric field intensity on the axis, avoiding complex integrations; In steady magnetic fields, for an infinitely long current-carrying thin metal plate, Ampère's circuital theorem is utilized, and the mirror symmetry of the magnetic field is employed to simplify the calculation of magnetic induction. The research shows that the symmetry method has the common advantages of "more concise in thinking and analysis, and more efficient in calculation and solution" in solving university physics problems. It can help researchers quickly identify problem-solving paths and reduce the complexity of problems. Keywords: Symmetry; University Physics; Gauss's Theorem; Ampère's Circuital Theorem Symmetry refers to the corresponding or proportional relationship between the various components within a whole. The origin of this concept can be traced back to the early productive practical activities of humans — in the process of observing and transforming the world, people first captured the existence of symmetry from life scenarios and natural phenomena. For example, regular circular geometric patterns, leaves with symmetrically distributed leaf veins, the symmetric left-right body structures of animals, and the axisymmetric layouts commonly seen in ancient Chinese architecture (such as the palace complex of the Forbidden City) are all intuitive manifestations of symmetry. Within the knowledge system of university physics, symmetry is a core concept that runs through multiple fields, widely existing in subdisciplines such as mechanics, electromagnetism, and quantum mechanics. Within the context of physics, the definition of symmetry is further refined: if a certain operation is performed on a physical system (i.e., the process of transitioning the system from its initial state to another state), and the core properties of the system after the operation are completely equivalent to those in its initial state without any changes, then the system is said to possess symmetry with respect to this operation.Put simply, the essence of symmetry can be understood as the property of a physical system that remains unchanged under specific changes. Whether it is spatial operations such as translation, rotation, or reflection, or temporal operations such as time translation, as long as the physical laws of the system (e.g., equations of motion, energy conservation relationships, etc.) remain unaffected after the operation, the system can be determined to have the corresponding symmetry. Symmetry occupies a crucial core position in the knowledge system of university physics. Its methods and ideas run through almost all core modules such as mechanics, thermodynamics, optics, and electromagnetism, serving as a key conceptual tool for linking different physics branches. Among these, the application of symmetry methods is particularly prominent and widespread in the field of electromagnetism. Whether it is the calculation of electric field intensity and magnetic induction intensity, or the analysis of the laws of electromagnetic induction, symmetry is often used as an important breakthrough. From the perspective of practical problem-solving, the value of symmetry is even more direct and notable. When solving complex physics problems, if one can skillfully utilize the symmetry of the system (such as spherical symmetry, cylindrical symmetry, planar symmetry, etc.), it can greatly simplify the physical model; at the same time, it can effectively reduce mathematical derivations and calculation steps, avoid the solution of complex integrals or systems of equations, and ultimately achieve the goal of obtaining results quickly and simply. Thanks to its features of simplifying models, optimizing solutions, and revealing essences, the symmetry method has transcended mere problem-solving techniques. It has become a fundamental thinking and tool for people to understand the structural composition of matter (such as the symmetric structure of crystals) and 56 Danish Scientific Journal No100, 2025 explore the laws of interactions between physical quantities, providing a key perspective for in-depth comprehension of the order and laws of the physical world. To intuitively demonstrate the convenience of the symmetry method in solving university physics problems, the following text will conduct an analysis with specific examples. First, we take a typical projectile motion problem in mechanics as the starting point, to briefly discuss the significant advantages demonstrated when using symmetry thinking to solve problems. 1. Application of Symmetry in Particle Kinematics Case 1: As shown in Figure 1, there is a well with a depth of H and a smooth inner wall, and its inner diameter is d. Now, a small ball is projected into the well from the wellhead along the diametrical direction of the well. During the falling process, the small ball undergoes n elastic collisions with the well wall. When it finally reaches the bottom of the well, its landing point and the initial projection point lie on the same straight line. Try to find the initial velocity 0 v of the small ball when it is projected. Figure 1: Schematic Diagram of a Well with Smooth Inner Walls and a Small Ball's n Elastic Collisions It can be known from the problem that the inner wall of the well is smooth, and the collision between the small ball and the well wall is an elastic collision. At this point, we can utilize the symmetry of motion and regard the motion of the small ball inside the well as projectile motion. As shown in Figure 2, the distance traveled by the small ball in the horizontal direction is nd. According to the laws of projectile motion: in the horizontal direction, there is 0 nd v t ; in the vertical direction, there is 2 2 1gtH . By combining these two equations, we can solve for Hg H nd v2 2 0 . Figure 2: Schematic Diagram of the Symmetry of the Small Ball's Motion, Equivalent Projectile Motion, and Solution of the Initial Velocity It can be seen from the above example in university physics mechanics that by skillfully applying the symmetry of motion, we equivalently transform the broken-line rebound trajectory of the small ball inside the well into a continuous projectile motion trajectory. Based on the characteristics of symmetric velocity direction and constant horizontal speed in elastic collisions, we achieve an essential simplification of the complex motion process. This processing method can directly strip away the "broken-line interference" caused by collisions, and transform the multi-process problem that originally requires segmental analysis into a single projectile motion model, truly realizing "simplifying the complex". It fully demonstrates the significant effectiveness of the symmetry method in simplifying physical processes and improving problem-solving efficiency. 2. Application of Symmetry in Electrostatic Fields The previous section has demonstrated the practical value of symmetry in solving university physics problems through the example of projectile motion in mechanics. In this example, we abstracted the small ball as the ideal model of a "particle" and further studied its motion laws inside the well — which essentially involves the exploration of the "mechanical motion laws of particles" in the field of mechanics. In the system of university physics, however, electromagnetism is another core branch; the "electromagnetic motion" it studies is another fundamental form of motion of matter, distinct from mechanical motion. Danish Scientific Journal No100, 2025 57 University physics usually analyzes the basic laws of electromagnetic motion from the perspective of "field". Similar to the rationalized modeling approach of "particle" in mechanics, electromagnetism also takes the "point charge" as the ideal model when studying electrostatic fields, and on this basis deduces the basic properties and laws of electrostatic fields. In the descriptive system of electrostatic fields, there are two core physical quantities: electric field intensity and electric potential. Among them, electric field intensity is a vector point function (it has both magnitude and direction, and its value changes with spatial position), and electric potential is a scalar point function (it only has magnitude, and its value also changes with spatial position). For a charged body at rest in an inertial frame of reference, the core logic for solving its electrostatic field problem lies in: if we can calculate the electric field intensity and electric potential at each point in the electrostatic field excited by this charged body through physical methods, then subsequent problems related to this electrostatic field can be easily solved. Case 2: As shown in Figure 3, consider a uniformly charged thin disk with a surface charge density of σ (where σ is a constant) and a radius of R. We now need to find the electric field intensity at any point on the axis that passes through the center of the disk and is perpendicular to its surface. Figure 3: Schematic Diagram of the Solution of the Electric Field Intensity on the Axis of a Uniformly Charged Thin Disk Gauss's Law is one of the important methods for solving electric field intensity in electromagnetism. The mathematical expression of Gauss's Law for electrostatic fields in vacuum is: 0 1 sE dS q     . The physical meaning of this expression can be clearly stated as: In vacuum, the electric flux of the electric field intensity through any closed surface (known as the Gaussian surface) is equal to the ratio of the algebraic sum of the charges enclosed by this Gaussian surface to the permittivity of free space.It should be particularly emphasized that the efficient application of Gauss's Law depends on the symmetry of the electric field — only when the electric field excited by a charged body has a regularly symmetric distribution (such as spherical symmetry, cylindrical symmetry, or planar symmetry) can a suitable Gaussian surface be selected. This selection ensures that the magnitude of the electric field intensity is the same everywhere on the Gaussian surface (or zero in some regions), and the direction of the electric field intensity is consistent with (or perpendicular to) the direction of the area element vector. In this way, the integral calculation of the electric field intensity flux is simplified. Therefore, Gauss's Law can be regarded as the core method for solving the electric field intensity using symmetry in electrostatic fields. The general steps for solving the electric field intensity using Gauss's Law are as follows: (1) Analyze the symmetry of the electric field: First, judge the charge distribution characteristics of the charged body (such as a uniformly charged spherical surface, an infinitely long uniformly charged cylindrical surface, and so on), then determine the symmetry type of the electric field it excites (spherical symmetry, cylindrical symmetry, etc.), and clarify the directional regularity of the electric field intensity and the characteristics of its magnitude distribution; (2) Select an appropriate Gaussian surface: Based on the symmetry of the electric field, select a Gaussian surface that matches the electric field distribution (e.g., a spherical Gaussian surface for a spherically symmetric electric field, and a coaxial cylindrical Gaussian surface for a cylindrically symmetric electric field), and ensure that the magnitude of the electric field intensity is the same everywhere on the Gaussian surface (or zero in a certain region), and that the angle between E and dS (area element vector) is constant (usually 0° or 90°, which facilitates integral calculation); (3) Calculate the electric field intensity flux and the enclosed charge: On one hand, calculate the electric field intensity flux through the Gaussian surface; on the other hand, based on the charge distribution of the charged body, calculate the algebraic sum of the enclosed charge inside the Gaussian surface; (4) Apply Gauss's Law to solve for the electric field intensity: Substitute the two calculation results mentioned above into the formula of Gauss's Law, and after rearrangement, you can solve for the magnitude of the electric field intensity. Then, combine it with the previously analyzed directional regularity of the electric field to finally determine the electric field intensity E ; 58 Danish Scientific Journal No100, 2025 When the distance x from the field point P to the center O of the thin disk is much smaller than R (where R is the radius of the disk), since the field point is extremely close to the disk, the edge effect of the disk can be neglected. At this time, the uniformly charged thin disk can be approximated as an "infinitely large uniformly charged plane". For an infinitely large uniformly charged plane, the electric field it excites has plane symmetry: the direction of the electric field intensity on both sides of the plane is perpendicular to the plane (for a positively charged plane, the electric field direction is away from the plane; for a negatively charged plane, it points towards the plane), and in the same plane parallel to the charged plane, the magnitude of the electric field intensity at each point is equal everywhere. Based on this symmetry feature, this problem can be solved efficiently by the symmetry method — Gauss's Law. As shown in Figure 4, based on the plane symmetry of the electric field, a cylindrical surface whose axis is perpendicular to the charged plane is selected as the Gaussian surface: the two circular bases of the cylinder are parallel to the charged plane and symmetric with respect to the plane (to ensure that the magnitude of the electric field intensity is equal at the two bases), and the lateral surface is perpendicular to the charged plane. Since the normal to the lateral surface of the cylinder is perpendicular to the electric field intensity, the electric field intensity flux through the lateral surface is zero. Furthermore, because the normals to the cylinder’s circular bases are parallel to the electric field intensity, and the magnitude of the electric field intensity is the same at both bases, the total electric field intensity flux through the two circular bases is 2ES (where S is the area of a single circular base). In addition, given that the surface charge density of the charged plane is σ, according to Gauss's Law, we have 0 2S ES    , that is, 0 2 E    . Figure 4: Schematic Diagram of the Selection of a Cylindrical Gaussian Surface Based on the Planar Symmetry of the Electric Field 3. Application of Symmetry in Steady Magnetic Fields From the physical examples in the previous text, we know that electrostatic fields are induced around stationary charges, and the analysis of such electric fields can be simplified using symmetry (e.g., Gauss's Law). Further research shows that when charges move to form a current, the field distribution around them changes — at this time, not only does an electric field exist, but a steady magnetic field is also induced (if the magnitude and direction of the current do not change with time, the corresponding magnetic field is a steady magnetic field). This phenomenon reveals the deep connection between electricity and magnetism: the two are not isolated but form a unified whole that is interconnected and mutually transformable (for example, the phenomenon of electromagnetic induction demonstrates magnetism generating electricity, while the magnetic effect of current demonstrates electricity generating magnetism). For this reason, in practical applications, scenarios involving electricity (such as circuits and electrical equipment) are often accompanied by the involvement of magnetism, and the analysis of steady magnetic fields has thus become an important part of the electromagnetism module in university physics. Case 3: As shown in Figure 5, there is an infinitely long thin metal plate with a width of 2d, through which a steady current I flows uniformly (the current direction is along the length of the metal plate). It is required to solve for the magnetic induction intensity B at a target field point P, where P lies on a plane that passes through the midline of the metal plate (i.e., the central line perpendicular to the width direction of the plate) and is perpendicular to the surface of the metal plate. Danish Scientific Journal No100, 2025 59 Figure 5: Schematic Diagram of the Infinitely Long Current-Carrying Thin Metal Plate and the Position of Target Field Point P In the analysis of steady magnetic fields, if the magnetic field distribution exhibits regular symmetry, the magnetic induction intensity can be efficiently solved using Ampère's Circuital Law — a approach highly similar to that of using Gauss's Law to solve for electric field intensity in electrostatic fields. The mathematical expression of Ampère's Circuital Law is: 0 LB dl I    (where L is any closed path, B is the magnetic induction intensity at each point on the path, dl is the line element vector of the path, 0  is the permeability of free space, and I  is the algebraic sum of the currents enclosed by the closed path L; the current direction is positive if it satisfies the right-hand screw rule with the direction of the loop’s traversal, and negative otherwise). The physical meaning of this theorem can be clearly stated as follows: In a steady magnetic field in vacuum, the line integral of the magnetic induction intensity B along any closed path (i.e., the circulation) is equal to the product of the permeability of free space and the algebraic sum of all currents enclosed by the closed path.It can be seen from this that the core prerequisite for solving the magnetic induction intensity using Ampère's Circuital Law is the symmetry of the magnetic field. Only in this way can an appropriate closed path (i.e., an Ampère loop) be selected—such that the magnitude of the magnetic induction intensity is uniform at all points on the loop (or zero in some sections), and its direction is consistent with (or perpendicular to) the direction of the loop’s line element. This simplifies the integral calculation of the circulation. Returning to Case 3, through theoretical analysis, we transform the aforementioned infinitely long current-carrying thin metal plate into a flat structure composed of a number of mutually parallel infinitely long current-carrying straight wires. At this point, the magnetic field induced by the current-carrying metal plate exhibits symmetry. Next, we use Ampère's Circuital Law to calculate the magnetic induction intensity B at any point outside the current-carrying plane. Since the magnetic field induced by the plane exhibits symmetry, the magnitude of the magnetic induction intensity B is equal at all points equidistant from the plane. Additionally, the direction of B on either side of the plane is parallel to the plane, while the directions of B on the two sides are opposite to each other. As shown in Figure 6, select a closed path abcda, where segments ab and cd are both parallel to B , and segment bc is perpendicular to B . According to Ampère's Circuital Law, we have: 00 b c d a a b c d L B dl B dl B dl B dl B dl I I ab                   Since 0 B ab B cd i ab       ,and ab cd ，therefore 0 2B ab i ab     ，that is 0 2 Bi   . 60 Danish Scientific Journal No100, 2025 Figure 6: Schematic Diagram of the Ampère Loop for the Solution of Magnetic Induction Intensity of a Current-Carrying Plane From the analysis of the above examples, a general conclusion can be drawn: When the symmetry method is applied to solving university physics problems, regardless of which branch of physics the problem involves, its core advantages are reflected in two aspects — namely, more concise thinking and analysis, and more efficient calculation and solution. Precisely because of this, in solving problems and conducting research in university physics, people often prioritize judging whether the physical scenario exhibits symmetry. If symmetric characteristics exist, a simpler solution path can be quickly identified, significantly reducing the complexity of the problem. Furthermore, the symmetry method is not only a "shortcut" for problemsolving; it also plays an irreplaceable and positive driving role in exploring physical models (e.g., equating complex motions to ideal models), uncovering physical mysteries (e.g., revealing the laws of interaction between fields and matter), and perceiving the inherent harmonious beauty of physics (e.g., the unity of laws brought about by symmetric distribution). References: 1. Ma, W. W., Xie, X. S., & Zhou, Y. Q. Physics (5th Edition). Beijing: Higher Education Press, 2006. 2. Ma, W. W., Chen, G. Q., & Chen, J. Study Guide for Physics (5th Edition). Beijing: Higher Education Press, 2006. 3. Tan, J. F., Li, H. Y., & Wang, F. X. Physics Tutoring and Detailed Solutions to Exercises. Jilin: Yanbian University Press, 2014. 4. Zhang, C. M., Liu, F. Y., & Zha, X. W. General Physics: University Physics. Shaanxi: Xi'an Jiaotong University Press, 2008. 5. Zhao, K. H. & Chen, X. M. Electromagnetism (2nd Edition). Beijing: Higher Education Press, 2006.